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Theorem vdwlem6 15633
Description: Lemma for vdw 15641. (Contributed by Mario Carneiro, 13-Sep-2014.)
Hypotheses
Ref Expression
vdwlem3.v (𝜑𝑉 ∈ ℕ)
vdwlem3.w (𝜑𝑊 ∈ ℕ)
vdwlem4.r (𝜑𝑅 ∈ Fin)
vdwlem4.h (𝜑𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)
vdwlem4.f 𝐹 = (𝑥 ∈ (1...𝑉) ↦ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))))
vdwlem7.m (𝜑𝑀 ∈ ℕ)
vdwlem7.g (𝜑𝐺:(1...𝑊)⟶𝑅)
vdwlem7.k (𝜑𝐾 ∈ (ℤ‘2))
vdwlem7.a (𝜑𝐴 ∈ ℕ)
vdwlem7.d (𝜑𝐷 ∈ ℕ)
vdwlem7.s (𝜑 → (𝐴(AP‘𝐾)𝐷) ⊆ (𝐹 “ {𝐺}))
vdwlem6.b (𝜑𝐵 ∈ ℕ)
vdwlem6.e (𝜑𝐸:(1...𝑀)⟶ℕ)
vdwlem6.s (𝜑 → ∀𝑖 ∈ (1...𝑀)((𝐵 + (𝐸𝑖))(AP‘𝐾)(𝐸𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}))
vdwlem6.j 𝐽 = (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝐵 + (𝐸𝑖))))
vdwlem6.r (𝜑 → (#‘ran 𝐽) = 𝑀)
vdwlem6.t 𝑇 = (𝐵 + (𝑊 · ((𝐴 + (𝑉𝐷)) − 1)))
vdwlem6.p 𝑃 = (𝑗 ∈ (1...(𝑀 + 1)) ↦ (if(𝑗 = (𝑀 + 1), 0, (𝐸𝑗)) + (𝑊 · 𝐷)))
Assertion
Ref Expression
vdwlem6 (𝜑 → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑖,𝑗,𝑥,𝑦,𝐺   𝑖,𝐾,𝑗,𝑥,𝑦   𝑖,𝐽,𝑗,𝑥   𝑃,𝑖,𝑥   𝜑,𝑖,𝑗,𝑥,𝑦   𝑅,𝑖,𝑥,𝑦   𝐵,𝑖,𝑗,𝑥,𝑦   𝑖,𝐻,𝑥,𝑦   𝑖,𝑀,𝑗,𝑥,𝑦   𝐷,𝑗,𝑥,𝑦   𝑖,𝐸,𝑗,𝑥,𝑦   𝑖,𝑊,𝑗,𝑥,𝑦   𝑇,𝑖,𝑥   𝑥,𝑉,𝑦
Allowed substitution hints:   𝐴(𝑖,𝑗)   𝐷(𝑖)   𝑃(𝑦,𝑗)   𝑅(𝑗)   𝑇(𝑦,𝑗)   𝐹(𝑥,𝑦,𝑖,𝑗)   𝐻(𝑗)   𝐽(𝑦)   𝑉(𝑖,𝑗)

Proof of Theorem vdwlem6
Dummy variables 𝑚 𝑛 𝑧 𝑎 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6168 . . . . . . 7 (𝐺‘(𝐵 + (𝐸𝑖))) ∈ V
2 vdwlem6.j . . . . . . 7 𝐽 = (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝐵 + (𝐸𝑖))))
31, 2fnmpti 5989 . . . . . 6 𝐽 Fn (1...𝑀)
4 fvelrnb 6210 . . . . . 6 (𝐽 Fn (1...𝑀) → ((𝐺𝐵) ∈ ran 𝐽 ↔ ∃𝑚 ∈ (1...𝑀)(𝐽𝑚) = (𝐺𝐵)))
53, 4ax-mp 5 . . . . 5 ((𝐺𝐵) ∈ ran 𝐽 ↔ ∃𝑚 ∈ (1...𝑀)(𝐽𝑚) = (𝐺𝐵))
6 vdwlem4.r . . . . . . . 8 (𝜑𝑅 ∈ Fin)
76adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽𝑚) = (𝐺𝐵))) → 𝑅 ∈ Fin)
8 vdwlem7.k . . . . . . . . 9 (𝜑𝐾 ∈ (ℤ‘2))
9 eluz2nn 11686 . . . . . . . . 9 (𝐾 ∈ (ℤ‘2) → 𝐾 ∈ ℕ)
108, 9syl 17 . . . . . . . 8 (𝜑𝐾 ∈ ℕ)
1110adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽𝑚) = (𝐺𝐵))) → 𝐾 ∈ ℕ)
12 vdwlem3.w . . . . . . . 8 (𝜑𝑊 ∈ ℕ)
1312adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽𝑚) = (𝐺𝐵))) → 𝑊 ∈ ℕ)
14 vdwlem7.g . . . . . . . 8 (𝜑𝐺:(1...𝑊)⟶𝑅)
1514adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽𝑚) = (𝐺𝐵))) → 𝐺:(1...𝑊)⟶𝑅)
16 vdwlem6.b . . . . . . . 8 (𝜑𝐵 ∈ ℕ)
1716adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽𝑚) = (𝐺𝐵))) → 𝐵 ∈ ℕ)
18 vdwlem7.m . . . . . . . 8 (𝜑𝑀 ∈ ℕ)
1918adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽𝑚) = (𝐺𝐵))) → 𝑀 ∈ ℕ)
20 vdwlem6.e . . . . . . . 8 (𝜑𝐸:(1...𝑀)⟶ℕ)
2120adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽𝑚) = (𝐺𝐵))) → 𝐸:(1...𝑀)⟶ℕ)
22 vdwlem6.s . . . . . . . 8 (𝜑 → ∀𝑖 ∈ (1...𝑀)((𝐵 + (𝐸𝑖))(AP‘𝐾)(𝐸𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}))
2322adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽𝑚) = (𝐺𝐵))) → ∀𝑖 ∈ (1...𝑀)((𝐵 + (𝐸𝑖))(AP‘𝐾)(𝐸𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}))
24 simprl 793 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽𝑚) = (𝐺𝐵))) → 𝑚 ∈ (1...𝑀))
25 simprr 795 . . . . . . . 8 ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽𝑚) = (𝐺𝐵))) → (𝐽𝑚) = (𝐺𝐵))
26 fveq2 6158 . . . . . . . . . . . 12 (𝑖 = 𝑚 → (𝐸𝑖) = (𝐸𝑚))
2726oveq2d 6631 . . . . . . . . . . 11 (𝑖 = 𝑚 → (𝐵 + (𝐸𝑖)) = (𝐵 + (𝐸𝑚)))
2827fveq2d 6162 . . . . . . . . . 10 (𝑖 = 𝑚 → (𝐺‘(𝐵 + (𝐸𝑖))) = (𝐺‘(𝐵 + (𝐸𝑚))))
29 fvex 6168 . . . . . . . . . 10 (𝐺‘(𝐵 + (𝐸𝑚))) ∈ V
3028, 2, 29fvmpt 6249 . . . . . . . . 9 (𝑚 ∈ (1...𝑀) → (𝐽𝑚) = (𝐺‘(𝐵 + (𝐸𝑚))))
3124, 30syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽𝑚) = (𝐺𝐵))) → (𝐽𝑚) = (𝐺‘(𝐵 + (𝐸𝑚))))
3225, 31eqtr3d 2657 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽𝑚) = (𝐺𝐵))) → (𝐺𝐵) = (𝐺‘(𝐵 + (𝐸𝑚))))
337, 11, 13, 15, 17, 19, 21, 23, 24, 32vdwlem1 15628 . . . . . 6 ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽𝑚) = (𝐺𝐵))) → (𝐾 + 1) MonoAP 𝐺)
3433rexlimdvaa 3027 . . . . 5 (𝜑 → (∃𝑚 ∈ (1...𝑀)(𝐽𝑚) = (𝐺𝐵) → (𝐾 + 1) MonoAP 𝐺))
355, 34syl5bi 232 . . . 4 (𝜑 → ((𝐺𝐵) ∈ ran 𝐽 → (𝐾 + 1) MonoAP 𝐺))
3635imp 445 . . 3 ((𝜑 ∧ (𝐺𝐵) ∈ ran 𝐽) → (𝐾 + 1) MonoAP 𝐺)
3736olcd 408 . 2 ((𝜑 ∧ (𝐺𝐵) ∈ ran 𝐽) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺))
38 vdwlem3.v . . . . . . 7 (𝜑𝑉 ∈ ℕ)
39 vdwlem4.h . . . . . . 7 (𝜑𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)
40 vdwlem4.f . . . . . . 7 𝐹 = (𝑥 ∈ (1...𝑉) ↦ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))))
41 vdwlem7.a . . . . . . 7 (𝜑𝐴 ∈ ℕ)
42 vdwlem7.d . . . . . . 7 (𝜑𝐷 ∈ ℕ)
43 vdwlem7.s . . . . . . 7 (𝜑 → (𝐴(AP‘𝐾)𝐷) ⊆ (𝐹 “ {𝐺}))
44 vdwlem6.r . . . . . . 7 (𝜑 → (#‘ran 𝐽) = 𝑀)
45 vdwlem6.t . . . . . . 7 𝑇 = (𝐵 + (𝑊 · ((𝐴 + (𝑉𝐷)) − 1)))
46 vdwlem6.p . . . . . . 7 𝑃 = (𝑗 ∈ (1...(𝑀 + 1)) ↦ (if(𝑗 = (𝑀 + 1), 0, (𝐸𝑗)) + (𝑊 · 𝐷)))
4738, 12, 6, 39, 40, 18, 14, 8, 41, 42, 43, 16, 20, 22, 2, 44, 45, 46vdwlem5 15632 . . . . . 6 (𝜑𝑇 ∈ ℕ)
4847adantr 481 . . . . 5 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → 𝑇 ∈ ℕ)
49 0nn0 11267 . . . . . . . . . 10 0 ∈ ℕ0
5049a1i 11 . . . . . . . . 9 ((((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) ∧ 𝑗 = (𝑀 + 1)) → 0 ∈ ℕ0)
51 nnuz 11683 . . . . . . . . . . . . . . . . 17 ℕ = (ℤ‘1)
5218, 51syl6eleq 2708 . . . . . . . . . . . . . . . 16 (𝜑𝑀 ∈ (ℤ‘1))
5352adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → 𝑀 ∈ (ℤ‘1))
54 elfzp1 12349 . . . . . . . . . . . . . . 15 (𝑀 ∈ (ℤ‘1) → (𝑗 ∈ (1...(𝑀 + 1)) ↔ (𝑗 ∈ (1...𝑀) ∨ 𝑗 = (𝑀 + 1))))
5553, 54syl 17 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → (𝑗 ∈ (1...(𝑀 + 1)) ↔ (𝑗 ∈ (1...𝑀) ∨ 𝑗 = (𝑀 + 1))))
5655biimpa 501 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) → (𝑗 ∈ (1...𝑀) ∨ 𝑗 = (𝑀 + 1)))
5756ord 392 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) → (¬ 𝑗 ∈ (1...𝑀) → 𝑗 = (𝑀 + 1)))
5857con1d 139 . . . . . . . . . . 11 (((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) → (¬ 𝑗 = (𝑀 + 1) → 𝑗 ∈ (1...𝑀)))
5958imp 445 . . . . . . . . . 10 ((((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) ∧ ¬ 𝑗 = (𝑀 + 1)) → 𝑗 ∈ (1...𝑀))
6020ad2antrr 761 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) → 𝐸:(1...𝑀)⟶ℕ)
6160ffvelrnda 6325 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) ∧ 𝑗 ∈ (1...𝑀)) → (𝐸𝑗) ∈ ℕ)
6261nnnn0d 11311 . . . . . . . . . 10 ((((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) ∧ 𝑗 ∈ (1...𝑀)) → (𝐸𝑗) ∈ ℕ0)
6359, 62syldan 487 . . . . . . . . 9 ((((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) ∧ ¬ 𝑗 = (𝑀 + 1)) → (𝐸𝑗) ∈ ℕ0)
6450, 63ifclda 4098 . . . . . . . 8 (((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) → if(𝑗 = (𝑀 + 1), 0, (𝐸𝑗)) ∈ ℕ0)
6512, 42nnmulcld 11028 . . . . . . . . 9 (𝜑 → (𝑊 · 𝐷) ∈ ℕ)
6665ad2antrr 761 . . . . . . . 8 (((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) → (𝑊 · 𝐷) ∈ ℕ)
67 nn0nnaddcl 11284 . . . . . . . 8 ((if(𝑗 = (𝑀 + 1), 0, (𝐸𝑗)) ∈ ℕ0 ∧ (𝑊 · 𝐷) ∈ ℕ) → (if(𝑗 = (𝑀 + 1), 0, (𝐸𝑗)) + (𝑊 · 𝐷)) ∈ ℕ)
6864, 66, 67syl2anc 692 . . . . . . 7 (((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) → (if(𝑗 = (𝑀 + 1), 0, (𝐸𝑗)) + (𝑊 · 𝐷)) ∈ ℕ)
6968, 46fmptd 6351 . . . . . 6 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → 𝑃:(1...(𝑀 + 1))⟶ℕ)
70 nnex 10986 . . . . . . 7 ℕ ∈ V
71 ovex 6643 . . . . . . 7 (1...(𝑀 + 1)) ∈ V
7270, 71elmap 7846 . . . . . 6 (𝑃 ∈ (ℕ ↑𝑚 (1...(𝑀 + 1))) ↔ 𝑃:(1...(𝑀 + 1))⟶ℕ)
7369, 72sylibr 224 . . . . 5 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → 𝑃 ∈ (ℕ ↑𝑚 (1...(𝑀 + 1))))
74 elfzp1 12349 . . . . . . . . . 10 (𝑀 ∈ (ℤ‘1) → (𝑖 ∈ (1...(𝑀 + 1)) ↔ (𝑖 ∈ (1...𝑀) ∨ 𝑖 = (𝑀 + 1))))
7552, 74syl 17 . . . . . . . . 9 (𝜑 → (𝑖 ∈ (1...(𝑀 + 1)) ↔ (𝑖 ∈ (1...𝑀) ∨ 𝑖 = (𝑀 + 1))))
7616adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑖 ∈ (1...𝑀)) → 𝐵 ∈ ℕ)
7776nncnd 10996 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑖 ∈ (1...𝑀)) → 𝐵 ∈ ℂ)
7877adantr 481 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐵 ∈ ℂ)
7920ffvelrnda 6325 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐸𝑖) ∈ ℕ)
8079nncnd 10996 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐸𝑖) ∈ ℂ)
8180adantr 481 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐸𝑖) ∈ ℂ)
8278, 81addcld 10019 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐵 + (𝐸𝑖)) ∈ ℂ)
83 nnm1nn0 11294 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 ∈ ℕ → (𝐴 − 1) ∈ ℕ0)
8441, 83syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝐴 − 1) ∈ ℕ0)
85 nn0nnaddcl 11284 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐴 − 1) ∈ ℕ0𝑉 ∈ ℕ) → ((𝐴 − 1) + 𝑉) ∈ ℕ)
8684, 38, 85syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝐴 − 1) + 𝑉) ∈ ℕ)
8712, 86nnmulcld 11028 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℕ)
8887nncnd 10996 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℂ)
8988ad2antrr 761 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℂ)
90 elfznn0 12390 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 ∈ (0...(𝐾 − 1)) → 𝑚 ∈ ℕ0)
9190adantl 482 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℕ0)
9291nn0cnd 11313 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℂ)
9392adantlr 750 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℂ)
9493, 81mulcld 10020 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · (𝐸𝑖)) ∈ ℂ)
9565nnnn0d 11311 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑊 · 𝐷) ∈ ℕ0)
9695adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · 𝐷) ∈ ℕ0)
9791, 96nn0mulcld 11316 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · (𝑊 · 𝐷)) ∈ ℕ0)
9897nn0cnd 11313 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · (𝑊 · 𝐷)) ∈ ℂ)
9998adantlr 750 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · (𝑊 · 𝐷)) ∈ ℂ)
10082, 89, 94, 99add4d 10224 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + ((𝑚 · (𝐸𝑖)) + (𝑚 · (𝑊 · 𝐷)))) = (((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) + ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑚 · (𝑊 · 𝐷)))))
10165nncnd 10996 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑊 · 𝐷) ∈ ℂ)
102101ad2antrr 761 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · 𝐷) ∈ ℂ)
10393, 81, 102adddid 10024 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · ((𝐸𝑖) + (𝑊 · 𝐷))) = ((𝑚 · (𝐸𝑖)) + (𝑚 · (𝑊 · 𝐷))))
104103oveq2d 6631 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸𝑖) + (𝑊 · 𝐷)))) = (((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + ((𝑚 · (𝐸𝑖)) + (𝑚 · (𝑊 · 𝐷)))))
10512nncnd 10996 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑊 ∈ ℂ)
106105adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝑊 ∈ ℂ)
10786nncnd 10996 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ((𝐴 − 1) + 𝑉) ∈ ℂ)
108107adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → ((𝐴 − 1) + 𝑉) ∈ ℂ)
10942nncnd 10996 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝐷 ∈ ℂ)
110109adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝐷 ∈ ℂ)
11192, 110mulcld 10020 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · 𝐷) ∈ ℂ)
112106, 108, 111adddid 10024 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · (((𝐴 − 1) + 𝑉) + (𝑚 · 𝐷))) = ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑊 · (𝑚 · 𝐷))))
11341nncnd 10996 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝐴 ∈ ℂ)
114113adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝐴 ∈ ℂ)
115 1cnd 10016 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 1 ∈ ℂ)
116114, 111, 115addsubd 10373 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → ((𝐴 + (𝑚 · 𝐷)) − 1) = ((𝐴 − 1) + (𝑚 · 𝐷)))
117116oveq1d 6630 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉) = (((𝐴 − 1) + (𝑚 · 𝐷)) + 𝑉))
11884nn0cnd 11313 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝐴 − 1) ∈ ℂ)
119118adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐴 − 1) ∈ ℂ)
12038nncnd 10996 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝑉 ∈ ℂ)
121120adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝑉 ∈ ℂ)
122119, 111, 121add32d 10223 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (((𝐴 − 1) + (𝑚 · 𝐷)) + 𝑉) = (((𝐴 − 1) + 𝑉) + (𝑚 · 𝐷)))
123117, 122eqtrd 2655 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉) = (((𝐴 − 1) + 𝑉) + (𝑚 · 𝐷)))
124123oveq2d 6631 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)) = (𝑊 · (((𝐴 − 1) + 𝑉) + (𝑚 · 𝐷))))
12592, 106, 110mul12d 10205 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · (𝑊 · 𝐷)) = (𝑊 · (𝑚 · 𝐷)))
126125oveq2d 6631 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑚 · (𝑊 · 𝐷))) = ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑊 · (𝑚 · 𝐷))))
127112, 124, 1263eqtr4d 2665 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)) = ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑚 · (𝑊 · 𝐷))))
128127adantlr 750 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)) = ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑚 · (𝑊 · 𝐷))))
129128oveq2d 6631 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))) = (((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) + ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑚 · (𝑊 · 𝐷)))))
130100, 104, 1293eqtr4d 2665 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸𝑖) + (𝑊 · 𝐷)))) = (((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))
13138ad2antrr 761 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑉 ∈ ℕ)
13212ad2antrr 761 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑊 ∈ ℕ)
13343adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐴(AP‘𝐾)𝐷) ⊆ (𝐹 “ {𝐺}))
134 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑚 · 𝐷))
135 oveq1 6622 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 = 𝑚 → (𝑛 · 𝐷) = (𝑚 · 𝐷))
136135oveq2d 6631 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 = 𝑚 → (𝐴 + (𝑛 · 𝐷)) = (𝐴 + (𝑚 · 𝐷)))
137136eqeq2d 2631 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = 𝑚 → ((𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷)) ↔ (𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑚 · 𝐷))))
138137rspcev 3299 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑚 ∈ (0...(𝐾 − 1)) ∧ (𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑚 · 𝐷))) → ∃𝑛 ∈ (0...(𝐾 − 1))(𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷)))
139134, 138mpan2 706 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 ∈ (0...(𝐾 − 1)) → ∃𝑛 ∈ (0...(𝐾 − 1))(𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷)))
14010nnnn0d 11311 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝐾 ∈ ℕ0)
141 vdwapval 15620 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐾 ∈ ℕ0𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → ((𝐴 + (𝑚 · 𝐷)) ∈ (𝐴(AP‘𝐾)𝐷) ↔ ∃𝑛 ∈ (0...(𝐾 − 1))(𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷))))
142140, 41, 42, 141syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → ((𝐴 + (𝑚 · 𝐷)) ∈ (𝐴(AP‘𝐾)𝐷) ↔ ∃𝑛 ∈ (0...(𝐾 − 1))(𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷))))
143142biimpar 502 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ∃𝑛 ∈ (0...(𝐾 − 1))(𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷))) → (𝐴 + (𝑚 · 𝐷)) ∈ (𝐴(AP‘𝐾)𝐷))
144139, 143sylan2 491 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐴 + (𝑚 · 𝐷)) ∈ (𝐴(AP‘𝐾)𝐷))
145133, 144sseldd 3589 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐴 + (𝑚 · 𝐷)) ∈ (𝐹 “ {𝐺}))
14638, 12, 6, 39, 40vdwlem4 15631 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝐹:(1...𝑉)⟶(𝑅𝑚 (1...𝑊)))
147 ffn 6012 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐹:(1...𝑉)⟶(𝑅𝑚 (1...𝑊)) → 𝐹 Fn (1...𝑉))
148146, 147syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐹 Fn (1...𝑉))
149 fniniseg 6304 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐹 Fn (1...𝑉) → ((𝐴 + (𝑚 · 𝐷)) ∈ (𝐹 “ {𝐺}) ↔ ((𝐴 + (𝑚 · 𝐷)) ∈ (1...𝑉) ∧ (𝐹‘(𝐴 + (𝑚 · 𝐷))) = 𝐺)))
150148, 149syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝐴 + (𝑚 · 𝐷)) ∈ (𝐹 “ {𝐺}) ↔ ((𝐴 + (𝑚 · 𝐷)) ∈ (1...𝑉) ∧ (𝐹‘(𝐴 + (𝑚 · 𝐷))) = 𝐺)))
151150biimpa 501 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝐴 + (𝑚 · 𝐷)) ∈ (𝐹 “ {𝐺})) → ((𝐴 + (𝑚 · 𝐷)) ∈ (1...𝑉) ∧ (𝐹‘(𝐴 + (𝑚 · 𝐷))) = 𝐺))
152145, 151syldan 487 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → ((𝐴 + (𝑚 · 𝐷)) ∈ (1...𝑉) ∧ (𝐹‘(𝐴 + (𝑚 · 𝐷))) = 𝐺))
153152simpld 475 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐴 + (𝑚 · 𝐷)) ∈ (1...𝑉))
154153adantlr 750 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐴 + (𝑚 · 𝐷)) ∈ (1...𝑉))
15522r19.21bi 2928 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝐵 + (𝐸𝑖))(AP‘𝐾)(𝐸𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}))
156155adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐵 + (𝐸𝑖))(AP‘𝐾)(𝐸𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}))
157 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) = ((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖)))
158 oveq1 6622 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 = 𝑚 → (𝑛 · (𝐸𝑖)) = (𝑚 · (𝐸𝑖)))
159158oveq2d 6631 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = 𝑚 → ((𝐵 + (𝐸𝑖)) + (𝑛 · (𝐸𝑖))) = ((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))))
160159eqeq2d 2631 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝑚 → (((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) = ((𝐵 + (𝐸𝑖)) + (𝑛 · (𝐸𝑖))) ↔ ((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) = ((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖)))))
161160rspcev 3299 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑚 ∈ (0...(𝐾 − 1)) ∧ ((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) = ((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖)))) → ∃𝑛 ∈ (0...(𝐾 − 1))((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) = ((𝐵 + (𝐸𝑖)) + (𝑛 · (𝐸𝑖))))
162157, 161mpan2 706 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ (0...(𝐾 − 1)) → ∃𝑛 ∈ (0...(𝐾 − 1))((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) = ((𝐵 + (𝐸𝑖)) + (𝑛 · (𝐸𝑖))))
16310adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑖 ∈ (1...𝑀)) → 𝐾 ∈ ℕ)
164163nnnn0d 11311 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑖 ∈ (1...𝑀)) → 𝐾 ∈ ℕ0)
16576, 79nnaddcld 11027 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐵 + (𝐸𝑖)) ∈ ℕ)
166 vdwapval 15620 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐾 ∈ ℕ0 ∧ (𝐵 + (𝐸𝑖)) ∈ ℕ ∧ (𝐸𝑖) ∈ ℕ) → (((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) ∈ ((𝐵 + (𝐸𝑖))(AP‘𝐾)(𝐸𝑖)) ↔ ∃𝑛 ∈ (0...(𝐾 − 1))((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) = ((𝐵 + (𝐸𝑖)) + (𝑛 · (𝐸𝑖)))))
167164, 165, 79, 166syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑖 ∈ (1...𝑀)) → (((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) ∈ ((𝐵 + (𝐸𝑖))(AP‘𝐾)(𝐸𝑖)) ↔ ∃𝑛 ∈ (0...(𝐾 − 1))((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) = ((𝐵 + (𝐸𝑖)) + (𝑛 · (𝐸𝑖)))))
168167biimpar 502 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑖 ∈ (1...𝑀)) ∧ ∃𝑛 ∈ (0...(𝐾 − 1))((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) = ((𝐵 + (𝐸𝑖)) + (𝑛 · (𝐸𝑖)))) → ((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) ∈ ((𝐵 + (𝐸𝑖))(AP‘𝐾)(𝐸𝑖)))
169162, 168sylan2 491 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) ∈ ((𝐵 + (𝐸𝑖))(AP‘𝐾)(𝐸𝑖)))
170156, 169sseldd 3589 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) ∈ (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}))
171 ffn 6012 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐺:(1...𝑊)⟶𝑅𝐺 Fn (1...𝑊))
17214, 171syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐺 Fn (1...𝑊))
173172adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑖 ∈ (1...𝑀)) → 𝐺 Fn (1...𝑊))
174 fniniseg 6304 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐺 Fn (1...𝑊) → (((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) ∈ (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}) ↔ (((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) ∈ (1...𝑊) ∧ (𝐺‘((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖)))) = (𝐺‘(𝐵 + (𝐸𝑖))))))
175173, 174syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑖 ∈ (1...𝑀)) → (((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) ∈ (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}) ↔ (((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) ∈ (1...𝑊) ∧ (𝐺‘((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖)))) = (𝐺‘(𝐵 + (𝐸𝑖))))))
176175biimpa 501 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (1...𝑀)) ∧ ((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) ∈ (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))})) → (((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) ∈ (1...𝑊) ∧ (𝐺‘((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖)))) = (𝐺‘(𝐵 + (𝐸𝑖)))))
177170, 176syldan 487 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) ∈ (1...𝑊) ∧ (𝐺‘((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖)))) = (𝐺‘(𝐵 + (𝐸𝑖)))))
178177simpld 475 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) ∈ (1...𝑊))
179131, 132, 154, 178vdwlem3 15630 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))) ∈ (1...(𝑊 · (2 · 𝑉))))
180130, 179eqeltrd 2698 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸𝑖) + (𝑊 · 𝐷)))) ∈ (1...(𝑊 · (2 · 𝑉))))
181 oveq1 6622 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = ((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) → (𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))) = (((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))
182181fveq2d 6162 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = ((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) → (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))) = (𝐻‘(((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))
183 eqid 2621 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))
184 fvex 6168 . . . . . . . . . . . . . . . . . . . 20 (𝐻‘(((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))) ∈ V
185182, 183, 184fvmpt 6249 . . . . . . . . . . . . . . . . . . 19 (((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) ∈ (1...𝑊) → ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))‘((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖)))) = (𝐻‘(((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))
186178, 185syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))‘((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖)))) = (𝐻‘(((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))
187177simprd 479 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐺‘((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖)))) = (𝐺‘(𝐵 + (𝐸𝑖))))
188152simprd 479 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐹‘(𝐴 + (𝑚 · 𝐷))) = 𝐺)
189 oveq1 6622 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 = (𝐴 + (𝑚 · 𝐷)) → (𝑥 − 1) = ((𝐴 + (𝑚 · 𝐷)) − 1))
190189oveq1d 6630 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 = (𝐴 + (𝑚 · 𝐷)) → ((𝑥 − 1) + 𝑉) = (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))
191190oveq2d 6631 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 = (𝐴 + (𝑚 · 𝐷)) → (𝑊 · ((𝑥 − 1) + 𝑉)) = (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))
192191oveq2d 6631 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = (𝐴 + (𝑚 · 𝐷)) → (𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))) = (𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))
193192fveq2d 6162 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = (𝐴 + (𝑚 · 𝐷)) → (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉)))) = (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))
194193mpteq2dv 4715 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = (𝐴 + (𝑚 · 𝐷)) → (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))))
195 ovex 6643 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1...𝑊) ∈ V
196195mptex 6451 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))) ∈ V
197194, 40, 196fvmpt 6249 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 + (𝑚 · 𝐷)) ∈ (1...𝑉) → (𝐹‘(𝐴 + (𝑚 · 𝐷))) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))))
198153, 197syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐹‘(𝐴 + (𝑚 · 𝐷))) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))))
199188, 198eqtr3d 2657 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝐺 = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))))
200199adantlr 750 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐺 = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))))
201200fveq1d 6160 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐺‘((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖)))) = ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))‘((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖)))))
202187, 201eqtr3d 2657 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐺‘(𝐵 + (𝐸𝑖))) = ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))‘((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖)))))
203130fveq2d 6162 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐻‘(((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸𝑖) + (𝑊 · 𝐷))))) = (𝐻‘(((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))
204186, 202, 2033eqtr4rd 2666 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐻‘(((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸𝑖) + (𝑊 · 𝐷))))) = (𝐺‘(𝐵 + (𝐸𝑖))))
205180, 204jca 554 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸𝑖) + (𝑊 · 𝐷)))) ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻‘(((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸𝑖) + (𝑊 · 𝐷))))) = (𝐺‘(𝐵 + (𝐸𝑖)))))
206 eleq1 2686 . . . . . . . . . . . . . . . . 17 (𝑥 = (((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸𝑖) + (𝑊 · 𝐷)))) → (𝑥 ∈ (1...(𝑊 · (2 · 𝑉))) ↔ (((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸𝑖) + (𝑊 · 𝐷)))) ∈ (1...(𝑊 · (2 · 𝑉)))))
207 fveq2 6158 . . . . . . . . . . . . . . . . . 18 (𝑥 = (((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸𝑖) + (𝑊 · 𝐷)))) → (𝐻𝑥) = (𝐻‘(((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸𝑖) + (𝑊 · 𝐷))))))
208207eqeq1d 2623 . . . . . . . . . . . . . . . . 17 (𝑥 = (((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸𝑖) + (𝑊 · 𝐷)))) → ((𝐻𝑥) = (𝐺‘(𝐵 + (𝐸𝑖))) ↔ (𝐻‘(((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸𝑖) + (𝑊 · 𝐷))))) = (𝐺‘(𝐵 + (𝐸𝑖)))))
209206, 208anbi12d 746 . . . . . . . . . . . . . . . 16 (𝑥 = (((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸𝑖) + (𝑊 · 𝐷)))) → ((𝑥 ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻𝑥) = (𝐺‘(𝐵 + (𝐸𝑖)))) ↔ ((((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸𝑖) + (𝑊 · 𝐷)))) ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻‘(((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸𝑖) + (𝑊 · 𝐷))))) = (𝐺‘(𝐵 + (𝐸𝑖))))))
210205, 209syl5ibrcom 237 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑥 = (((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸𝑖) + (𝑊 · 𝐷)))) → (𝑥 ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻𝑥) = (𝐺‘(𝐵 + (𝐸𝑖))))))
211210rexlimdva 3026 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀)) → (∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = (((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸𝑖) + (𝑊 · 𝐷)))) → (𝑥 ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻𝑥) = (𝐺‘(𝐵 + (𝐸𝑖))))))
21287adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℕ)
213165, 212nnaddcld 11027 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ ℕ)
21465adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑊 · 𝐷) ∈ ℕ)
21579, 214nnaddcld 11027 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝐸𝑖) + (𝑊 · 𝐷)) ∈ ℕ)
216 vdwapval 15620 . . . . . . . . . . . . . . 15 ((𝐾 ∈ ℕ0 ∧ ((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ ℕ ∧ ((𝐸𝑖) + (𝑊 · 𝐷)) ∈ ℕ) → (𝑥 ∈ (((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)((𝐸𝑖) + (𝑊 · 𝐷))) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = (((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸𝑖) + (𝑊 · 𝐷))))))
217164, 213, 215, 216syl3anc 1323 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑥 ∈ (((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)((𝐸𝑖) + (𝑊 · 𝐷))) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = (((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸𝑖) + (𝑊 · 𝐷))))))
218 ffn 6012 . . . . . . . . . . . . . . . . 17 (𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅𝐻 Fn (1...(𝑊 · (2 · 𝑉))))
21939, 218syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝐻 Fn (1...(𝑊 · (2 · 𝑉))))
220219adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → 𝐻 Fn (1...(𝑊 · (2 · 𝑉))))
221 fniniseg 6304 . . . . . . . . . . . . . . 15 (𝐻 Fn (1...(𝑊 · (2 · 𝑉))) → (𝑥 ∈ (𝐻 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}) ↔ (𝑥 ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻𝑥) = (𝐺‘(𝐵 + (𝐸𝑖))))))
222220, 221syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑥 ∈ (𝐻 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}) ↔ (𝑥 ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻𝑥) = (𝐺‘(𝐵 + (𝐸𝑖))))))
223211, 217, 2223imtr4d 283 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑥 ∈ (((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)((𝐸𝑖) + (𝑊 · 𝐷))) → 𝑥 ∈ (𝐻 “ {(𝐺‘(𝐵 + (𝐸𝑖)))})))
224223ssrdv 3594 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → (((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)((𝐸𝑖) + (𝑊 · 𝐷))) ⊆ (𝐻 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}))
225 ssun1 3760 . . . . . . . . . . . . . . . . . . 19 (1...𝑀) ⊆ ((1...𝑀) ∪ {(𝑀 + 1)})
226 fzsuc 12346 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ (ℤ‘1) → (1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)}))
22752, 226syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)}))
228225, 227syl5sseqr 3639 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1...𝑀) ⊆ (1...(𝑀 + 1)))
229228sselda 3588 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑖 ∈ (1...(𝑀 + 1)))
230 eqeq1 2625 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑖 → (𝑗 = (𝑀 + 1) ↔ 𝑖 = (𝑀 + 1)))
231 fveq2 6158 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑖 → (𝐸𝑗) = (𝐸𝑖))
232230, 231ifbieq2d 4089 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑖 → if(𝑗 = (𝑀 + 1), 0, (𝐸𝑗)) = if(𝑖 = (𝑀 + 1), 0, (𝐸𝑖)))
233232oveq1d 6630 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑖 → (if(𝑗 = (𝑀 + 1), 0, (𝐸𝑗)) + (𝑊 · 𝐷)) = (if(𝑖 = (𝑀 + 1), 0, (𝐸𝑖)) + (𝑊 · 𝐷)))
234 ovex 6643 . . . . . . . . . . . . . . . . . 18 (if(𝑖 = (𝑀 + 1), 0, (𝐸𝑖)) + (𝑊 · 𝐷)) ∈ V
235233, 46, 234fvmpt 6249 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (1...(𝑀 + 1)) → (𝑃𝑖) = (if(𝑖 = (𝑀 + 1), 0, (𝐸𝑖)) + (𝑊 · 𝐷)))
236229, 235syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑃𝑖) = (if(𝑖 = (𝑀 + 1), 0, (𝐸𝑖)) + (𝑊 · 𝐷)))
237 elfzle2 12303 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (1...𝑀) → 𝑖𝑀)
23818nnred 10995 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑀 ∈ ℝ)
239238ltp1d 10914 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑀 < (𝑀 + 1))
240 peano2re 10169 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑀 ∈ ℝ → (𝑀 + 1) ∈ ℝ)
241238, 240syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑀 + 1) ∈ ℝ)
242238, 241ltnled 10144 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑀 < (𝑀 + 1) ↔ ¬ (𝑀 + 1) ≤ 𝑀))
243239, 242mpbid 222 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ¬ (𝑀 + 1) ≤ 𝑀)
244 breq1 4626 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 = (𝑀 + 1) → (𝑖𝑀 ↔ (𝑀 + 1) ≤ 𝑀))
245244notbid 308 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = (𝑀 + 1) → (¬ 𝑖𝑀 ↔ ¬ (𝑀 + 1) ≤ 𝑀))
246243, 245syl5ibrcom 237 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑖 = (𝑀 + 1) → ¬ 𝑖𝑀))
247246con2d 129 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑖𝑀 → ¬ 𝑖 = (𝑀 + 1)))
248237, 247syl5 34 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑖 ∈ (1...𝑀) → ¬ 𝑖 = (𝑀 + 1)))
249248imp 445 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (1...𝑀)) → ¬ 𝑖 = (𝑀 + 1))
250 iffalse 4073 . . . . . . . . . . . . . . . . . 18 𝑖 = (𝑀 + 1) → if(𝑖 = (𝑀 + 1), 0, (𝐸𝑖)) = (𝐸𝑖))
251250oveq1d 6630 . . . . . . . . . . . . . . . . 17 𝑖 = (𝑀 + 1) → (if(𝑖 = (𝑀 + 1), 0, (𝐸𝑖)) + (𝑊 · 𝐷)) = ((𝐸𝑖) + (𝑊 · 𝐷)))
252249, 251syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑀)) → (if(𝑖 = (𝑀 + 1), 0, (𝐸𝑖)) + (𝑊 · 𝐷)) = ((𝐸𝑖) + (𝑊 · 𝐷)))
253236, 252eqtrd 2655 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑃𝑖) = ((𝐸𝑖) + (𝑊 · 𝐷)))
254253oveq2d 6631 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑇 + (𝑃𝑖)) = (𝑇 + ((𝐸𝑖) + (𝑊 · 𝐷))))
25547nncnd 10996 . . . . . . . . . . . . . . . 16 (𝜑𝑇 ∈ ℂ)
256255adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑇 ∈ ℂ)
257101adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑊 · 𝐷) ∈ ℂ)
258256, 80, 257add12d 10222 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑇 + ((𝐸𝑖) + (𝑊 · 𝐷))) = ((𝐸𝑖) + (𝑇 + (𝑊 · 𝐷))))
25945oveq1i 6625 . . . . . . . . . . . . . . . . . 18 (𝑇 + (𝑊 · 𝐷)) = ((𝐵 + (𝑊 · ((𝐴 + (𝑉𝐷)) − 1))) + (𝑊 · 𝐷))
26016nncnd 10996 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐵 ∈ ℂ)
261120, 109subcld 10352 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑉𝐷) ∈ ℂ)
262113, 261addcld 10019 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝐴 + (𝑉𝐷)) ∈ ℂ)
263 ax-1cn 9954 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ ℂ
264 subcl 10240 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴 + (𝑉𝐷)) ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 + (𝑉𝐷)) − 1) ∈ ℂ)
265262, 263, 264sylancl 693 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝐴 + (𝑉𝐷)) − 1) ∈ ℂ)
266105, 265mulcld 10020 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑊 · ((𝐴 + (𝑉𝐷)) − 1)) ∈ ℂ)
267260, 266, 101addassd 10022 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝐵 + (𝑊 · ((𝐴 + (𝑉𝐷)) − 1))) + (𝑊 · 𝐷)) = (𝐵 + ((𝑊 · ((𝐴 + (𝑉𝐷)) − 1)) + (𝑊 · 𝐷))))
268105, 265, 109adddid 10024 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑊 · (((𝐴 + (𝑉𝐷)) − 1) + 𝐷)) = ((𝑊 · ((𝐴 + (𝑉𝐷)) − 1)) + (𝑊 · 𝐷)))
269113, 261, 109addassd 10022 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → ((𝐴 + (𝑉𝐷)) + 𝐷) = (𝐴 + ((𝑉𝐷) + 𝐷)))
270120, 109npcand 10356 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → ((𝑉𝐷) + 𝐷) = 𝑉)
271270oveq2d 6631 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝐴 + ((𝑉𝐷) + 𝐷)) = (𝐴 + 𝑉))
272269, 271eqtrd 2655 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ((𝐴 + (𝑉𝐷)) + 𝐷) = (𝐴 + 𝑉))
273272oveq1d 6630 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (((𝐴 + (𝑉𝐷)) + 𝐷) − 1) = ((𝐴 + 𝑉) − 1))
274 1cnd 10016 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → 1 ∈ ℂ)
275262, 109, 274addsubd 10373 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (((𝐴 + (𝑉𝐷)) + 𝐷) − 1) = (((𝐴 + (𝑉𝐷)) − 1) + 𝐷))
276113, 120, 274addsubd 10373 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝐴 + 𝑉) − 1) = ((𝐴 − 1) + 𝑉))
277273, 275, 2763eqtr3d 2663 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (((𝐴 + (𝑉𝐷)) − 1) + 𝐷) = ((𝐴 − 1) + 𝑉))
278277oveq2d 6631 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑊 · (((𝐴 + (𝑉𝐷)) − 1) + 𝐷)) = (𝑊 · ((𝐴 − 1) + 𝑉)))
279268, 278eqtr3d 2657 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑊 · ((𝐴 + (𝑉𝐷)) − 1)) + (𝑊 · 𝐷)) = (𝑊 · ((𝐴 − 1) + 𝑉)))
280279oveq2d 6631 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐵 + ((𝑊 · ((𝐴 + (𝑉𝐷)) − 1)) + (𝑊 · 𝐷))) = (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))))
281267, 280eqtrd 2655 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝐵 + (𝑊 · ((𝐴 + (𝑉𝐷)) − 1))) + (𝑊 · 𝐷)) = (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))))
282259, 281syl5eq 2667 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑇 + (𝑊 · 𝐷)) = (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))))
283282oveq2d 6631 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝐸𝑖) + (𝑇 + (𝑊 · 𝐷))) = ((𝐸𝑖) + (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
284283adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝐸𝑖) + (𝑇 + (𝑊 · 𝐷))) = ((𝐸𝑖) + (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
28588adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℂ)
28680, 77, 285addassd 10022 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → (((𝐸𝑖) + 𝐵) + (𝑊 · ((𝐴 − 1) + 𝑉))) = ((𝐸𝑖) + (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
28780, 77addcomd 10198 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝐸𝑖) + 𝐵) = (𝐵 + (𝐸𝑖)))
288287oveq1d 6630 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → (((𝐸𝑖) + 𝐵) + (𝑊 · ((𝐴 − 1) + 𝑉))) = ((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))))
289284, 286, 2883eqtr2d 2661 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝐸𝑖) + (𝑇 + (𝑊 · 𝐷))) = ((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))))
290254, 258, 2893eqtrd 2659 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑇 + (𝑃𝑖)) = ((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))))
291290, 253oveq12d 6633 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) = (((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)((𝐸𝑖) + (𝑊 · 𝐷))))
292 cnvimass 5454 . . . . . . . . . . . . . . . . . . 19 (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}) ⊆ dom 𝐺
293 fdm 6018 . . . . . . . . . . . . . . . . . . . 20 (𝐺:(1...𝑊)⟶𝑅 → dom 𝐺 = (1...𝑊))
29414, 293syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → dom 𝐺 = (1...𝑊))
295292, 294syl5sseq 3638 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}) ⊆ (1...𝑊))
296295adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}) ⊆ (1...𝑊))
297 vdwapid1 15622 . . . . . . . . . . . . . . . . . . 19 ((𝐾 ∈ ℕ ∧ (𝐵 + (𝐸𝑖)) ∈ ℕ ∧ (𝐸𝑖) ∈ ℕ) → (𝐵 + (𝐸𝑖)) ∈ ((𝐵 + (𝐸𝑖))(AP‘𝐾)(𝐸𝑖)))
298163, 165, 79, 297syl3anc 1323 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐵 + (𝐸𝑖)) ∈ ((𝐵 + (𝐸𝑖))(AP‘𝐾)(𝐸𝑖)))
299155, 298sseldd 3589 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐵 + (𝐸𝑖)) ∈ (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}))
300296, 299sseldd 3589 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐵 + (𝐸𝑖)) ∈ (1...𝑊))
301 oveq1 6622 . . . . . . . . . . . . . . . . . 18 (𝑦 = (𝐵 + (𝐸𝑖)) → (𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))) = ((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))))
302301fveq2d 6162 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝐵 + (𝐸𝑖)) → (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))) = (𝐻‘((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))))
303 eqid 2621 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))))) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
304 fvex 6168 . . . . . . . . . . . . . . . . 17 (𝐻‘((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))) ∈ V
305302, 303, 304fvmpt 6249 . . . . . . . . . . . . . . . 16 ((𝐵 + (𝐸𝑖)) ∈ (1...𝑊) → ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))‘(𝐵 + (𝐸𝑖))) = (𝐻‘((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))))
306300, 305syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))‘(𝐵 + (𝐸𝑖))) = (𝐻‘((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))))
307 vdwapid1 15622 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐾 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → 𝐴 ∈ (𝐴(AP‘𝐾)𝐷))
30810, 41, 42, 307syl3anc 1323 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐴 ∈ (𝐴(AP‘𝐾)𝐷))
30943, 308sseldd 3589 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐴 ∈ (𝐹 “ {𝐺}))
310 fniniseg 6304 . . . . . . . . . . . . . . . . . . . . 21 (𝐹 Fn (1...𝑉) → (𝐴 ∈ (𝐹 “ {𝐺}) ↔ (𝐴 ∈ (1...𝑉) ∧ (𝐹𝐴) = 𝐺)))
311148, 310syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝐴 ∈ (𝐹 “ {𝐺}) ↔ (𝐴 ∈ (1...𝑉) ∧ (𝐹𝐴) = 𝐺)))
312309, 311mpbid 222 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐴 ∈ (1...𝑉) ∧ (𝐹𝐴) = 𝐺))
313312simprd 479 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐹𝐴) = 𝐺)
314312simpld 475 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐴 ∈ (1...𝑉))
315 oveq1 6622 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = 𝐴 → (𝑥 − 1) = (𝐴 − 1))
316315oveq1d 6630 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝐴 → ((𝑥 − 1) + 𝑉) = ((𝐴 − 1) + 𝑉))
317316oveq2d 6631 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝐴 → (𝑊 · ((𝑥 − 1) + 𝑉)) = (𝑊 · ((𝐴 − 1) + 𝑉)))
318317oveq2d 6631 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝐴 → (𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))) = (𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))))
319318fveq2d 6162 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝐴 → (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉)))) = (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
320319mpteq2dv 4715 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝐴 → (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))))))
321195mptex 6451 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))))) ∈ V
322320, 40, 321fvmpt 6249 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ (1...𝑉) → (𝐹𝐴) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))))))
323314, 322syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐹𝐴) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))))))
324313, 323eqtr3d 2657 . . . . . . . . . . . . . . . . 17 (𝜑𝐺 = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))))))
325324fveq1d 6160 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐺‘(𝐵 + (𝐸𝑖))) = ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))‘(𝐵 + (𝐸𝑖))))
326325adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐺‘(𝐵 + (𝐸𝑖))) = ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))‘(𝐵 + (𝐸𝑖))))
327290fveq2d 6162 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻‘(𝑇 + (𝑃𝑖))) = (𝐻‘((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))))
328306, 326, 3273eqtr4rd 2666 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻‘(𝑇 + (𝑃𝑖))) = (𝐺‘(𝐵 + (𝐸𝑖))))
329328sneqd 4167 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑀)) → {(𝐻‘(𝑇 + (𝑃𝑖)))} = {(𝐺‘(𝐵 + (𝐸𝑖)))})
330329imaeq2d 5435 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))}) = (𝐻 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}))
331224, 291, 3303sstr4d 3633 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))}))
332331ex 450 . . . . . . . . . 10 (𝜑 → (𝑖 ∈ (1...𝑀) → ((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))})))
333260adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝐵 ∈ ℂ)
33488adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℂ)
335333, 334, 98addassd 10022 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) = (𝐵 + ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑚 · (𝑊 · 𝐷)))))
336127oveq2d 6631 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))) = (𝐵 + ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑚 · (𝑊 · 𝐷)))))
337335, 336eqtr4d 2658 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) = (𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))
33838adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝑉 ∈ ℕ)
33912adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝑊 ∈ ℕ)
340 eluzfz1 12306 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑀 ∈ (ℤ‘1) → 1 ∈ (1...𝑀))
34152, 340syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → 1 ∈ (1...𝑀))
342 ne0i 3903 . . . . . . . . . . . . . . . . . . . . . . . 24 (1 ∈ (1...𝑀) → (1...𝑀) ≠ ∅)
343341, 342syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (1...𝑀) ≠ ∅)
344 elfzuz3 12297 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐵 + (𝐸𝑖)) ∈ (1...𝑊) → 𝑊 ∈ (ℤ‘(𝐵 + (𝐸𝑖))))
345300, 344syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑊 ∈ (ℤ‘(𝐵 + (𝐸𝑖))))
34616nnzd 11441 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑𝐵 ∈ ℤ)
347 uzid 11662 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐵 ∈ ℤ → 𝐵 ∈ (ℤ𝐵))
348346, 347syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝐵 ∈ (ℤ𝐵))
349348adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑖 ∈ (1...𝑀)) → 𝐵 ∈ (ℤ𝐵))
35079nnnn0d 11311 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐸𝑖) ∈ ℕ0)
351 uzaddcl 11704 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐵 ∈ (ℤ𝐵) ∧ (𝐸𝑖) ∈ ℕ0) → (𝐵 + (𝐸𝑖)) ∈ (ℤ𝐵))
352349, 350, 351syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐵 + (𝐸𝑖)) ∈ (ℤ𝐵))
353 uztrn 11664 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑊 ∈ (ℤ‘(𝐵 + (𝐸𝑖))) ∧ (𝐵 + (𝐸𝑖)) ∈ (ℤ𝐵)) → 𝑊 ∈ (ℤ𝐵))
354345, 352, 353syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑊 ∈ (ℤ𝐵))
355 eluzle 11660 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑊 ∈ (ℤ𝐵) → 𝐵𝑊)
356354, 355syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑖 ∈ (1...𝑀)) → 𝐵𝑊)
357356ralrimiva 2962 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ∀𝑖 ∈ (1...𝑀)𝐵𝑊)
358 r19.2z 4038 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...𝑀) ≠ ∅ ∧ ∀𝑖 ∈ (1...𝑀)𝐵𝑊) → ∃𝑖 ∈ (1...𝑀)𝐵𝑊)
359343, 357, 358syl2anc 692 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ∃𝑖 ∈ (1...𝑀)𝐵𝑊)
360 idd 24 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (1...𝑀) → (𝐵𝑊𝐵𝑊))
361360rexlimiv 3022 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑖 ∈ (1...𝑀)𝐵𝑊𝐵𝑊)
362359, 361syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐵𝑊)
36312nnzd 11441 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑊 ∈ ℤ)
364 fznn 12366 . . . . . . . . . . . . . . . . . . . . . 22 (𝑊 ∈ ℤ → (𝐵 ∈ (1...𝑊) ↔ (𝐵 ∈ ℕ ∧ 𝐵𝑊)))
365363, 364syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐵 ∈ (1...𝑊) ↔ (𝐵 ∈ ℕ ∧ 𝐵𝑊)))
36616, 362, 365mpbir2and 956 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐵 ∈ (1...𝑊))
367366adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝐵 ∈ (1...𝑊))
368338, 339, 153, 367vdwlem3 15630 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))) ∈ (1...(𝑊 · (2 · 𝑉))))
369337, 368eqeltrd 2698 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) ∈ (1...(𝑊 · (2 · 𝑉))))
370 oveq1 6622 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝐵 → (𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))) = (𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))
371370fveq2d 6162 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝐵 → (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))) = (𝐻‘(𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))
372 fvex 6168 . . . . . . . . . . . . . . . . . . . 20 (𝐻‘(𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))) ∈ V
373371, 183, 372fvmpt 6249 . . . . . . . . . . . . . . . . . . 19 (𝐵 ∈ (1...𝑊) → ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))‘𝐵) = (𝐻‘(𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))
374367, 373syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))‘𝐵) = (𝐻‘(𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))
375199fveq1d 6160 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐺𝐵) = ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))‘𝐵))
376337fveq2d 6162 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐻‘((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷)))) = (𝐻‘(𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))
377374, 375, 3763eqtr4rd 2666 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐻‘((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷)))) = (𝐺𝐵))
378369, 377jca 554 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻‘((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷)))) = (𝐺𝐵)))
379 eleq1 2686 . . . . . . . . . . . . . . . . 17 (𝑧 = ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) → (𝑧 ∈ (1...(𝑊 · (2 · 𝑉))) ↔ ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) ∈ (1...(𝑊 · (2 · 𝑉)))))
380 fveq2 6158 . . . . . . . . . . . . . . . . . 18 (𝑧 = ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) → (𝐻𝑧) = (𝐻‘((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷)))))
381380eqeq1d 2623 . . . . . . . . . . . . . . . . 17 (𝑧 = ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) → ((𝐻𝑧) = (𝐺𝐵) ↔ (𝐻‘((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷)))) = (𝐺𝐵)))
382379, 381anbi12d 746 . . . . . . . . . . . . . . . 16 (𝑧 = ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) → ((𝑧 ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻𝑧) = (𝐺𝐵)) ↔ (((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻‘((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷)))) = (𝐺𝐵))))
383378, 382syl5ibrcom 237 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝑧 = ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) → (𝑧 ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻𝑧) = (𝐺𝐵))))
384383rexlimdva 3026 . . . . . . . . . . . . . 14 (𝜑 → (∃𝑚 ∈ (0...(𝐾 − 1))𝑧 = ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) → (𝑧 ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻𝑧) = (𝐺𝐵))))
38516, 87nnaddcld 11027 . . . . . . . . . . . . . . 15 (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ ℕ)
386 vdwapval 15620 . . . . . . . . . . . . . . 15 ((𝐾 ∈ ℕ0 ∧ (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ ℕ ∧ (𝑊 · 𝐷) ∈ ℕ) → (𝑧 ∈ ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)(𝑊 · 𝐷)) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑧 = ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷)))))
387140, 385, 65, 386syl3anc 1323 . . . . . . . . . . . . . 14 (𝜑 → (𝑧 ∈ ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)(𝑊 · 𝐷)) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑧 = ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷)))))
388 fniniseg 6304 . . . . . . . . . . . . . . 15 (𝐻 Fn (1...(𝑊 · (2 · 𝑉))) → (𝑧 ∈ (𝐻 “ {(𝐺𝐵)}) ↔ (𝑧 ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻𝑧) = (𝐺𝐵))))
389219, 388syl 17 . . . . . . . . . . . . . 14 (𝜑 → (𝑧 ∈ (𝐻 “ {(𝐺𝐵)}) ↔ (𝑧 ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻𝑧) = (𝐺𝐵))))
390384, 387, 3893imtr4d 283 . . . . . . . . . . . . 13 (𝜑 → (𝑧 ∈ ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)(𝑊 · 𝐷)) → 𝑧 ∈ (𝐻 “ {(𝐺𝐵)})))
391390ssrdv 3594 . . . . . . . . . . . 12 (𝜑 → ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)(𝑊 · 𝐷)) ⊆ (𝐻 “ {(𝐺𝐵)}))
39218peano2nnd 10997 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑀 + 1) ∈ ℕ)
393392, 51syl6eleq 2708 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑀 + 1) ∈ (ℤ‘1))
394 eluzfz2 12307 . . . . . . . . . . . . . . . . . 18 ((𝑀 + 1) ∈ (ℤ‘1) → (𝑀 + 1) ∈ (1...(𝑀 + 1)))
395393, 394syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑀 + 1) ∈ (1...(𝑀 + 1)))
396 iftrue 4070 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑀 + 1) → if(𝑗 = (𝑀 + 1), 0, (𝐸𝑗)) = 0)
397396oveq1d 6630 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑀 + 1) → (if(𝑗 = (𝑀 + 1), 0, (𝐸𝑗)) + (𝑊 · 𝐷)) = (0 + (𝑊 · 𝐷)))
398 ovex 6643 . . . . . . . . . . . . . . . . . 18 (0 + (𝑊 · 𝐷)) ∈ V
399397, 46, 398fvmpt 6249 . . . . . . . . . . . . . . . . 17 ((𝑀 + 1) ∈ (1...(𝑀 + 1)) → (𝑃‘(𝑀 + 1)) = (0 + (𝑊 · 𝐷)))
400395, 399syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑃‘(𝑀 + 1)) = (0 + (𝑊 · 𝐷)))
401101addid2d 10197 . . . . . . . . . . . . . . . 16 (𝜑 → (0 + (𝑊 · 𝐷)) = (𝑊 · 𝐷))
402400, 401eqtrd 2655 . . . . . . . . . . . . . . 15 (𝜑 → (𝑃‘(𝑀 + 1)) = (𝑊 · 𝐷))
403402oveq2d 6631 . . . . . . . . . . . . . 14 (𝜑 → (𝑇 + (𝑃‘(𝑀 + 1))) = (𝑇 + (𝑊 · 𝐷)))
404403, 282eqtrd 2655 . . . . . . . . . . . . 13 (𝜑 → (𝑇 + (𝑃‘(𝑀 + 1))) = (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))))
405404, 402oveq12d 6633 . . . . . . . . . . . 12 (𝜑 → ((𝑇 + (𝑃‘(𝑀 + 1)))(AP‘𝐾)(𝑃‘(𝑀 + 1))) = ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)(𝑊 · 𝐷)))
406 oveq1 6622 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝐵 → (𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))) = (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))))
407406fveq2d 6162 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐵 → (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))) = (𝐻‘(𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
408 fvex 6168 . . . . . . . . . . . . . . . . 17 (𝐻‘(𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))) ∈ V
409407, 303, 408fvmpt 6249 . . . . . . . . . . . . . . . 16 (𝐵 ∈ (1...𝑊) → ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))‘𝐵) = (𝐻‘(𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
410366, 409syl 17 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))‘𝐵) = (𝐻‘(𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
411324fveq1d 6160 . . . . . . . . . . . . . . 15 (𝜑 → (𝐺𝐵) = ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))‘𝐵))
412404fveq2d 6162 . . . . . . . . . . . . . . 15 (𝜑 → (𝐻‘(𝑇 + (𝑃‘(𝑀 + 1)))) = (𝐻‘(𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
413410, 411, 4123eqtr4rd 2666 . . . . . . . . . . . . . 14 (𝜑 → (𝐻‘(𝑇 + (𝑃‘(𝑀 + 1)))) = (𝐺𝐵))
414413sneqd 4167 . . . . . . . . . . . . 13 (𝜑 → {(𝐻‘(𝑇 + (𝑃‘(𝑀 + 1))))} = {(𝐺𝐵)})
415414imaeq2d 5435 . . . . . . . . . . . 12 (𝜑 → (𝐻 “ {(𝐻‘(𝑇 + (𝑃‘(𝑀 + 1))))}) = (𝐻 “ {(𝐺𝐵)}))
416391, 405, 4153sstr4d 3633 . . . . . . . . . . 11 (𝜑 → ((𝑇 + (𝑃‘(𝑀 + 1)))(AP‘𝐾)(𝑃‘(𝑀 + 1))) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃‘(𝑀 + 1))))}))
417 fveq2 6158 . . . . . . . . . . . . . 14 (𝑖 = (𝑀 + 1) → (𝑃𝑖) = (𝑃‘(𝑀 + 1)))
418417oveq2d 6631 . . . . . . . . . . . . 13 (𝑖 = (𝑀 + 1) → (𝑇 + (𝑃𝑖)) = (𝑇 + (𝑃‘(𝑀 + 1))))
419418, 417oveq12d 6633 . . . . . . . . . . . 12 (𝑖 = (𝑀 + 1) → ((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) = ((𝑇 + (𝑃‘(𝑀 + 1)))(AP‘𝐾)(𝑃‘(𝑀 + 1))))
420418fveq2d 6162 . . . . . . . . . . . . . 14 (𝑖 = (𝑀 + 1) → (𝐻‘(𝑇 + (𝑃𝑖))) = (𝐻‘(𝑇 + (𝑃‘(𝑀 + 1)))))
421420sneqd 4167 . . . . . . . . . . . . 13 (𝑖 = (𝑀 + 1) → {(𝐻‘(𝑇 + (𝑃𝑖)))} = {(𝐻‘(𝑇 + (𝑃‘(𝑀 + 1))))})
422421imaeq2d 5435 . . . . . . . . . . . 12 (𝑖 = (𝑀 + 1) → (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))}) = (𝐻 “ {(𝐻‘(𝑇 + (𝑃‘(𝑀 + 1))))}))
423419, 422sseq12d 3619 . . . . . . . . . . 11 (𝑖 = (𝑀 + 1) → (((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))}) ↔ ((𝑇 + (𝑃‘(𝑀 + 1)))(AP‘𝐾)(𝑃‘(𝑀 + 1))) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃‘(𝑀 + 1))))})))
424416, 423syl5ibrcom 237 . . . . . . . . . 10 (𝜑 → (𝑖 = (𝑀 + 1) → ((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))})))
425332, 424jaod 395 . . . . . . . . 9 (𝜑 → ((𝑖 ∈ (1...𝑀) ∨ 𝑖 = (𝑀 + 1)) → ((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))})))
42675, 425sylbid 230 . . . . . . . 8 (𝜑 → (𝑖 ∈ (1...(𝑀 + 1)) → ((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))})))
427426ralrimiv 2961 . . . . . . 7 (𝜑 → ∀𝑖 ∈ (1...(𝑀 + 1))((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))}))
428427adantr 481 . . . . . 6 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → ∀𝑖 ∈ (1...(𝑀 + 1))((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))}))
429227rexeqdv 3138 . . . . . . . . . . . 12 (𝜑 → (∃𝑖 ∈ (1...(𝑀 + 1))𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ↔ ∃𝑖 ∈ ((1...𝑀) ∪ {(𝑀 + 1)})𝑥 = (𝐻‘(𝑇 + (𝑃𝑖)))))
430 rexun 3777 . . . . . . . . . . . . 13 (∃𝑖 ∈ ((1...𝑀) ∪ {(𝑀 + 1)})𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ↔ (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ∨ ∃𝑖 ∈ {(𝑀 + 1)}𝑥 = (𝐻‘(𝑇 + (𝑃𝑖)))))
431328eqeq2d 2631 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ↔ 𝑥 = (𝐺‘(𝐵 + (𝐸𝑖)))))
432431rexbidva 3044 . . . . . . . . . . . . . 14 (𝜑 → (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ↔ ∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸𝑖)))))
433 ovex 6643 . . . . . . . . . . . . . . . 16 (𝑀 + 1) ∈ V
434420eqeq2d 2631 . . . . . . . . . . . . . . . 16 (𝑖 = (𝑀 + 1) → (𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ↔ 𝑥 = (𝐻‘(𝑇 + (𝑃‘(𝑀 + 1))))))
435433, 434rexsn 4201 . . . . . . . . . . . . . . 15 (∃𝑖 ∈ {(𝑀 + 1)}𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ↔ 𝑥 = (𝐻‘(𝑇 + (𝑃‘(𝑀 + 1)))))
436413eqeq2d 2631 . . . . . . . . . . . . . . 15 (𝜑 → (𝑥 = (𝐻‘(𝑇 + (𝑃‘(𝑀 + 1)))) ↔ 𝑥 = (𝐺𝐵)))
437435, 436syl5bb 272 . . . . . . . . . . . . . 14 (𝜑 → (∃𝑖 ∈ {(𝑀 + 1)}𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ↔ 𝑥 = (𝐺𝐵)))
438432, 437orbi12d 745 . . . . . . . . . . . . 13 (𝜑 → ((∃𝑖 ∈ (1...𝑀)𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ∨ ∃𝑖 ∈ {(𝑀 + 1)}𝑥 = (𝐻‘(𝑇 + (𝑃𝑖)))) ↔ (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸𝑖))) ∨ 𝑥 = (𝐺𝐵))))
439430, 438syl5bb 272 . . . . . . . . . . . 12 (𝜑 → (∃𝑖 ∈ ((1...𝑀) ∪ {(𝑀 + 1)})𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ↔ (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸𝑖))) ∨ 𝑥 = (𝐺𝐵))))
440429, 439bitrd 268 . . . . . . . . . . 11 (𝜑 → (∃𝑖 ∈ (1...(𝑀 + 1))𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ↔ (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸𝑖))) ∨ 𝑥 = (𝐺𝐵))))
441440adantr 481 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → (∃𝑖 ∈ (1...(𝑀 + 1))𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ↔ (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸𝑖))) ∨ 𝑥 = (𝐺𝐵))))
442441abbidv 2738 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → {𝑥 ∣ ∃𝑖 ∈ (1...(𝑀 + 1))𝑥 = (𝐻‘(𝑇 + (𝑃𝑖)))} = {𝑥 ∣ (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸𝑖))) ∨ 𝑥 = (𝐺𝐵))})
443 eqid 2621 . . . . . . . . . 10 (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃𝑖)))) = (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃𝑖))))
444443rnmpt 5341 . . . . . . . . 9 ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃𝑖)))) = {𝑥 ∣ ∃𝑖 ∈ (1...(𝑀 + 1))𝑥 = (𝐻‘(𝑇 + (𝑃𝑖)))}
4452rnmpt 5341 . . . . . . . . . . 11 ran 𝐽 = {𝑥 ∣ ∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸𝑖)))}
446 df-sn 4156 . . . . . . . . . . 11 {(𝐺𝐵)} = {𝑥𝑥 = (𝐺𝐵)}
447445, 446uneq12i 3749 . . . . . . . . . 10 (ran 𝐽 ∪ {(𝐺𝐵)}) = ({𝑥 ∣ ∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸𝑖)))} ∪ {𝑥𝑥 = (𝐺𝐵)})
448 unab 3876 . . . . . . . . . 10 ({𝑥 ∣ ∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸𝑖)))} ∪ {𝑥𝑥 = (𝐺𝐵)}) = {𝑥 ∣ (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸𝑖))) ∨ 𝑥 = (𝐺𝐵))}
449447, 448eqtri 2643 . . . . . . . . 9 (ran 𝐽 ∪ {(𝐺𝐵)}) = {𝑥 ∣ (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸𝑖))) ∨ 𝑥 = (𝐺𝐵))}
450442, 444, 4493eqtr4g 2680 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃𝑖)))) = (ran 𝐽 ∪ {(𝐺𝐵)}))
451450fveq2d 6162 . . . . . . 7 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃𝑖))))) = (#‘(ran 𝐽 ∪ {(𝐺𝐵)})))
452 fzfi 12727 . . . . . . . . . 10 (1...𝑀) ∈ Fin
453 dffn4 6088 . . . . . . . . . . 11 (𝐽 Fn (1...𝑀) ↔ 𝐽:(1...𝑀)–onto→ran 𝐽)
4543, 453mpbi 220 . . . . . . . . . 10 𝐽:(1...𝑀)–onto→ran 𝐽
455 fofi 8212 . . . . . . . . . 10 (((1...𝑀) ∈ Fin ∧ 𝐽:(1...𝑀)–onto→ran 𝐽) → ran 𝐽 ∈ Fin)
456452, 454, 455mp2an 707 . . . . . . . . 9 ran 𝐽 ∈ Fin
457456a1i 11 . . . . . . . 8 (𝜑 → ran 𝐽 ∈ Fin)
458 fvex 6168 . . . . . . . . 9 (𝐺𝐵) ∈ V
459 hashunsng 13137 . . . . . . . . 9 ((𝐺𝐵) ∈ V → ((ran 𝐽 ∈ Fin ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → (#‘(ran 𝐽 ∪ {(𝐺𝐵)})) = ((#‘ran 𝐽) + 1)))
460458, 459ax-mp 5 . . . . . . . 8 ((ran 𝐽 ∈ Fin ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → (#‘(ran 𝐽 ∪ {(𝐺𝐵)})) = ((#‘ran 𝐽) + 1))
461457, 460sylan 488 . . . . . . 7 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → (#‘(ran 𝐽 ∪ {(𝐺𝐵)})) = ((#‘ran 𝐽) + 1))
46244adantr 481 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → (#‘ran 𝐽) = 𝑀)
463462oveq1d 6630 . . . . . . 7 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → ((#‘ran 𝐽) + 1) = (𝑀 + 1))
464451, 461, 4633eqtrd 2659 . . . . . 6 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃𝑖))))) = (𝑀 + 1))
465428, 464jca 554 . . . . 5 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → (∀𝑖 ∈ (1...(𝑀 + 1))((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃𝑖))))) = (𝑀 + 1)))
466 oveq1 6622 . . . . . . . . . 10 (𝑎 = 𝑇 → (𝑎 + (𝑑𝑖)) = (𝑇 + (𝑑𝑖)))
467466oveq1d 6630 . . . . . . . . 9 (𝑎 = 𝑇 → ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) = ((𝑇 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)))
468466fveq2d 6162 . . . . . . . . . . 11 (𝑎 = 𝑇 → (𝐻‘(𝑎 + (𝑑𝑖))) = (𝐻‘(𝑇 + (𝑑𝑖))))
469468sneqd 4167 . . . . . . . . . 10 (𝑎 = 𝑇 → {(𝐻‘(𝑎 + (𝑑𝑖)))} = {(𝐻‘(𝑇 + (𝑑𝑖)))})
470469imaeq2d 5435 . . . . . . . . 9 (𝑎 = 𝑇 → (𝐻 “ {(𝐻‘(𝑎 + (𝑑𝑖)))}) = (𝐻 “ {(𝐻‘(𝑇 + (𝑑𝑖)))}))
471467, 470sseq12d 3619 . . . . . . . 8 (𝑎 = 𝑇 → (((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑎 + (𝑑𝑖)))}) ↔ ((𝑇 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑑𝑖)))})))
472471ralbidv 2982 . . . . . . 7 (𝑎 = 𝑇 → (∀𝑖 ∈ (1...(𝑀 + 1))((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑎 + (𝑑𝑖)))}) ↔ ∀𝑖 ∈ (1...(𝑀 + 1))((𝑇 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑑𝑖)))})))
473468mpteq2dv 4715 . . . . . . . . . 10 (𝑎 = 𝑇 → (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑎 + (𝑑𝑖)))) = (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑𝑖)))))
474473rneqd 5323 . . . . . . . . 9 (𝑎 = 𝑇 → ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑎 + (𝑑𝑖)))) = ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑𝑖)))))
475474fveq2d 6162 . . . . . . . 8 (𝑎 = 𝑇 → (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑎 + (𝑑𝑖))))) = (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑𝑖))))))
476475eqeq1d 2623 . . . . . . 7 (𝑎 = 𝑇 → ((#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑎 + (𝑑𝑖))))) = (𝑀 + 1) ↔ (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑𝑖))))) = (𝑀 + 1)))
477472, 476anbi12d 746 . . . . . 6 (𝑎 = 𝑇 → ((∀𝑖 ∈ (1...(𝑀 + 1))((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑎 + (𝑑𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑎 + (𝑑𝑖))))) = (𝑀 + 1)) ↔ (∀𝑖 ∈ (1...(𝑀 + 1))((𝑇 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑑𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑𝑖))))) = (𝑀 + 1))))
478 fveq1 6157 . . . . . . . . . . 11 (𝑑 = 𝑃 → (𝑑𝑖) = (𝑃𝑖))
479478oveq2d 6631 . . . . . . . . . 10 (𝑑 = 𝑃 → (𝑇 + (𝑑𝑖)) = (𝑇 + (𝑃𝑖)))
480479, 478oveq12d 6633 . . . . . . . . 9 (𝑑 = 𝑃 → ((𝑇 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) = ((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)))
481479fveq2d 6162 . . . . . . . . . . 11 (𝑑 = 𝑃 → (𝐻‘(𝑇 + (𝑑𝑖))) = (𝐻‘(𝑇 + (𝑃𝑖))))
482481sneqd 4167 . . . . . . . . . 10 (𝑑 = 𝑃 → {(𝐻‘(𝑇 + (𝑑𝑖)))} = {(𝐻‘(𝑇 + (𝑃𝑖)))})
483482imaeq2d 5435 . . . . . . . . 9 (𝑑 = 𝑃 → (𝐻 “ {(𝐻‘(𝑇 + (𝑑𝑖)))}) = (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))}))
484480, 483sseq12d 3619 . . . . . . . 8 (𝑑 = 𝑃 → (((𝑇 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑑𝑖)))}) ↔ ((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))})))
485484ralbidv 2982 . . . . . . 7 (𝑑 = 𝑃 → (∀𝑖 ∈ (1...(𝑀 + 1))((𝑇 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑑𝑖)))}) ↔ ∀𝑖 ∈ (1...(𝑀 + 1))((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))})))
486481mpteq2dv 4715 . . . . . . . . . 10 (𝑑 = 𝑃 → (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑𝑖)))) = (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃𝑖)))))
487486rneqd 5323 . . . . . . . . 9 (𝑑 = 𝑃 → ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑𝑖)))) = ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃𝑖)))))
488487fveq2d 6162 . . . . . . . 8 (𝑑 = 𝑃 → (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑𝑖))))) = (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃𝑖))))))
489488eqeq1d 2623 . . . . . . 7 (𝑑 = 𝑃 → ((#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑𝑖))))) = (𝑀 + 1) ↔ (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃𝑖))))) = (𝑀 + 1)))
490485, 489anbi12d 746 . . . . . 6 (𝑑 = 𝑃 → ((∀𝑖 ∈ (1...(𝑀 + 1))((𝑇 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑑𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑𝑖))))) = (𝑀 + 1)) ↔ (∀𝑖 ∈ (1...(𝑀 + 1))((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃𝑖))))) = (𝑀 + 1))))
491477, 490rspc2ev 3313 . . . . 5 ((𝑇 ∈ ℕ ∧ 𝑃 ∈ (ℕ ↑𝑚 (1...(𝑀 + 1))) ∧ (∀𝑖 ∈ (1...(𝑀 + 1))((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃𝑖))))) = (𝑀 + 1))) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑𝑚 (1...(𝑀 + 1)))(∀𝑖 ∈ (1...(𝑀 + 1))((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑎 + (𝑑𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑎 + (𝑑𝑖))))) = (𝑀 + 1)))
49248, 73, 465, 491syl3anc 1323 . . . 4 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑𝑚 (1...(𝑀 + 1)))(∀𝑖 ∈ (1...(𝑀 + 1))((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑎 + (𝑑𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑎 + (𝑑𝑖))))) = (𝑀 + 1)))
493 ovex 6643 . . . . 5 (1...(𝑊 · (2 · 𝑉))) ∈ V
49410adantr 481 . . . . . 6 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → 𝐾 ∈ ℕ)
495494nnnn0d 11311 . . . . 5 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → 𝐾 ∈ ℕ0)
49639adantr 481 . . . . 5 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → 𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)
49718adantr 481 . . . . . 6 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → 𝑀 ∈ ℕ)
498497peano2nnd 10997 . . . . 5 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → (𝑀 + 1) ∈ ℕ)
499 eqid 2621 . . . . 5 (1...(𝑀 + 1)) = (1...(𝑀 + 1))
500493, 495, 496, 498, 499vdwpc 15627 . . . 4 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑𝑚 (1...(𝑀 + 1)))(∀𝑖 ∈ (1...(𝑀 + 1))((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑎 + (𝑑𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑎 + (𝑑𝑖))))) = (𝑀 + 1))))
501492, 500mpbird 247 . . 3 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → ⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻)
502501orcd 407 . 2 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺))
50337, 502pm2.61dan 831 1 (𝜑 → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wcel 1987  {cab 2607  wne 2790  wral 2908  wrex 2909  Vcvv 3190  cun 3558  wss 3560  c0 3897  ifcif 4064  {csn 4155  cop 4161   class class class wbr 4623  cmpt 4683  ccnv 5083  dom cdm 5084  ran crn 5085  cima 5087   Fn wfn 5852  wf 5853  ontowfo 5855  cfv 5857  (class class class)co 6615  𝑚 cmap 7817  Fincfn 7915  cc 9894  cr 9895  0cc0 9896  1c1 9897   + caddc 9899   · cmul 9901   < clt 10034  cle 10035  cmin 10226  cn 10980  2c2 11030  0cn0 11252  cz 11337  cuz 11647  ...cfz 12284  #chash 13073  APcvdwa 15612   MonoAP cvdwm 15613   PolyAP cvdwp 15614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-card 8725  df-cda 8950  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-n0 11253  df-z 11338  df-uz 11648  df-rp 11793  df-fz 12285  df-hash 13074  df-vdwap 15615  df-vdwmc 15616  df-vdwpc 15617
This theorem is referenced by:  vdwlem7  15634
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