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Theorem vdwlem6 15914
Description: Lemma for vdw 15922. (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 6428 . . . . . . 7 (𝐺‘(𝐵 + (𝐸𝑖))) ∈ V
2 vdwlem6.j . . . . . . 7 𝐽 = (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝐵 + (𝐸𝑖))))
31, 2fnmpti 6240 . . . . . 6 𝐽 Fn (1...𝑀)
4 fvelrnb 6471 . . . . . 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 468 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽𝑚) = (𝐺𝐵))) → 𝑅 ∈ Fin)
8 vdwlem7.k . . . . . . . . 9 (𝜑𝐾 ∈ (ℤ‘2))
9 eluz2nn 11951 . . . . . . . . 9 (𝐾 ∈ (ℤ‘2) → 𝐾 ∈ ℕ)
108, 9syl 17 . . . . . . . 8 (𝜑𝐾 ∈ ℕ)
1110adantr 468 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽𝑚) = (𝐺𝐵))) → 𝐾 ∈ ℕ)
12 vdwlem3.w . . . . . . . 8 (𝜑𝑊 ∈ ℕ)
1312adantr 468 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽𝑚) = (𝐺𝐵))) → 𝑊 ∈ ℕ)
14 vdwlem7.g . . . . . . . 8 (𝜑𝐺:(1...𝑊)⟶𝑅)
1514adantr 468 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽𝑚) = (𝐺𝐵))) → 𝐺:(1...𝑊)⟶𝑅)
16 vdwlem6.b . . . . . . . 8 (𝜑𝐵 ∈ ℕ)
1716adantr 468 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽𝑚) = (𝐺𝐵))) → 𝐵 ∈ ℕ)
18 vdwlem7.m . . . . . . . 8 (𝜑𝑀 ∈ ℕ)
1918adantr 468 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽𝑚) = (𝐺𝐵))) → 𝑀 ∈ ℕ)
20 vdwlem6.e . . . . . . . 8 (𝜑𝐸:(1...𝑀)⟶ℕ)
2120adantr 468 . . . . . . 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 468 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽𝑚) = (𝐺𝐵))) → ∀𝑖 ∈ (1...𝑀)((𝐵 + (𝐸𝑖))(AP‘𝐾)(𝐸𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}))
24 simprl 778 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽𝑚) = (𝐺𝐵))) → 𝑚 ∈ (1...𝑀))
25 simprr 780 . . . . . . . 8 ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽𝑚) = (𝐺𝐵))) → (𝐽𝑚) = (𝐺𝐵))
26 fveq2 6415 . . . . . . . . . . . 12 (𝑖 = 𝑚 → (𝐸𝑖) = (𝐸𝑚))
2726oveq2d 6897 . . . . . . . . . . 11 (𝑖 = 𝑚 → (𝐵 + (𝐸𝑖)) = (𝐵 + (𝐸𝑚)))
2827fveq2d 6419 . . . . . . . . . 10 (𝑖 = 𝑚 → (𝐺‘(𝐵 + (𝐸𝑖))) = (𝐺‘(𝐵 + (𝐸𝑚))))
29 fvex 6428 . . . . . . . . . 10 (𝐺‘(𝐵 + (𝐸𝑚))) ∈ V
3028, 2, 29fvmpt 6510 . . . . . . . . 9 (𝑚 ∈ (1...𝑀) → (𝐽𝑚) = (𝐺‘(𝐵 + (𝐸𝑚))))
3124, 30syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽𝑚) = (𝐺𝐵))) → (𝐽𝑚) = (𝐺‘(𝐵 + (𝐸𝑚))))
3225, 31eqtr3d 2853 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽𝑚) = (𝐺𝐵))) → (𝐺𝐵) = (𝐺‘(𝐵 + (𝐸𝑚))))
337, 11, 13, 15, 17, 19, 21, 23, 24, 32vdwlem1 15909 . . . . . 6 ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽𝑚) = (𝐺𝐵))) → (𝐾 + 1) MonoAP 𝐺)
3433rexlimdvaa 3231 . . . . 5 (𝜑 → (∃𝑚 ∈ (1...𝑀)(𝐽𝑚) = (𝐺𝐵) → (𝐾 + 1) MonoAP 𝐺))
355, 34syl5bi 233 . . . 4 (𝜑 → ((𝐺𝐵) ∈ ran 𝐽 → (𝐾 + 1) MonoAP 𝐺))
3635imp 395 . . 3 ((𝜑 ∧ (𝐺𝐵) ∈ ran 𝐽) → (𝐾 + 1) MonoAP 𝐺)
3736olcd 892 . 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 15913 . . . . . 6 (𝜑𝑇 ∈ ℕ)
4847adantr 468 . . . . 5 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → 𝑇 ∈ ℕ)
49 0nn0 11581 . . . . . . . . . 10 0 ∈ ℕ0
5049a1i 11 . . . . . . . . 9 ((((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) ∧ 𝑗 = (𝑀 + 1)) → 0 ∈ ℕ0)
51 nnuz 11948 . . . . . . . . . . . . . . . . 17 ℕ = (ℤ‘1)
5218, 51syl6eleq 2906 . . . . . . . . . . . . . . . 16 (𝜑𝑀 ∈ (ℤ‘1))
5352adantr 468 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → 𝑀 ∈ (ℤ‘1))
54 elfzp1 12621 . . . . . . . . . . . . . . 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 464 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) → (𝑗 ∈ (1...𝑀) ∨ 𝑗 = (𝑀 + 1)))
5756ord 882 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) → (¬ 𝑗 ∈ (1...𝑀) → 𝑗 = (𝑀 + 1)))
5857con1d 141 . . . . . . . . . . 11 (((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) → (¬ 𝑗 = (𝑀 + 1) → 𝑗 ∈ (1...𝑀)))
5958imp 395 . . . . . . . . . 10 ((((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) ∧ ¬ 𝑗 = (𝑀 + 1)) → 𝑗 ∈ (1...𝑀))
6020ad2antrr 708 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) → 𝐸:(1...𝑀)⟶ℕ)
6160ffvelrnda 6588 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) ∧ 𝑗 ∈ (1...𝑀)) → (𝐸𝑗) ∈ ℕ)
6261nnnn0d 11624 . . . . . . . . . 10 ((((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) ∧ 𝑗 ∈ (1...𝑀)) → (𝐸𝑗) ∈ ℕ0)
6359, 62syldan 581 . . . . . . . . 9 ((((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) ∧ ¬ 𝑗 = (𝑀 + 1)) → (𝐸𝑗) ∈ ℕ0)
6450, 63ifclda 4324 . . . . . . . 8 (((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) → if(𝑗 = (𝑀 + 1), 0, (𝐸𝑗)) ∈ ℕ0)
6512, 42nnmulcld 11361 . . . . . . . . 9 (𝜑 → (𝑊 · 𝐷) ∈ ℕ)
6665ad2antrr 708 . . . . . . . 8 (((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) → (𝑊 · 𝐷) ∈ ℕ)
67 nn0nnaddcl 11597 . . . . . . . 8 ((if(𝑗 = (𝑀 + 1), 0, (𝐸𝑗)) ∈ ℕ0 ∧ (𝑊 · 𝐷) ∈ ℕ) → (if(𝑗 = (𝑀 + 1), 0, (𝐸𝑗)) + (𝑊 · 𝐷)) ∈ ℕ)
6864, 66, 67syl2anc 575 . . . . . . 7 (((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) → (if(𝑗 = (𝑀 + 1), 0, (𝐸𝑗)) + (𝑊 · 𝐷)) ∈ ℕ)
6968, 46fmptd 6613 . . . . . 6 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → 𝑃:(1...(𝑀 + 1))⟶ℕ)
70 nnex 11318 . . . . . . 7 ℕ ∈ V
71 ovex 6913 . . . . . . 7 (1...(𝑀 + 1)) ∈ V
7270, 71elmap 8128 . . . . . 6 (𝑃 ∈ (ℕ ↑𝑚 (1...(𝑀 + 1))) ↔ 𝑃:(1...(𝑀 + 1))⟶ℕ)
7369, 72sylibr 225 . . . . 5 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → 𝑃 ∈ (ℕ ↑𝑚 (1...(𝑀 + 1))))
74 elfzp1 12621 . . . . . . . . . 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 468 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑖 ∈ (1...𝑀)) → 𝐵 ∈ ℕ)
7776nncnd 11328 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑖 ∈ (1...𝑀)) → 𝐵 ∈ ℂ)
7877adantr 468 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐵 ∈ ℂ)
7920ffvelrnda 6588 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐸𝑖) ∈ ℕ)
8079nncnd 11328 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐸𝑖) ∈ ℂ)
8180adantr 468 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐸𝑖) ∈ ℂ)
8278, 81addcld 10351 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐵 + (𝐸𝑖)) ∈ ℂ)
83 nnm1nn0 11607 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 ∈ ℕ → (𝐴 − 1) ∈ ℕ0)
8441, 83syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝐴 − 1) ∈ ℕ0)
85 nn0nnaddcl 11597 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐴 − 1) ∈ ℕ0𝑉 ∈ ℕ) → ((𝐴 − 1) + 𝑉) ∈ ℕ)
8684, 38, 85syl2anc 575 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝐴 − 1) + 𝑉) ∈ ℕ)
8712, 86nnmulcld 11361 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℕ)
8887nncnd 11328 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℂ)
8988ad2antrr 708 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℂ)
90 elfznn0 12663 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 ∈ (0...(𝐾 − 1)) → 𝑚 ∈ ℕ0)
9190adantl 469 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℕ0)
9291nn0cnd 11626 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℂ)
9392adantlr 697 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℂ)
9493, 81mulcld 10352 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · (𝐸𝑖)) ∈ ℂ)
9565nnnn0d 11624 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑊 · 𝐷) ∈ ℕ0)
9695adantr 468 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · 𝐷) ∈ ℕ0)
9791, 96nn0mulcld 11629 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · (𝑊 · 𝐷)) ∈ ℕ0)
9897nn0cnd 11626 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · (𝑊 · 𝐷)) ∈ ℂ)
9998adantlr 697 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · (𝑊 · 𝐷)) ∈ ℂ)
10082, 89, 94, 99add4d 10556 . . . . . . . . . . . . . . . . . . 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 11328 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑊 · 𝐷) ∈ ℂ)
102101ad2antrr 708 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · 𝐷) ∈ ℂ)
10393, 81, 102adddid 10356 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · ((𝐸𝑖) + (𝑊 · 𝐷))) = ((𝑚 · (𝐸𝑖)) + (𝑚 · (𝑊 · 𝐷))))
104103oveq2d 6897 . . . . . . . . . . . . . . . . . . 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 11328 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑊 ∈ ℂ)
106105adantr 468 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝑊 ∈ ℂ)
10786nncnd 11328 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ((𝐴 − 1) + 𝑉) ∈ ℂ)
108107adantr 468 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → ((𝐴 − 1) + 𝑉) ∈ ℂ)
10942nncnd 11328 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝐷 ∈ ℂ)
110109adantr 468 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝐷 ∈ ℂ)
11192, 110mulcld 10352 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · 𝐷) ∈ ℂ)
112106, 108, 111adddid 10356 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · (((𝐴 − 1) + 𝑉) + (𝑚 · 𝐷))) = ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑊 · (𝑚 · 𝐷))))
11341nncnd 11328 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝐴 ∈ ℂ)
114113adantr 468 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝐴 ∈ ℂ)
115 1cnd 10327 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 1 ∈ ℂ)
116114, 111, 115addsubd 10705 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → ((𝐴 + (𝑚 · 𝐷)) − 1) = ((𝐴 − 1) + (𝑚 · 𝐷)))
117116oveq1d 6896 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉) = (((𝐴 − 1) + (𝑚 · 𝐷)) + 𝑉))
11884nn0cnd 11626 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝐴 − 1) ∈ ℂ)
119118adantr 468 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐴 − 1) ∈ ℂ)
12038nncnd 11328 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝑉 ∈ ℂ)
121120adantr 468 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝑉 ∈ ℂ)
122119, 111, 121add32d 10555 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (((𝐴 − 1) + (𝑚 · 𝐷)) + 𝑉) = (((𝐴 − 1) + 𝑉) + (𝑚 · 𝐷)))
123117, 122eqtrd 2851 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉) = (((𝐴 − 1) + 𝑉) + (𝑚 · 𝐷)))
124123oveq2d 6897 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)) = (𝑊 · (((𝐴 − 1) + 𝑉) + (𝑚 · 𝐷))))
12592, 106, 110mul12d 10537 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · (𝑊 · 𝐷)) = (𝑊 · (𝑚 · 𝐷)))
126125oveq2d 6897 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑚 · (𝑊 · 𝐷))) = ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑊 · (𝑚 · 𝐷))))
127112, 124, 1263eqtr4d 2861 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)) = ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑚 · (𝑊 · 𝐷))))
128127adantlr 697 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)) = ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑚 · (𝑊 · 𝐷))))
129128oveq2d 6897 . . . . . . . . . . . . . . . . . . 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 2861 . . . . . . . . . . . . . . . . . 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 708 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑉 ∈ ℕ)
13212ad2antrr 708 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑊 ∈ ℕ)
13343adantr 468 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐴(AP‘𝐾)𝐷) ⊆ (𝐹 “ {𝐺}))
134 eqid 2817 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑚 · 𝐷))
135 oveq1 6888 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 = 𝑚 → (𝑛 · 𝐷) = (𝑚 · 𝐷))
136135oveq2d 6897 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = 𝑚 → (𝐴 + (𝑛 · 𝐷)) = (𝐴 + (𝑚 · 𝐷)))
137136rspceeqv 3531 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑚 ∈ (0...(𝐾 − 1)) ∧ (𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑚 · 𝐷))) → ∃𝑛 ∈ (0...(𝐾 − 1))(𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷)))
138134, 137mpan2 674 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 ∈ (0...(𝐾 − 1)) → ∃𝑛 ∈ (0...(𝐾 − 1))(𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷)))
13910nnnn0d 11624 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝐾 ∈ ℕ0)
140 vdwapval 15901 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐾 ∈ ℕ0𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → ((𝐴 + (𝑚 · 𝐷)) ∈ (𝐴(AP‘𝐾)𝐷) ↔ ∃𝑛 ∈ (0...(𝐾 − 1))(𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷))))
141139, 41, 42, 140syl3anc 1483 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → ((𝐴 + (𝑚 · 𝐷)) ∈ (𝐴(AP‘𝐾)𝐷) ↔ ∃𝑛 ∈ (0...(𝐾 − 1))(𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷))))
142141biimpar 465 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ∃𝑛 ∈ (0...(𝐾 − 1))(𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷))) → (𝐴 + (𝑚 · 𝐷)) ∈ (𝐴(AP‘𝐾)𝐷))
143138, 142sylan2 582 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐴 + (𝑚 · 𝐷)) ∈ (𝐴(AP‘𝐾)𝐷))
144133, 143sseldd 3810 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐴 + (𝑚 · 𝐷)) ∈ (𝐹 “ {𝐺}))
14538, 12, 6, 39, 40vdwlem4 15912 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝐹:(1...𝑉)⟶(𝑅𝑚 (1...𝑊)))
146145ffnd 6264 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐹 Fn (1...𝑉))
147 fniniseg 6567 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐹 Fn (1...𝑉) → ((𝐴 + (𝑚 · 𝐷)) ∈ (𝐹 “ {𝐺}) ↔ ((𝐴 + (𝑚 · 𝐷)) ∈ (1...𝑉) ∧ (𝐹‘(𝐴 + (𝑚 · 𝐷))) = 𝐺)))
148146, 147syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝐴 + (𝑚 · 𝐷)) ∈ (𝐹 “ {𝐺}) ↔ ((𝐴 + (𝑚 · 𝐷)) ∈ (1...𝑉) ∧ (𝐹‘(𝐴 + (𝑚 · 𝐷))) = 𝐺)))
149148biimpa 464 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝐴 + (𝑚 · 𝐷)) ∈ (𝐹 “ {𝐺})) → ((𝐴 + (𝑚 · 𝐷)) ∈ (1...𝑉) ∧ (𝐹‘(𝐴 + (𝑚 · 𝐷))) = 𝐺))
150144, 149syldan 581 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → ((𝐴 + (𝑚 · 𝐷)) ∈ (1...𝑉) ∧ (𝐹‘(𝐴 + (𝑚 · 𝐷))) = 𝐺))
151150simpld 484 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐴 + (𝑚 · 𝐷)) ∈ (1...𝑉))
152151adantlr 697 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐴 + (𝑚 · 𝐷)) ∈ (1...𝑉))
15322r19.21bi 3131 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝐵 + (𝐸𝑖))(AP‘𝐾)(𝐸𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}))
154153adantr 468 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐵 + (𝐸𝑖))(AP‘𝐾)(𝐸𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}))
155 eqid 2817 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) = ((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖)))
156 oveq1 6888 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = 𝑚 → (𝑛 · (𝐸𝑖)) = (𝑚 · (𝐸𝑖)))
157156oveq2d 6897 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝑚 → ((𝐵 + (𝐸𝑖)) + (𝑛 · (𝐸𝑖))) = ((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))))
158157rspceeqv 3531 . . . . . . . . . . . . . . . . . . . . . . . 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))))
159155, 158mpan2 674 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ (0...(𝐾 − 1)) → ∃𝑛 ∈ (0...(𝐾 − 1))((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) = ((𝐵 + (𝐸𝑖)) + (𝑛 · (𝐸𝑖))))
16010adantr 468 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑖 ∈ (1...𝑀)) → 𝐾 ∈ ℕ)
161160nnnn0d 11624 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑖 ∈ (1...𝑀)) → 𝐾 ∈ ℕ0)
16276, 79nnaddcld 11360 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐵 + (𝐸𝑖)) ∈ ℕ)
163 vdwapval 15901 . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
164161, 162, 79, 163syl3anc 1483 . . . . . . . . . . . . . . . . . . . . . . . 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)))))
165164biimpar 465 . . . . . . . . . . . . . . . . . . . . . . 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)))
166159, 165sylan2 582 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) ∈ ((𝐵 + (𝐸𝑖))(AP‘𝐾)(𝐸𝑖)))
167154, 166sseldd 3810 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) ∈ (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}))
16814ffnd 6264 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐺 Fn (1...𝑊))
169168adantr 468 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑖 ∈ (1...𝑀)) → 𝐺 Fn (1...𝑊))
170 fniniseg 6567 . . . . . . . . . . . . . . . . . . . . . . 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))))))
171169, 170syl 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))))))
172171biimpa 464 . . . . . . . . . . . . . . . . . . . . 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)))))
173167, 172syldan 581 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) ∈ (1...𝑊) ∧ (𝐺‘((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖)))) = (𝐺‘(𝐵 + (𝐸𝑖)))))
174173simpld 484 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) ∈ (1...𝑊))
175131, 132, 152, 174vdwlem3 15911 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))) ∈ (1...(𝑊 · (2 · 𝑉))))
176130, 175eqeltrd 2896 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸𝑖) + (𝑊 · 𝐷)))) ∈ (1...(𝑊 · (2 · 𝑉))))
177 fvoveq1 6904 . . . . . . . . . . . . . . . . . . . 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)))))
178 eqid 2817 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))
179 fvex 6428 . . . . . . . . . . . . . . . . . . . 20 (𝐻‘(((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))) ∈ V
180177, 178, 179fvmpt 6510 . . . . . . . . . . . . . . . . . . 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)))))
181174, 180syl 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)))))
182173simprd 485 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐺‘((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖)))) = (𝐺‘(𝐵 + (𝐸𝑖))))
183150simprd 485 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐹‘(𝐴 + (𝑚 · 𝐷))) = 𝐺)
184 oveq1 6888 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 = (𝐴 + (𝑚 · 𝐷)) → (𝑥 − 1) = ((𝐴 + (𝑚 · 𝐷)) − 1))
185184oveq1d 6896 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 = (𝐴 + (𝑚 · 𝐷)) → ((𝑥 − 1) + 𝑉) = (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))
186185oveq2d 6897 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 = (𝐴 + (𝑚 · 𝐷)) → (𝑊 · ((𝑥 − 1) + 𝑉)) = (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))
187186oveq2d 6897 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = (𝐴 + (𝑚 · 𝐷)) → (𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))) = (𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))
188187fveq2d 6419 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = (𝐴 + (𝑚 · 𝐷)) → (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉)))) = (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))
189188mpteq2dv 4950 . . . . . . . . . . . . . . . . . . . . . . . 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))))))
190 ovex 6913 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1...𝑊) ∈ V
191190mptex 6718 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))) ∈ V
192189, 40, 191fvmpt 6510 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 + (𝑚 · 𝐷)) ∈ (1...𝑉) → (𝐹‘(𝐴 + (𝑚 · 𝐷))) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))))
193151, 192syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐹‘(𝐴 + (𝑚 · 𝐷))) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))))
194183, 193eqtr3d 2853 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝐺 = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))))
195194adantlr 697 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐺 = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))))
196195fveq1d 6417 . . . . . . . . . . . . . . . . . . 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)))))
197182, 196eqtr3d 2853 . . . . . . . . . . . . . . . . . 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)))))
198130fveq2d 6419 . . . . . . . . . . . . . . . . . 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)))))
199181, 197, 1983eqtr4rd 2862 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐻‘(((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸𝑖) + (𝑊 · 𝐷))))) = (𝐺‘(𝐵 + (𝐸𝑖))))
200176, 199jca 503 . . . . . . . . . . . . . . . 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)))))
201 eleq1 2884 . . . . . . . . . . . . . . . . 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)))))
202 fveqeq2 6424 . . . . . . . . . . . . . . . . 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)))))
203201, 202anbi12d 618 . . . . . . . . . . . . . . . 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))))))
204200, 203syl5ibrcom 238 . . . . . . . . . . . . . . 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))))))
205204rexlimdva 3230 . . . . . . . . . . . . . 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))))))
20687adantr 468 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℕ)
207162, 206nnaddcld 11360 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ ℕ)
20865adantr 468 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑊 · 𝐷) ∈ ℕ)
20979, 208nnaddcld 11360 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝐸𝑖) + (𝑊 · 𝐷)) ∈ ℕ)
210 vdwapval 15901 . . . . . . . . . . . . . . 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))))))
211161, 207, 209, 210syl3anc 1483 . . . . . . . . . . . . . 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))))))
21239ffnd 6264 . . . . . . . . . . . . . . . 16 (𝜑𝐻 Fn (1...(𝑊 · (2 · 𝑉))))
213212adantr 468 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → 𝐻 Fn (1...(𝑊 · (2 · 𝑉))))
214 fniniseg 6567 . . . . . . . . . . . . . . 15 (𝐻 Fn (1...(𝑊 · (2 · 𝑉))) → (𝑥 ∈ (𝐻 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}) ↔ (𝑥 ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻𝑥) = (𝐺‘(𝐵 + (𝐸𝑖))))))
215213, 214syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑥 ∈ (𝐻 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}) ↔ (𝑥 ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻𝑥) = (𝐺‘(𝐵 + (𝐸𝑖))))))
216205, 211, 2153imtr4d 285 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑥 ∈ (((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)((𝐸𝑖) + (𝑊 · 𝐷))) → 𝑥 ∈ (𝐻 “ {(𝐺‘(𝐵 + (𝐸𝑖)))})))
217216ssrdv 3815 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → (((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)((𝐸𝑖) + (𝑊 · 𝐷))) ⊆ (𝐻 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}))
218 ssun1 3986 . . . . . . . . . . . . . . . . . . 19 (1...𝑀) ⊆ ((1...𝑀) ∪ {(𝑀 + 1)})
219 fzsuc 12618 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ (ℤ‘1) → (1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)}))
22052, 219syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)}))
221218, 220syl5sseqr 3862 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1...𝑀) ⊆ (1...(𝑀 + 1)))
222221sselda 3809 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑖 ∈ (1...(𝑀 + 1)))
223 eqeq1 2821 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑖 → (𝑗 = (𝑀 + 1) ↔ 𝑖 = (𝑀 + 1)))
224 fveq2 6415 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑖 → (𝐸𝑗) = (𝐸𝑖))
225223, 224ifbieq2d 4315 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑖 → if(𝑗 = (𝑀 + 1), 0, (𝐸𝑗)) = if(𝑖 = (𝑀 + 1), 0, (𝐸𝑖)))
226225oveq1d 6896 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑖 → (if(𝑗 = (𝑀 + 1), 0, (𝐸𝑗)) + (𝑊 · 𝐷)) = (if(𝑖 = (𝑀 + 1), 0, (𝐸𝑖)) + (𝑊 · 𝐷)))
227 ovex 6913 . . . . . . . . . . . . . . . . . 18 (if(𝑖 = (𝑀 + 1), 0, (𝐸𝑖)) + (𝑊 · 𝐷)) ∈ V
228226, 46, 227fvmpt 6510 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (1...(𝑀 + 1)) → (𝑃𝑖) = (if(𝑖 = (𝑀 + 1), 0, (𝐸𝑖)) + (𝑊 · 𝐷)))
229222, 228syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑃𝑖) = (if(𝑖 = (𝑀 + 1), 0, (𝐸𝑖)) + (𝑊 · 𝐷)))
230 elfzle2 12575 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (1...𝑀) → 𝑖𝑀)
23118nnred 11327 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑀 ∈ ℝ)
232231ltp1d 11246 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑀 < (𝑀 + 1))
233 peano2re 10501 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑀 ∈ ℝ → (𝑀 + 1) ∈ ℝ)
234231, 233syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑀 + 1) ∈ ℝ)
235231, 234ltnled 10476 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑀 < (𝑀 + 1) ↔ ¬ (𝑀 + 1) ≤ 𝑀))
236232, 235mpbid 223 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ¬ (𝑀 + 1) ≤ 𝑀)
237 breq1 4858 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 = (𝑀 + 1) → (𝑖𝑀 ↔ (𝑀 + 1) ≤ 𝑀))
238237notbid 309 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = (𝑀 + 1) → (¬ 𝑖𝑀 ↔ ¬ (𝑀 + 1) ≤ 𝑀))
239236, 238syl5ibrcom 238 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑖 = (𝑀 + 1) → ¬ 𝑖𝑀))
240239con2d 131 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑖𝑀 → ¬ 𝑖 = (𝑀 + 1)))
241230, 240syl5 34 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑖 ∈ (1...𝑀) → ¬ 𝑖 = (𝑀 + 1)))
242241imp 395 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (1...𝑀)) → ¬ 𝑖 = (𝑀 + 1))
243 iffalse 4299 . . . . . . . . . . . . . . . . . 18 𝑖 = (𝑀 + 1) → if(𝑖 = (𝑀 + 1), 0, (𝐸𝑖)) = (𝐸𝑖))
244243oveq1d 6896 . . . . . . . . . . . . . . . . 17 𝑖 = (𝑀 + 1) → (if(𝑖 = (𝑀 + 1), 0, (𝐸𝑖)) + (𝑊 · 𝐷)) = ((𝐸𝑖) + (𝑊 · 𝐷)))
245242, 244syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑀)) → (if(𝑖 = (𝑀 + 1), 0, (𝐸𝑖)) + (𝑊 · 𝐷)) = ((𝐸𝑖) + (𝑊 · 𝐷)))
246229, 245eqtrd 2851 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑃𝑖) = ((𝐸𝑖) + (𝑊 · 𝐷)))
247246oveq2d 6897 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑇 + (𝑃𝑖)) = (𝑇 + ((𝐸𝑖) + (𝑊 · 𝐷))))
24847nncnd 11328 . . . . . . . . . . . . . . . 16 (𝜑𝑇 ∈ ℂ)
249248adantr 468 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑇 ∈ ℂ)
250101adantr 468 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑊 · 𝐷) ∈ ℂ)
251249, 80, 250add12d 10554 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑇 + ((𝐸𝑖) + (𝑊 · 𝐷))) = ((𝐸𝑖) + (𝑇 + (𝑊 · 𝐷))))
25245oveq1i 6891 . . . . . . . . . . . . . . . . . 18 (𝑇 + (𝑊 · 𝐷)) = ((𝐵 + (𝑊 · ((𝐴 + (𝑉𝐷)) − 1))) + (𝑊 · 𝐷))
25316nncnd 11328 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐵 ∈ ℂ)
254120, 109subcld 10684 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑉𝐷) ∈ ℂ)
255113, 254addcld 10351 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝐴 + (𝑉𝐷)) ∈ ℂ)
256 ax-1cn 10286 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ ℂ
257 subcl 10572 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴 + (𝑉𝐷)) ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 + (𝑉𝐷)) − 1) ∈ ℂ)
258255, 256, 257sylancl 576 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝐴 + (𝑉𝐷)) − 1) ∈ ℂ)
259105, 258mulcld 10352 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑊 · ((𝐴 + (𝑉𝐷)) − 1)) ∈ ℂ)
260253, 259, 101addassd 10354 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝐵 + (𝑊 · ((𝐴 + (𝑉𝐷)) − 1))) + (𝑊 · 𝐷)) = (𝐵 + ((𝑊 · ((𝐴 + (𝑉𝐷)) − 1)) + (𝑊 · 𝐷))))
261105, 258, 109adddid 10356 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑊 · (((𝐴 + (𝑉𝐷)) − 1) + 𝐷)) = ((𝑊 · ((𝐴 + (𝑉𝐷)) − 1)) + (𝑊 · 𝐷)))
262113, 254, 109addassd 10354 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → ((𝐴 + (𝑉𝐷)) + 𝐷) = (𝐴 + ((𝑉𝐷) + 𝐷)))
263120, 109npcand 10688 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → ((𝑉𝐷) + 𝐷) = 𝑉)
264263oveq2d 6897 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝐴 + ((𝑉𝐷) + 𝐷)) = (𝐴 + 𝑉))
265262, 264eqtrd 2851 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ((𝐴 + (𝑉𝐷)) + 𝐷) = (𝐴 + 𝑉))
266265oveq1d 6896 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (((𝐴 + (𝑉𝐷)) + 𝐷) − 1) = ((𝐴 + 𝑉) − 1))
267 1cnd 10327 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → 1 ∈ ℂ)
268255, 109, 267addsubd 10705 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (((𝐴 + (𝑉𝐷)) + 𝐷) − 1) = (((𝐴 + (𝑉𝐷)) − 1) + 𝐷))
269113, 120, 267addsubd 10705 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝐴 + 𝑉) − 1) = ((𝐴 − 1) + 𝑉))
270266, 268, 2693eqtr3d 2859 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (((𝐴 + (𝑉𝐷)) − 1) + 𝐷) = ((𝐴 − 1) + 𝑉))
271270oveq2d 6897 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑊 · (((𝐴 + (𝑉𝐷)) − 1) + 𝐷)) = (𝑊 · ((𝐴 − 1) + 𝑉)))
272261, 271eqtr3d 2853 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑊 · ((𝐴 + (𝑉𝐷)) − 1)) + (𝑊 · 𝐷)) = (𝑊 · ((𝐴 − 1) + 𝑉)))
273272oveq2d 6897 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐵 + ((𝑊 · ((𝐴 + (𝑉𝐷)) − 1)) + (𝑊 · 𝐷))) = (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))))
274260, 273eqtrd 2851 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝐵 + (𝑊 · ((𝐴 + (𝑉𝐷)) − 1))) + (𝑊 · 𝐷)) = (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))))
275252, 274syl5eq 2863 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑇 + (𝑊 · 𝐷)) = (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))))
276275oveq2d 6897 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝐸𝑖) + (𝑇 + (𝑊 · 𝐷))) = ((𝐸𝑖) + (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
277276adantr 468 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝐸𝑖) + (𝑇 + (𝑊 · 𝐷))) = ((𝐸𝑖) + (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
27888adantr 468 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℂ)
27980, 77, 278addassd 10354 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → (((𝐸𝑖) + 𝐵) + (𝑊 · ((𝐴 − 1) + 𝑉))) = ((𝐸𝑖) + (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
28080, 77addcomd 10530 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝐸𝑖) + 𝐵) = (𝐵 + (𝐸𝑖)))
281280oveq1d 6896 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → (((𝐸𝑖) + 𝐵) + (𝑊 · ((𝐴 − 1) + 𝑉))) = ((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))))
282277, 279, 2813eqtr2d 2857 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝐸𝑖) + (𝑇 + (𝑊 · 𝐷))) = ((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))))
283247, 251, 2823eqtrd 2855 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑇 + (𝑃𝑖)) = ((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))))
284283, 246oveq12d 6899 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) = (((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)((𝐸𝑖) + (𝑊 · 𝐷))))
285 cnvimass 5706 . . . . . . . . . . . . . . . . . . 19 (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}) ⊆ dom 𝐺
286285, 14fssdm 6279 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}) ⊆ (1...𝑊))
287286adantr 468 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}) ⊆ (1...𝑊))
288 vdwapid1 15903 . . . . . . . . . . . . . . . . . . 19 ((𝐾 ∈ ℕ ∧ (𝐵 + (𝐸𝑖)) ∈ ℕ ∧ (𝐸𝑖) ∈ ℕ) → (𝐵 + (𝐸𝑖)) ∈ ((𝐵 + (𝐸𝑖))(AP‘𝐾)(𝐸𝑖)))
289160, 162, 79, 288syl3anc 1483 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐵 + (𝐸𝑖)) ∈ ((𝐵 + (𝐸𝑖))(AP‘𝐾)(𝐸𝑖)))
290153, 289sseldd 3810 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐵 + (𝐸𝑖)) ∈ (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}))
291287, 290sseldd 3810 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐵 + (𝐸𝑖)) ∈ (1...𝑊))
292 fvoveq1 6904 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝐵 + (𝐸𝑖)) → (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))) = (𝐻‘((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))))
293 eqid 2817 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))))) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
294 fvex 6428 . . . . . . . . . . . . . . . . 17 (𝐻‘((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))) ∈ V
295292, 293, 294fvmpt 6510 . . . . . . . . . . . . . . . 16 ((𝐵 + (𝐸𝑖)) ∈ (1...𝑊) → ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))‘(𝐵 + (𝐸𝑖))) = (𝐻‘((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))))
296291, 295syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))‘(𝐵 + (𝐸𝑖))) = (𝐻‘((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))))
297 vdwapid1 15903 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐾 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → 𝐴 ∈ (𝐴(AP‘𝐾)𝐷))
29810, 41, 42, 297syl3anc 1483 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐴 ∈ (𝐴(AP‘𝐾)𝐷))
29943, 298sseldd 3810 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐴 ∈ (𝐹 “ {𝐺}))
300 fniniseg 6567 . . . . . . . . . . . . . . . . . . . . 21 (𝐹 Fn (1...𝑉) → (𝐴 ∈ (𝐹 “ {𝐺}) ↔ (𝐴 ∈ (1...𝑉) ∧ (𝐹𝐴) = 𝐺)))
301146, 300syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝐴 ∈ (𝐹 “ {𝐺}) ↔ (𝐴 ∈ (1...𝑉) ∧ (𝐹𝐴) = 𝐺)))
302299, 301mpbid 223 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐴 ∈ (1...𝑉) ∧ (𝐹𝐴) = 𝐺))
303302simprd 485 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐹𝐴) = 𝐺)
304302simpld 484 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐴 ∈ (1...𝑉))
305 oveq1 6888 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = 𝐴 → (𝑥 − 1) = (𝐴 − 1))
306305oveq1d 6896 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝐴 → ((𝑥 − 1) + 𝑉) = ((𝐴 − 1) + 𝑉))
307306oveq2d 6897 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝐴 → (𝑊 · ((𝑥 − 1) + 𝑉)) = (𝑊 · ((𝐴 − 1) + 𝑉)))
308307oveq2d 6897 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝐴 → (𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))) = (𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))))
309308fveq2d 6419 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝐴 → (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉)))) = (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
310309mpteq2dv 4950 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝐴 → (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))))))
311190mptex 6718 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))))) ∈ V
312310, 40, 311fvmpt 6510 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ (1...𝑉) → (𝐹𝐴) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))))))
313304, 312syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐹𝐴) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))))))
314303, 313eqtr3d 2853 . . . . . . . . . . . . . . . . 17 (𝜑𝐺 = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))))))
315314fveq1d 6417 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐺‘(𝐵 + (𝐸𝑖))) = ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))‘(𝐵 + (𝐸𝑖))))
316315adantr 468 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐺‘(𝐵 + (𝐸𝑖))) = ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))‘(𝐵 + (𝐸𝑖))))
317283fveq2d 6419 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻‘(𝑇 + (𝑃𝑖))) = (𝐻‘((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))))
318296, 316, 3173eqtr4rd 2862 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻‘(𝑇 + (𝑃𝑖))) = (𝐺‘(𝐵 + (𝐸𝑖))))
319318sneqd 4393 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑀)) → {(𝐻‘(𝑇 + (𝑃𝑖)))} = {(𝐺‘(𝐵 + (𝐸𝑖)))})
320319imaeq2d 5687 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))}) = (𝐻 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}))
321217, 284, 3203sstr4d 3856 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))}))
322321ex 399 . . . . . . . . . 10 (𝜑 → (𝑖 ∈ (1...𝑀) → ((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))})))
323253adantr 468 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝐵 ∈ ℂ)
32488adantr 468 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℂ)
325323, 324, 98addassd 10354 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) = (𝐵 + ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑚 · (𝑊 · 𝐷)))))
326127oveq2d 6897 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))) = (𝐵 + ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑚 · (𝑊 · 𝐷)))))
327325, 326eqtr4d 2854 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) = (𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))
32838adantr 468 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝑉 ∈ ℕ)
32912adantr 468 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝑊 ∈ ℕ)
330 eluzfz1 12578 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑀 ∈ (ℤ‘1) → 1 ∈ (1...𝑀))
33152, 330syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → 1 ∈ (1...𝑀))
332331ne0d 4134 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (1...𝑀) ≠ ∅)
333 elfzuz3 12569 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐵 + (𝐸𝑖)) ∈ (1...𝑊) → 𝑊 ∈ (ℤ‘(𝐵 + (𝐸𝑖))))
334291, 333syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑊 ∈ (ℤ‘(𝐵 + (𝐸𝑖))))
33516nnzd 11754 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑𝐵 ∈ ℤ)
336 uzid 11926 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐵 ∈ ℤ → 𝐵 ∈ (ℤ𝐵))
337335, 336syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝐵 ∈ (ℤ𝐵))
338337adantr 468 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑖 ∈ (1...𝑀)) → 𝐵 ∈ (ℤ𝐵))
33979nnnn0d 11624 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐸𝑖) ∈ ℕ0)
340 uzaddcl 11969 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐵 ∈ (ℤ𝐵) ∧ (𝐸𝑖) ∈ ℕ0) → (𝐵 + (𝐸𝑖)) ∈ (ℤ𝐵))
341338, 339, 340syl2anc 575 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐵 + (𝐸𝑖)) ∈ (ℤ𝐵))
342 uztrn 11928 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑊 ∈ (ℤ‘(𝐵 + (𝐸𝑖))) ∧ (𝐵 + (𝐸𝑖)) ∈ (ℤ𝐵)) → 𝑊 ∈ (ℤ𝐵))
343334, 341, 342syl2anc 575 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑊 ∈ (ℤ𝐵))
344 eluzle 11924 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑊 ∈ (ℤ𝐵) → 𝐵𝑊)
345343, 344syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑖 ∈ (1...𝑀)) → 𝐵𝑊)
346345ralrimiva 3165 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ∀𝑖 ∈ (1...𝑀)𝐵𝑊)
347 r19.2z 4266 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...𝑀) ≠ ∅ ∧ ∀𝑖 ∈ (1...𝑀)𝐵𝑊) → ∃𝑖 ∈ (1...𝑀)𝐵𝑊)
348332, 346, 347syl2anc 575 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ∃𝑖 ∈ (1...𝑀)𝐵𝑊)
349 idd 24 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (1...𝑀) → (𝐵𝑊𝐵𝑊))
350349rexlimiv 3226 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑖 ∈ (1...𝑀)𝐵𝑊𝐵𝑊)
351348, 350syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐵𝑊)
35212nnzd 11754 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑊 ∈ ℤ)
353 fznn 12638 . . . . . . . . . . . . . . . . . . . . . 22 (𝑊 ∈ ℤ → (𝐵 ∈ (1...𝑊) ↔ (𝐵 ∈ ℕ ∧ 𝐵𝑊)))
354352, 353syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐵 ∈ (1...𝑊) ↔ (𝐵 ∈ ℕ ∧ 𝐵𝑊)))
35516, 351, 354mpbir2and 695 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐵 ∈ (1...𝑊))
356355adantr 468 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝐵 ∈ (1...𝑊))
357328, 329, 151, 356vdwlem3 15911 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))) ∈ (1...(𝑊 · (2 · 𝑉))))
358327, 357eqeltrd 2896 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) ∈ (1...(𝑊 · (2 · 𝑉))))
359 fvoveq1 6904 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝐵 → (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))) = (𝐻‘(𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))
360 fvex 6428 . . . . . . . . . . . . . . . . . . . 20 (𝐻‘(𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))) ∈ V
361359, 178, 360fvmpt 6510 . . . . . . . . . . . . . . . . . . 19 (𝐵 ∈ (1...𝑊) → ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))‘𝐵) = (𝐻‘(𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))
362356, 361syl 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)))))
363194fveq1d 6417 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐺𝐵) = ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))‘𝐵))
364327fveq2d 6419 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐻‘((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷)))) = (𝐻‘(𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))
365362, 363, 3643eqtr4rd 2862 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐻‘((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷)))) = (𝐺𝐵))
366358, 365jca 503 . . . . . . . . . . . . . . . 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)))
367 eleq1 2884 . . . . . . . . . . . . . . . . 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)))))
368 fveqeq2 6424 . . . . . . . . . . . . . . . . 17 (𝑧 = ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) → ((𝐻𝑧) = (𝐺𝐵) ↔ (𝐻‘((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷)))) = (𝐺𝐵)))
369367, 368anbi12d 618 . . . . . . . . . . . . . . . 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))))
370366, 369syl5ibrcom 238 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝑧 = ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) → (𝑧 ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻𝑧) = (𝐺𝐵))))
371370rexlimdva 3230 . . . . . . . . . . . . . 14 (𝜑 → (∃𝑚 ∈ (0...(𝐾 − 1))𝑧 = ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) → (𝑧 ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻𝑧) = (𝐺𝐵))))
37216, 87nnaddcld 11360 . . . . . . . . . . . . . . 15 (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ ℕ)
373 vdwapval 15901 . . . . . . . . . . . . . . 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)))))
374139, 372, 65, 373syl3anc 1483 . . . . . . . . . . . . . 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)))))
375 fniniseg 6567 . . . . . . . . . . . . . . 15 (𝐻 Fn (1...(𝑊 · (2 · 𝑉))) → (𝑧 ∈ (𝐻 “ {(𝐺𝐵)}) ↔ (𝑧 ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻𝑧) = (𝐺𝐵))))
376212, 375syl 17 . . . . . . . . . . . . . 14 (𝜑 → (𝑧 ∈ (𝐻 “ {(𝐺𝐵)}) ↔ (𝑧 ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻𝑧) = (𝐺𝐵))))
377371, 374, 3763imtr4d 285 . . . . . . . . . . . . 13 (𝜑 → (𝑧 ∈ ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)(𝑊 · 𝐷)) → 𝑧 ∈ (𝐻 “ {(𝐺𝐵)})))
378377ssrdv 3815 . . . . . . . . . . . 12 (𝜑 → ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)(𝑊 · 𝐷)) ⊆ (𝐻 “ {(𝐺𝐵)}))
37918peano2nnd 11329 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑀 + 1) ∈ ℕ)
380379, 51syl6eleq 2906 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑀 + 1) ∈ (ℤ‘1))
381 eluzfz2 12579 . . . . . . . . . . . . . . . . . 18 ((𝑀 + 1) ∈ (ℤ‘1) → (𝑀 + 1) ∈ (1...(𝑀 + 1)))
382380, 381syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑀 + 1) ∈ (1...(𝑀 + 1)))
383 iftrue 4296 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑀 + 1) → if(𝑗 = (𝑀 + 1), 0, (𝐸𝑗)) = 0)
384383oveq1d 6896 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑀 + 1) → (if(𝑗 = (𝑀 + 1), 0, (𝐸𝑗)) + (𝑊 · 𝐷)) = (0 + (𝑊 · 𝐷)))
385 ovex 6913 . . . . . . . . . . . . . . . . . 18 (0 + (𝑊 · 𝐷)) ∈ V
386384, 46, 385fvmpt 6510 . . . . . . . . . . . . . . . . 17 ((𝑀 + 1) ∈ (1...(𝑀 + 1)) → (𝑃‘(𝑀 + 1)) = (0 + (𝑊 · 𝐷)))
387382, 386syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑃‘(𝑀 + 1)) = (0 + (𝑊 · 𝐷)))
388101addid2d 10529 . . . . . . . . . . . . . . . 16 (𝜑 → (0 + (𝑊 · 𝐷)) = (𝑊 · 𝐷))
389387, 388eqtrd 2851 . . . . . . . . . . . . . . 15 (𝜑 → (𝑃‘(𝑀 + 1)) = (𝑊 · 𝐷))
390389oveq2d 6897 . . . . . . . . . . . . . 14 (𝜑 → (𝑇 + (𝑃‘(𝑀 + 1))) = (𝑇 + (𝑊 · 𝐷)))
391390, 275eqtrd 2851 . . . . . . . . . . . . 13 (𝜑 → (𝑇 + (𝑃‘(𝑀 + 1))) = (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))))
392391, 389oveq12d 6899 . . . . . . . . . . . 12 (𝜑 → ((𝑇 + (𝑃‘(𝑀 + 1)))(AP‘𝐾)(𝑃‘(𝑀 + 1))) = ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)(𝑊 · 𝐷)))
393 fvoveq1 6904 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐵 → (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))) = (𝐻‘(𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
394 fvex 6428 . . . . . . . . . . . . . . . . 17 (𝐻‘(𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))) ∈ V
395393, 293, 394fvmpt 6510 . . . . . . . . . . . . . . . 16 (𝐵 ∈ (1...𝑊) → ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))‘𝐵) = (𝐻‘(𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
396355, 395syl 17 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))‘𝐵) = (𝐻‘(𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
397314fveq1d 6417 . . . . . . . . . . . . . . 15 (𝜑 → (𝐺𝐵) = ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))‘𝐵))
398391fveq2d 6419 . . . . . . . . . . . . . . 15 (𝜑 → (𝐻‘(𝑇 + (𝑃‘(𝑀 + 1)))) = (𝐻‘(𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
399396, 397, 3983eqtr4rd 2862 . . . . . . . . . . . . . 14 (𝜑 → (𝐻‘(𝑇 + (𝑃‘(𝑀 + 1)))) = (𝐺𝐵))
400399sneqd 4393 . . . . . . . . . . . . 13 (𝜑 → {(𝐻‘(𝑇 + (𝑃‘(𝑀 + 1))))} = {(𝐺𝐵)})
401400imaeq2d 5687 . . . . . . . . . . . 12 (𝜑 → (𝐻 “ {(𝐻‘(𝑇 + (𝑃‘(𝑀 + 1))))}) = (𝐻 “ {(𝐺𝐵)}))
402378, 392, 4013sstr4d 3856 . . . . . . . . . . 11 (𝜑 → ((𝑇 + (𝑃‘(𝑀 + 1)))(AP‘𝐾)(𝑃‘(𝑀 + 1))) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃‘(𝑀 + 1))))}))
403 fveq2 6415 . . . . . . . . . . . . . 14 (𝑖 = (𝑀 + 1) → (𝑃𝑖) = (𝑃‘(𝑀 + 1)))
404403oveq2d 6897 . . . . . . . . . . . . 13 (𝑖 = (𝑀 + 1) → (𝑇 + (𝑃𝑖)) = (𝑇 + (𝑃‘(𝑀 + 1))))
405404, 403oveq12d 6899 . . . . . . . . . . . 12 (𝑖 = (𝑀 + 1) → ((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) = ((𝑇 + (𝑃‘(𝑀 + 1)))(AP‘𝐾)(𝑃‘(𝑀 + 1))))
406404fveq2d 6419 . . . . . . . . . . . . . 14 (𝑖 = (𝑀 + 1) → (𝐻‘(𝑇 + (𝑃𝑖))) = (𝐻‘(𝑇 + (𝑃‘(𝑀 + 1)))))
407406sneqd 4393 . . . . . . . . . . . . 13 (𝑖 = (𝑀 + 1) → {(𝐻‘(𝑇 + (𝑃𝑖)))} = {(𝐻‘(𝑇 + (𝑃‘(𝑀 + 1))))})
408407imaeq2d 5687 . . . . . . . . . . . 12 (𝑖 = (𝑀 + 1) → (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))}) = (𝐻 “ {(𝐻‘(𝑇 + (𝑃‘(𝑀 + 1))))}))
409405, 408sseq12d 3842 . . . . . . . . . . 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))))})))
410402, 409syl5ibrcom 238 . . . . . . . . . 10 (𝜑 → (𝑖 = (𝑀 + 1) → ((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))})))
411322, 410jaod 877 . . . . . . . . 9 (𝜑 → ((𝑖 ∈ (1...𝑀) ∨ 𝑖 = (𝑀 + 1)) → ((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))})))
41275, 411sylbid 231 . . . . . . . 8 (𝜑 → (𝑖 ∈ (1...(𝑀 + 1)) → ((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))})))
413412ralrimiv 3164 . . . . . . 7 (𝜑 → ∀𝑖 ∈ (1...(𝑀 + 1))((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))}))
414413adantr 468 . . . . . 6 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → ∀𝑖 ∈ (1...(𝑀 + 1))((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))}))
415220rexeqdv 3345 . . . . . . . . . . . 12 (𝜑 → (∃𝑖 ∈ (1...(𝑀 + 1))𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ↔ ∃𝑖 ∈ ((1...𝑀) ∪ {(𝑀 + 1)})𝑥 = (𝐻‘(𝑇 + (𝑃𝑖)))))
416 rexun 4003 . . . . . . . . . . . . 13 (∃𝑖 ∈ ((1...𝑀) ∪ {(𝑀 + 1)})𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ↔ (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ∨ ∃𝑖 ∈ {(𝑀 + 1)}𝑥 = (𝐻‘(𝑇 + (𝑃𝑖)))))
417318eqeq2d 2827 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ↔ 𝑥 = (𝐺‘(𝐵 + (𝐸𝑖)))))
418417rexbidva 3248 . . . . . . . . . . . . . 14 (𝜑 → (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ↔ ∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸𝑖)))))
419 ovex 6913 . . . . . . . . . . . . . . . 16 (𝑀 + 1) ∈ V
420406eqeq2d 2827 . . . . . . . . . . . . . . . 16 (𝑖 = (𝑀 + 1) → (𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ↔ 𝑥 = (𝐻‘(𝑇 + (𝑃‘(𝑀 + 1))))))
421419, 420rexsn 4427 . . . . . . . . . . . . . . 15 (∃𝑖 ∈ {(𝑀 + 1)}𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ↔ 𝑥 = (𝐻‘(𝑇 + (𝑃‘(𝑀 + 1)))))
422399eqeq2d 2827 . . . . . . . . . . . . . . 15 (𝜑 → (𝑥 = (𝐻‘(𝑇 + (𝑃‘(𝑀 + 1)))) ↔ 𝑥 = (𝐺𝐵)))
423421, 422syl5bb 274 . . . . . . . . . . . . . 14 (𝜑 → (∃𝑖 ∈ {(𝑀 + 1)}𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ↔ 𝑥 = (𝐺𝐵)))
424418, 423orbi12d 933 . . . . . . . . . . . . 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))))
425416, 424syl5bb 274 . . . . . . . . . . . 12 (𝜑 → (∃𝑖 ∈ ((1...𝑀) ∪ {(𝑀 + 1)})𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ↔ (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸𝑖))) ∨ 𝑥 = (𝐺𝐵))))
426415, 425bitrd 270 . . . . . . . . . . 11 (𝜑 → (∃𝑖 ∈ (1...(𝑀 + 1))𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ↔ (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸𝑖))) ∨ 𝑥 = (𝐺𝐵))))
427426adantr 468 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → (∃𝑖 ∈ (1...(𝑀 + 1))𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ↔ (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸𝑖))) ∨ 𝑥 = (𝐺𝐵))))
428427abbidv 2936 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → {𝑥 ∣ ∃𝑖 ∈ (1...(𝑀 + 1))𝑥 = (𝐻‘(𝑇 + (𝑃𝑖)))} = {𝑥 ∣ (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸𝑖))) ∨ 𝑥 = (𝐺𝐵))})
429 eqid 2817 . . . . . . . . . 10 (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃𝑖)))) = (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃𝑖))))
430429rnmpt 5583 . . . . . . . . 9 ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃𝑖)))) = {𝑥 ∣ ∃𝑖 ∈ (1...(𝑀 + 1))𝑥 = (𝐻‘(𝑇 + (𝑃𝑖)))}
4312rnmpt 5583 . . . . . . . . . . 11 ran 𝐽 = {𝑥 ∣ ∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸𝑖)))}
432 df-sn 4382 . . . . . . . . . . 11 {(𝐺𝐵)} = {𝑥𝑥 = (𝐺𝐵)}
433431, 432uneq12i 3975 . . . . . . . . . 10 (ran 𝐽 ∪ {(𝐺𝐵)}) = ({𝑥 ∣ ∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸𝑖)))} ∪ {𝑥𝑥 = (𝐺𝐵)})
434 unab 4106 . . . . . . . . . 10 ({𝑥 ∣ ∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸𝑖)))} ∪ {𝑥𝑥 = (𝐺𝐵)}) = {𝑥 ∣ (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸𝑖))) ∨ 𝑥 = (𝐺𝐵))}
435433, 434eqtri 2839 . . . . . . . . 9 (ran 𝐽 ∪ {(𝐺𝐵)}) = {𝑥 ∣ (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸𝑖))) ∨ 𝑥 = (𝐺𝐵))}
436428, 430, 4353eqtr4g 2876 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃𝑖)))) = (ran 𝐽 ∪ {(𝐺𝐵)}))
437436fveq2d 6419 . . . . . . 7 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → (♯‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃𝑖))))) = (♯‘(ran 𝐽 ∪ {(𝐺𝐵)})))
438 fzfi 13002 . . . . . . . . . 10 (1...𝑀) ∈ Fin
439 dffn4 6344 . . . . . . . . . . 11 (𝐽 Fn (1...𝑀) ↔ 𝐽:(1...𝑀)–onto→ran 𝐽)
4403, 439mpbi 221 . . . . . . . . . 10 𝐽:(1...𝑀)–onto→ran 𝐽
441 fofi 8498 . . . . . . . . . 10 (((1...𝑀) ∈ Fin ∧ 𝐽:(1...𝑀)–onto→ran 𝐽) → ran 𝐽 ∈ Fin)
442438, 440, 441mp2an 675 . . . . . . . . 9 ran 𝐽 ∈ Fin
443442a1i 11 . . . . . . . 8 (𝜑 → ran 𝐽 ∈ Fin)
444 fvex 6428 . . . . . . . . 9 (𝐺𝐵) ∈ V
445 hashunsng 13406 . . . . . . . . 9 ((𝐺𝐵) ∈ V → ((ran 𝐽 ∈ Fin ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → (♯‘(ran 𝐽 ∪ {(𝐺𝐵)})) = ((♯‘ran 𝐽) + 1)))
446444, 445ax-mp 5 . . . . . . . 8 ((ran 𝐽 ∈ Fin ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → (♯‘(ran 𝐽 ∪ {(𝐺𝐵)})) = ((♯‘ran 𝐽) + 1))
447443, 446sylan 571 . . . . . . 7 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → (♯‘(ran 𝐽 ∪ {(𝐺𝐵)})) = ((♯‘ran 𝐽) + 1))
44844adantr 468 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → (♯‘ran 𝐽) = 𝑀)
449448oveq1d 6896 . . . . . . 7 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → ((♯‘ran 𝐽) + 1) = (𝑀 + 1))
450437, 447, 4493eqtrd 2855 . . . . . 6 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → (♯‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃𝑖))))) = (𝑀 + 1))
451414, 450jca 503 . . . . 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)))
452 oveq1 6888 . . . . . . . . . 10 (𝑎 = 𝑇 → (𝑎 + (𝑑𝑖)) = (𝑇 + (𝑑𝑖)))
453452oveq1d 6896 . . . . . . . . 9 (𝑎 = 𝑇 → ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) = ((𝑇 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)))
454 fvoveq1 6904 . . . . . . . . . . 11 (𝑎 = 𝑇 → (𝐻‘(𝑎 + (𝑑𝑖))) = (𝐻‘(𝑇 + (𝑑𝑖))))
455454sneqd 4393 . . . . . . . . . 10 (𝑎 = 𝑇 → {(𝐻‘(𝑎 + (𝑑𝑖)))} = {(𝐻‘(𝑇 + (𝑑𝑖)))})
456455imaeq2d 5687 . . . . . . . . 9 (𝑎 = 𝑇 → (𝐻 “ {(𝐻‘(𝑎 + (𝑑𝑖)))}) = (𝐻 “ {(𝐻‘(𝑇 + (𝑑𝑖)))}))
457453, 456sseq12d 3842 . . . . . . . 8 (𝑎 = 𝑇 → (((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑎 + (𝑑𝑖)))}) ↔ ((𝑇 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑑𝑖)))})))
458457ralbidv 3185 . . . . . . 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)))})))
459454mpteq2dv 4950 . . . . . . . . 9 (𝑎 = 𝑇 → (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑎 + (𝑑𝑖)))) = (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑𝑖)))))
460459rneqd 5565 . . . . . . . 8 (𝑎 = 𝑇 → ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑎 + (𝑑𝑖)))) = ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑𝑖)))))
461460fveqeq2d 6423 . . . . . . 7 (𝑎 = 𝑇 → ((♯‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑎 + (𝑑𝑖))))) = (𝑀 + 1) ↔ (♯‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑𝑖))))) = (𝑀 + 1)))
462458, 461anbi12d 618 . . . . . 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))))
463 fveq1 6414 . . . . . . . . . . 11 (𝑑 = 𝑃 → (𝑑𝑖) = (𝑃𝑖))
464463oveq2d 6897 . . . . . . . . . 10 (𝑑 = 𝑃 → (𝑇 + (𝑑𝑖)) = (𝑇 + (𝑃𝑖)))
465464, 463oveq12d 6899 . . . . . . . . 9 (𝑑 = 𝑃 → ((𝑇 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) = ((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)))
466464fveq2d 6419 . . . . . . . . . . 11 (𝑑 = 𝑃 → (𝐻‘(𝑇 + (𝑑𝑖))) = (𝐻‘(𝑇 + (𝑃𝑖))))
467466sneqd 4393 . . . . . . . . . 10 (𝑑 = 𝑃 → {(𝐻‘(𝑇 + (𝑑𝑖)))} = {(𝐻‘(𝑇 + (𝑃𝑖)))})
468467imaeq2d 5687 . . . . . . . . 9 (𝑑 = 𝑃 → (𝐻 “ {(𝐻‘(𝑇 + (𝑑𝑖)))}) = (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))}))
469465, 468sseq12d 3842 . . . . . . . 8 (𝑑 = 𝑃 → (((𝑇 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑑𝑖)))}) ↔ ((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))})))
470469ralbidv 3185 . . . . . . 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)))})))
471466mpteq2dv 4950 . . . . . . . . 9 (𝑑 = 𝑃 → (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑𝑖)))) = (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃𝑖)))))
472471rneqd 5565 . . . . . . . 8 (𝑑 = 𝑃 → ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑𝑖)))) = ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃𝑖)))))
473472fveqeq2d 6423 . . . . . . 7 (𝑑 = 𝑃 → ((♯‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑𝑖))))) = (𝑀 + 1) ↔ (♯‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃𝑖))))) = (𝑀 + 1)))
474470, 473anbi12d 618 . . . . . 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))))
475462, 474rspc2ev 3528 . . . . 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)))
47648, 73, 451, 475syl3anc 1483 . . . 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)))
477 ovex 6913 . . . . 5 (1...(𝑊 · (2 · 𝑉))) ∈ V
47810adantr 468 . . . . . 6 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → 𝐾 ∈ ℕ)
479478nnnn0d 11624 . . . . 5 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → 𝐾 ∈ ℕ0)
48039adantr 468 . . . . 5 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → 𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)
48118adantr 468 . . . . . 6 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → 𝑀 ∈ ℕ)
482481peano2nnd 11329 . . . . 5 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → (𝑀 + 1) ∈ ℕ)
483 eqid 2817 . . . . 5 (1...(𝑀 + 1)) = (1...(𝑀 + 1))
484477, 479, 480, 482, 483vdwpc 15908 . . . 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))))
485476, 484mpbird 248 . . 3 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → ⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻)
486485orcd 891 . 2 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺))
48737, 486pm2.61dan 838 1 (𝜑 → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 865   = wceq 1637  wcel 2157  {cab 2803  wne 2989  wral 3107  wrex 3108  Vcvv 3402  cun 3778  wss 3780  c0 4127  ifcif 4290  {csn 4381  cop 4387   class class class wbr 4855  cmpt 4934  ccnv 5321  ran crn 5323  cima 5325   Fn wfn 6103  wf 6104  ontowfo 6106  cfv 6108  (class class class)co 6881  𝑚 cmap 8099  Fincfn 8199  cc 10226  cr 10227  0cc0 10228  1c1 10229   + caddc 10231   · cmul 10233   < clt 10366  cle 10367  cmin 10558  cn 11312  2c2 11363  0cn0 11566  cz 11650  cuz 11911  ...cfz 12556  chash 13344  APcvdwa 15893   MonoAP cvdwm 15894   PolyAP cvdwp 15895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4975  ax-sep 4986  ax-nul 4994  ax-pow 5046  ax-pr 5107  ax-un 7186  ax-cnex 10284  ax-resscn 10285  ax-1cn 10286  ax-icn 10287  ax-addcl 10288  ax-addrcl 10289  ax-mulcl 10290  ax-mulrcl 10291  ax-mulcom 10292  ax-addass 10293  ax-mulass 10294  ax-distr 10295  ax-i2m1 10296  ax-1ne0 10297  ax-1rid 10298  ax-rnegex 10299  ax-rrecex 10300  ax-cnre 10301  ax-pre-lttri 10302  ax-pre-lttrn 10303  ax-pre-ltadd 10304  ax-pre-mulgt0 10305
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-pss 3796  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-tp 4386  df-op 4388  df-uni 4642  df-int 4681  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-tr 4958  df-id 5230  df-eprel 5235  df-po 5243  df-so 5244  df-fr 5281  df-we 5283  df-xp 5328  df-rel 5329  df-cnv 5330  df-co 5331  df-dm 5332  df-rn 5333  df-res 5334  df-ima 5335  df-pred 5904  df-ord 5950  df-on 5951  df-lim 5952  df-suc 5953  df-iota 6071  df-fun 6110  df-fn 6111  df-f 6112  df-f1 6113  df-fo 6114  df-f1o 6115  df-fv 6116  df-riota 6842  df-ov 6884  df-oprab 6885  df-mpt2 6886  df-om 7303  df-1st 7405  df-2nd 7406  df-wrecs 7649  df-recs 7711  df-rdg 7749  df-1o 7803  df-oadd 7807  df-er 7986  df-map 8101  df-en 8200  df-dom 8201  df-sdom 8202  df-fin 8203  df-card 9055  df-cda 9282  df-pnf 10368  df-mnf 10369  df-xr 10370  df-ltxr 10371  df-le 10372  df-sub 10560  df-neg 10561  df-nn 11313  df-2 11371  df-n0 11567  df-z 11651  df-uz 11912  df-rp 12054  df-fz 12557  df-hash 13345  df-vdwap 15896  df-vdwmc 15897  df-vdwpc 15898
This theorem is referenced by:  vdwlem7  15915
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