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Theorem vdwlem6 16915
Description: Lemma for vdw 16923. (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 6901 . . . . . . 7 (𝐺‘(𝐵 + (𝐸𝑖))) ∈ V
2 vdwlem6.j . . . . . . 7 𝐽 = (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝐵 + (𝐸𝑖))))
31, 2fnmpti 6690 . . . . . 6 𝐽 Fn (1...𝑀)
4 fvelrnb 6949 . . . . . 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 12864 . . . . . . . . 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 769 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽𝑚) = (𝐺𝐵))) → 𝑚 ∈ (1...𝑀))
25 simprr 771 . . . . . . . 8 ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽𝑚) = (𝐺𝐵))) → (𝐽𝑚) = (𝐺𝐵))
26 fveq2 6888 . . . . . . . . . . . 12 (𝑖 = 𝑚 → (𝐸𝑖) = (𝐸𝑚))
2726oveq2d 7421 . . . . . . . . . . 11 (𝑖 = 𝑚 → (𝐵 + (𝐸𝑖)) = (𝐵 + (𝐸𝑚)))
2827fveq2d 6892 . . . . . . . . . 10 (𝑖 = 𝑚 → (𝐺‘(𝐵 + (𝐸𝑖))) = (𝐺‘(𝐵 + (𝐸𝑚))))
29 fvex 6901 . . . . . . . . . 10 (𝐺‘(𝐵 + (𝐸𝑚))) ∈ V
3028, 2, 29fvmpt 6995 . . . . . . . . 9 (𝑚 ∈ (1...𝑀) → (𝐽𝑚) = (𝐺‘(𝐵 + (𝐸𝑚))))
3124, 30syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽𝑚) = (𝐺𝐵))) → (𝐽𝑚) = (𝐺‘(𝐵 + (𝐸𝑚))))
3225, 31eqtr3d 2774 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽𝑚) = (𝐺𝐵))) → (𝐺𝐵) = (𝐺‘(𝐵 + (𝐸𝑚))))
337, 11, 13, 15, 17, 19, 21, 23, 24, 32vdwlem1 16910 . . . . . 6 ((𝜑 ∧ (𝑚 ∈ (1...𝑀) ∧ (𝐽𝑚) = (𝐺𝐵))) → (𝐾 + 1) MonoAP 𝐺)
3433rexlimdvaa 3156 . . . . 5 (𝜑 → (∃𝑚 ∈ (1...𝑀)(𝐽𝑚) = (𝐺𝐵) → (𝐾 + 1) MonoAP 𝐺))
355, 34biimtrid 241 . . . 4 (𝜑 → ((𝐺𝐵) ∈ ran 𝐽 → (𝐾 + 1) MonoAP 𝐺))
3635imp 407 . . 3 ((𝜑 ∧ (𝐺𝐵) ∈ ran 𝐽) → (𝐾 + 1) MonoAP 𝐺)
3736olcd 872 . 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 16914 . . . . . 6 (𝜑𝑇 ∈ ℕ)
4847adantr 481 . . . . 5 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → 𝑇 ∈ ℕ)
49 0nn0 12483 . . . . . . . . . 10 0 ∈ ℕ0
5049a1i 11 . . . . . . . . 9 ((((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) ∧ 𝑗 = (𝑀 + 1)) → 0 ∈ ℕ0)
51 nnuz 12861 . . . . . . . . . . . . . . . . 17 ℕ = (ℤ‘1)
5218, 51eleqtrdi 2843 . . . . . . . . . . . . . . . 16 (𝜑𝑀 ∈ (ℤ‘1))
5352adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → 𝑀 ∈ (ℤ‘1))
54 elfzp1 13547 . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) → (𝑗 ∈ (1...𝑀) ∨ 𝑗 = (𝑀 + 1)))
5756ord 862 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) → (¬ 𝑗 ∈ (1...𝑀) → 𝑗 = (𝑀 + 1)))
5857con1d 145 . . . . . . . . . . 11 (((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) → (¬ 𝑗 = (𝑀 + 1) → 𝑗 ∈ (1...𝑀)))
5958imp 407 . . . . . . . . . 10 ((((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) ∧ ¬ 𝑗 = (𝑀 + 1)) → 𝑗 ∈ (1...𝑀))
6020ad2antrr 724 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) → 𝐸:(1...𝑀)⟶ℕ)
6160ffvelcdmda 7083 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) ∧ 𝑗 ∈ (1...𝑀)) → (𝐸𝑗) ∈ ℕ)
6261nnnn0d 12528 . . . . . . . . . 10 ((((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) ∧ 𝑗 ∈ (1...𝑀)) → (𝐸𝑗) ∈ ℕ0)
6359, 62syldan 591 . . . . . . . . 9 ((((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) ∧ ¬ 𝑗 = (𝑀 + 1)) → (𝐸𝑗) ∈ ℕ0)
6450, 63ifclda 4562 . . . . . . . 8 (((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) → if(𝑗 = (𝑀 + 1), 0, (𝐸𝑗)) ∈ ℕ0)
6512, 42nnmulcld 12261 . . . . . . . . 9 (𝜑 → (𝑊 · 𝐷) ∈ ℕ)
6665ad2antrr 724 . . . . . . . 8 (((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) → (𝑊 · 𝐷) ∈ ℕ)
67 nn0nnaddcl 12499 . . . . . . . 8 ((if(𝑗 = (𝑀 + 1), 0, (𝐸𝑗)) ∈ ℕ0 ∧ (𝑊 · 𝐷) ∈ ℕ) → (if(𝑗 = (𝑀 + 1), 0, (𝐸𝑗)) + (𝑊 · 𝐷)) ∈ ℕ)
6864, 66, 67syl2anc 584 . . . . . . 7 (((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) ∧ 𝑗 ∈ (1...(𝑀 + 1))) → (if(𝑗 = (𝑀 + 1), 0, (𝐸𝑗)) + (𝑊 · 𝐷)) ∈ ℕ)
6968, 46fmptd 7110 . . . . . 6 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → 𝑃:(1...(𝑀 + 1))⟶ℕ)
70 nnex 12214 . . . . . . 7 ℕ ∈ V
71 ovex 7438 . . . . . . 7 (1...(𝑀 + 1)) ∈ V
7270, 71elmap 8861 . . . . . 6 (𝑃 ∈ (ℕ ↑m (1...(𝑀 + 1))) ↔ 𝑃:(1...(𝑀 + 1))⟶ℕ)
7369, 72sylibr 233 . . . . 5 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → 𝑃 ∈ (ℕ ↑m (1...(𝑀 + 1))))
74 elfzp1 13547 . . . . . . . . . 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 12224 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑖 ∈ (1...𝑀)) → 𝐵 ∈ ℂ)
7877adantr 481 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐵 ∈ ℂ)
7920ffvelcdmda 7083 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐸𝑖) ∈ ℕ)
8079nncnd 12224 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐸𝑖) ∈ ℂ)
8180adantr 481 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐸𝑖) ∈ ℂ)
8278, 81addcld 11229 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐵 + (𝐸𝑖)) ∈ ℂ)
83 nnm1nn0 12509 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 ∈ ℕ → (𝐴 − 1) ∈ ℕ0)
8441, 83syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝐴 − 1) ∈ ℕ0)
85 nn0nnaddcl 12499 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐴 − 1) ∈ ℕ0𝑉 ∈ ℕ) → ((𝐴 − 1) + 𝑉) ∈ ℕ)
8684, 38, 85syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝐴 − 1) + 𝑉) ∈ ℕ)
8712, 86nnmulcld 12261 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℕ)
8887nncnd 12224 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℂ)
8988ad2antrr 724 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℂ)
90 elfznn0 13590 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 ∈ (0...(𝐾 − 1)) → 𝑚 ∈ ℕ0)
9190adantl 482 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℕ0)
9291nn0cnd 12530 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℂ)
9392adantlr 713 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℂ)
9493, 81mulcld 11230 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · (𝐸𝑖)) ∈ ℂ)
9565nnnn0d 12528 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑊 · 𝐷) ∈ ℕ0)
9695adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · 𝐷) ∈ ℕ0)
9791, 96nn0mulcld 12533 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · (𝑊 · 𝐷)) ∈ ℕ0)
9897nn0cnd 12530 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · (𝑊 · 𝐷)) ∈ ℂ)
9998adantlr 713 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · (𝑊 · 𝐷)) ∈ ℂ)
10082, 89, 94, 99add4d 11438 . . . . . . . . . . . . . . . . . . 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 12224 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑊 · 𝐷) ∈ ℂ)
102101ad2antrr 724 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · 𝐷) ∈ ℂ)
10393, 81, 102adddid 11234 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · ((𝐸𝑖) + (𝑊 · 𝐷))) = ((𝑚 · (𝐸𝑖)) + (𝑚 · (𝑊 · 𝐷))))
104103oveq2d 7421 . . . . . . . . . . . . . . . . . . 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 12224 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑊 ∈ ℂ)
106105adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝑊 ∈ ℂ)
10786nncnd 12224 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ((𝐴 − 1) + 𝑉) ∈ ℂ)
108107adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → ((𝐴 − 1) + 𝑉) ∈ ℂ)
10942nncnd 12224 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝐷 ∈ ℂ)
110109adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝐷 ∈ ℂ)
11192, 110mulcld 11230 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · 𝐷) ∈ ℂ)
112106, 108, 111adddid 11234 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · (((𝐴 − 1) + 𝑉) + (𝑚 · 𝐷))) = ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑊 · (𝑚 · 𝐷))))
11341nncnd 12224 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝐴 ∈ ℂ)
114113adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝐴 ∈ ℂ)
115 1cnd 11205 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 1 ∈ ℂ)
116114, 111, 115addsubd 11588 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → ((𝐴 + (𝑚 · 𝐷)) − 1) = ((𝐴 − 1) + (𝑚 · 𝐷)))
117116oveq1d 7420 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉) = (((𝐴 − 1) + (𝑚 · 𝐷)) + 𝑉))
11884nn0cnd 12530 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝐴 − 1) ∈ ℂ)
119118adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐴 − 1) ∈ ℂ)
12038nncnd 12224 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝑉 ∈ ℂ)
121120adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝑉 ∈ ℂ)
122119, 111, 121add32d 11437 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (((𝐴 − 1) + (𝑚 · 𝐷)) + 𝑉) = (((𝐴 − 1) + 𝑉) + (𝑚 · 𝐷)))
123117, 122eqtrd 2772 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉) = (((𝐴 − 1) + 𝑉) + (𝑚 · 𝐷)))
124123oveq2d 7421 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)) = (𝑊 · (((𝐴 − 1) + 𝑉) + (𝑚 · 𝐷))))
12592, 106, 110mul12d 11419 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · (𝑊 · 𝐷)) = (𝑊 · (𝑚 · 𝐷)))
126125oveq2d 7421 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑚 · (𝑊 · 𝐷))) = ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑊 · (𝑚 · 𝐷))))
127112, 124, 1263eqtr4d 2782 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)) = ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑚 · (𝑊 · 𝐷))))
128127adantlr 713 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)) = ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑚 · (𝑊 · 𝐷))))
129128oveq2d 7421 . . . . . . . . . . . . . . . . . . 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 2782 . . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑉 ∈ ℕ)
13212ad2antrr 724 . . . . . . . . . . . . . . . . . . 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 2732 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑚 · 𝐷))
135 oveq1 7412 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 = 𝑚 → (𝑛 · 𝐷) = (𝑚 · 𝐷))
136135oveq2d 7421 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = 𝑚 → (𝐴 + (𝑛 · 𝐷)) = (𝐴 + (𝑚 · 𝐷)))
137136rspceeqv 3632 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑚 ∈ (0...(𝐾 − 1)) ∧ (𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑚 · 𝐷))) → ∃𝑛 ∈ (0...(𝐾 − 1))(𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷)))
138134, 137mpan2 689 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 ∈ (0...(𝐾 − 1)) → ∃𝑛 ∈ (0...(𝐾 − 1))(𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷)))
13910nnnn0d 12528 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝐾 ∈ ℕ0)
140 vdwapval 16902 . . . . . . . . . . . . . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → ((𝐴 + (𝑚 · 𝐷)) ∈ (𝐴(AP‘𝐾)𝐷) ↔ ∃𝑛 ∈ (0...(𝐾 − 1))(𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷))))
142141biimpar 478 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ∃𝑛 ∈ (0...(𝐾 − 1))(𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷))) → (𝐴 + (𝑚 · 𝐷)) ∈ (𝐴(AP‘𝐾)𝐷))
143138, 142sylan2 593 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐴 + (𝑚 · 𝐷)) ∈ (𝐴(AP‘𝐾)𝐷))
144133, 143sseldd 3982 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐴 + (𝑚 · 𝐷)) ∈ (𝐹 “ {𝐺}))
14538, 12, 6, 39, 40vdwlem4 16913 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝐹:(1...𝑉)⟶(𝑅m (1...𝑊)))
146145ffnd 6715 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐹 Fn (1...𝑉))
147 fniniseg 7058 . . . . . . . . . . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝐴 + (𝑚 · 𝐷)) ∈ (𝐹 “ {𝐺})) → ((𝐴 + (𝑚 · 𝐷)) ∈ (1...𝑉) ∧ (𝐹‘(𝐴 + (𝑚 · 𝐷))) = 𝐺))
150144, 149syldan 591 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → ((𝐴 + (𝑚 · 𝐷)) ∈ (1...𝑉) ∧ (𝐹‘(𝐴 + (𝑚 · 𝐷))) = 𝐺))
151150simpld 495 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐴 + (𝑚 · 𝐷)) ∈ (1...𝑉))
152151adantlr 713 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐴 + (𝑚 · 𝐷)) ∈ (1...𝑉))
15322r19.21bi 3248 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝐵 + (𝐸𝑖))(AP‘𝐾)(𝐸𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}))
154153adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐵 + (𝐸𝑖))(AP‘𝐾)(𝐸𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}))
155 eqid 2732 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) = ((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖)))
156 oveq1 7412 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = 𝑚 → (𝑛 · (𝐸𝑖)) = (𝑚 · (𝐸𝑖)))
157156oveq2d 7421 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝑚 → ((𝐵 + (𝐸𝑖)) + (𝑛 · (𝐸𝑖))) = ((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))))
158157rspceeqv 3632 . . . . . . . . . . . . . . . . . . . . . . . 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 689 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ (0...(𝐾 − 1)) → ∃𝑛 ∈ (0...(𝐾 − 1))((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) = ((𝐵 + (𝐸𝑖)) + (𝑛 · (𝐸𝑖))))
16010adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑖 ∈ (1...𝑀)) → 𝐾 ∈ ℕ)
161160nnnn0d 12528 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑖 ∈ (1...𝑀)) → 𝐾 ∈ ℕ0)
16276, 79nnaddcld 12260 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐵 + (𝐸𝑖)) ∈ ℕ)
163 vdwapval 16902 . . . . . . . . . . . . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . . . . . . . . . . . . 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 478 . . . . . . . . . . . . . . . . . . . . . . 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 593 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) ∈ ((𝐵 + (𝐸𝑖))(AP‘𝐾)(𝐸𝑖)))
167154, 166sseldd 3982 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) ∈ (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}))
16814ffnd 6715 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐺 Fn (1...𝑊))
169168adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑖 ∈ (1...𝑀)) → 𝐺 Fn (1...𝑊))
170 fniniseg 7058 . . . . . . . . . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . . . . . . . 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 591 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) ∈ (1...𝑊) ∧ (𝐺‘((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖)))) = (𝐺‘(𝐵 + (𝐸𝑖)))))
174173simpld 495 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) ∈ (1...𝑊))
175131, 132, 152, 174vdwlem3 16912 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))) ∈ (1...(𝑊 · (2 · 𝑉))))
176130, 175eqeltrd 2833 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸𝑖) + (𝑊 · 𝐷)))) ∈ (1...(𝑊 · (2 · 𝑉))))
177 fvoveq1 7428 . . . . . . . . . . . . . . . . . . . 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 2732 . . . . . . . . . . . . . . . . . . . 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 6901 . . . . . . . . . . . . . . . . . . . 20 (𝐻‘(((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖))) + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))) ∈ V
180177, 178, 179fvmpt 6995 . . . . . . . . . . . . . . . . . . 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 496 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐺‘((𝐵 + (𝐸𝑖)) + (𝑚 · (𝐸𝑖)))) = (𝐺‘(𝐵 + (𝐸𝑖))))
183150simprd 496 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐹‘(𝐴 + (𝑚 · 𝐷))) = 𝐺)
184 oveq1 7412 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 = (𝐴 + (𝑚 · 𝐷)) → (𝑥 − 1) = ((𝐴 + (𝑚 · 𝐷)) − 1))
185184oveq1d 7420 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 = (𝐴 + (𝑚 · 𝐷)) → ((𝑥 − 1) + 𝑉) = (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))
186185oveq2d 7421 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 = (𝐴 + (𝑚 · 𝐷)) → (𝑊 · ((𝑥 − 1) + 𝑉)) = (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))
187186oveq2d 7421 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = (𝐴 + (𝑚 · 𝐷)) → (𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))) = (𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))
188187fveq2d 6892 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = (𝐴 + (𝑚 · 𝐷)) → (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉)))) = (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))
189188mpteq2dv 5249 . . . . . . . . . . . . . . . . . . . . . . . 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 7438 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1...𝑊) ∈ V
191190mptex 7221 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))) ∈ V
192189, 40, 191fvmpt 6995 . . . . . . . . . . . . . . . . . . . . . . 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 2774 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝐺 = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))))
195194adantlr 713 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐺 = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))))
196195fveq1d 6890 . . . . . . . . . . . . . . . . . . 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 2774 . . . . . . . . . . . . . . . . . 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 6892 . . . . . . . . . . . . . . . . . 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 2783 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐻‘(((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · ((𝐸𝑖) + (𝑊 · 𝐷))))) = (𝐺‘(𝐵 + (𝐸𝑖))))
200176, 199jca 512 . . . . . . . . . . . . . . . 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 2821 . . . . . . . . . . . . . . . . 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 6897 . . . . . . . . . . . . . . . . 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 631 . . . . . . . . . . . . . . . 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 246 . . . . . . . . . . . . . . 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 3155 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℕ)
207162, 206nnaddcld 12260 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ ℕ)
20865adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑊 · 𝐷) ∈ ℕ)
20979, 208nnaddcld 12260 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝐸𝑖) + (𝑊 · 𝐷)) ∈ ℕ)
210 vdwapval 16902 . . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . . 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 6715 . . . . . . . . . . . . . . . 16 (𝜑𝐻 Fn (1...(𝑊 · (2 · 𝑉))))
213212adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → 𝐻 Fn (1...(𝑊 · (2 · 𝑉))))
214 fniniseg 7058 . . . . . . . . . . . . . . 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 293 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑥 ∈ (((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)((𝐸𝑖) + (𝑊 · 𝐷))) → 𝑥 ∈ (𝐻 “ {(𝐺‘(𝐵 + (𝐸𝑖)))})))
217216ssrdv 3987 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → (((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)((𝐸𝑖) + (𝑊 · 𝐷))) ⊆ (𝐻 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}))
218 ssun1 4171 . . . . . . . . . . . . . . . . . . 19 (1...𝑀) ⊆ ((1...𝑀) ∪ {(𝑀 + 1)})
219 fzsuc 13544 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ (ℤ‘1) → (1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)}))
22052, 219syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)}))
221218, 220sseqtrrid 4034 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1...𝑀) ⊆ (1...(𝑀 + 1)))
222221sselda 3981 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑖 ∈ (1...(𝑀 + 1)))
223 eqeq1 2736 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑖 → (𝑗 = (𝑀 + 1) ↔ 𝑖 = (𝑀 + 1)))
224 fveq2 6888 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑖 → (𝐸𝑗) = (𝐸𝑖))
225223, 224ifbieq2d 4553 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑖 → if(𝑗 = (𝑀 + 1), 0, (𝐸𝑗)) = if(𝑖 = (𝑀 + 1), 0, (𝐸𝑖)))
226225oveq1d 7420 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑖 → (if(𝑗 = (𝑀 + 1), 0, (𝐸𝑗)) + (𝑊 · 𝐷)) = (if(𝑖 = (𝑀 + 1), 0, (𝐸𝑖)) + (𝑊 · 𝐷)))
227 ovex 7438 . . . . . . . . . . . . . . . . . 18 (if(𝑖 = (𝑀 + 1), 0, (𝐸𝑖)) + (𝑊 · 𝐷)) ∈ V
228226, 46, 227fvmpt 6995 . . . . . . . . . . . . . . . . 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)))
23018nnred 12223 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑀 ∈ ℝ)
231230ltp1d 12140 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑀 < (𝑀 + 1))
232 peano2re 11383 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑀 ∈ ℝ → (𝑀 + 1) ∈ ℝ)
233230, 232syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑀 + 1) ∈ ℝ)
234230, 233ltnled 11357 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑀 < (𝑀 + 1) ↔ ¬ (𝑀 + 1) ≤ 𝑀))
235231, 234mpbid 231 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ¬ (𝑀 + 1) ≤ 𝑀)
236 breq1 5150 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = (𝑀 + 1) → (𝑖𝑀 ↔ (𝑀 + 1) ≤ 𝑀))
237236notbid 317 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = (𝑀 + 1) → (¬ 𝑖𝑀 ↔ ¬ (𝑀 + 1) ≤ 𝑀))
238235, 237syl5ibrcom 246 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑖 = (𝑀 + 1) → ¬ 𝑖𝑀))
239238con2d 134 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑖𝑀 → ¬ 𝑖 = (𝑀 + 1)))
240 elfzle2 13501 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (1...𝑀) → 𝑖𝑀)
241239, 240impel 506 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (1...𝑀)) → ¬ 𝑖 = (𝑀 + 1))
242 iffalse 4536 . . . . . . . . . . . . . . . . . 18 𝑖 = (𝑀 + 1) → if(𝑖 = (𝑀 + 1), 0, (𝐸𝑖)) = (𝐸𝑖))
243242oveq1d 7420 . . . . . . . . . . . . . . . . 17 𝑖 = (𝑀 + 1) → (if(𝑖 = (𝑀 + 1), 0, (𝐸𝑖)) + (𝑊 · 𝐷)) = ((𝐸𝑖) + (𝑊 · 𝐷)))
244241, 243syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑀)) → (if(𝑖 = (𝑀 + 1), 0, (𝐸𝑖)) + (𝑊 · 𝐷)) = ((𝐸𝑖) + (𝑊 · 𝐷)))
245229, 244eqtrd 2772 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑃𝑖) = ((𝐸𝑖) + (𝑊 · 𝐷)))
246245oveq2d 7421 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑇 + (𝑃𝑖)) = (𝑇 + ((𝐸𝑖) + (𝑊 · 𝐷))))
24747nncnd 12224 . . . . . . . . . . . . . . . 16 (𝜑𝑇 ∈ ℂ)
248247adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑇 ∈ ℂ)
249101adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑊 · 𝐷) ∈ ℂ)
250248, 80, 249add12d 11436 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑇 + ((𝐸𝑖) + (𝑊 · 𝐷))) = ((𝐸𝑖) + (𝑇 + (𝑊 · 𝐷))))
25145oveq1i 7415 . . . . . . . . . . . . . . . . . 18 (𝑇 + (𝑊 · 𝐷)) = ((𝐵 + (𝑊 · ((𝐴 + (𝑉𝐷)) − 1))) + (𝑊 · 𝐷))
25216nncnd 12224 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐵 ∈ ℂ)
253120, 109subcld 11567 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑉𝐷) ∈ ℂ)
254113, 253addcld 11229 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝐴 + (𝑉𝐷)) ∈ ℂ)
255 ax-1cn 11164 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ ℂ
256 subcl 11455 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴 + (𝑉𝐷)) ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 + (𝑉𝐷)) − 1) ∈ ℂ)
257254, 255, 256sylancl 586 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝐴 + (𝑉𝐷)) − 1) ∈ ℂ)
258105, 257mulcld 11230 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑊 · ((𝐴 + (𝑉𝐷)) − 1)) ∈ ℂ)
259252, 258, 101addassd 11232 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝐵 + (𝑊 · ((𝐴 + (𝑉𝐷)) − 1))) + (𝑊 · 𝐷)) = (𝐵 + ((𝑊 · ((𝐴 + (𝑉𝐷)) − 1)) + (𝑊 · 𝐷))))
260105, 257, 109adddid 11234 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑊 · (((𝐴 + (𝑉𝐷)) − 1) + 𝐷)) = ((𝑊 · ((𝐴 + (𝑉𝐷)) − 1)) + (𝑊 · 𝐷)))
261113, 253, 109addassd 11232 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → ((𝐴 + (𝑉𝐷)) + 𝐷) = (𝐴 + ((𝑉𝐷) + 𝐷)))
262120, 109npcand 11571 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → ((𝑉𝐷) + 𝐷) = 𝑉)
263262oveq2d 7421 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝐴 + ((𝑉𝐷) + 𝐷)) = (𝐴 + 𝑉))
264261, 263eqtrd 2772 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ((𝐴 + (𝑉𝐷)) + 𝐷) = (𝐴 + 𝑉))
265264oveq1d 7420 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (((𝐴 + (𝑉𝐷)) + 𝐷) − 1) = ((𝐴 + 𝑉) − 1))
266 1cnd 11205 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → 1 ∈ ℂ)
267254, 109, 266addsubd 11588 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (((𝐴 + (𝑉𝐷)) + 𝐷) − 1) = (((𝐴 + (𝑉𝐷)) − 1) + 𝐷))
268113, 120, 266addsubd 11588 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝐴 + 𝑉) − 1) = ((𝐴 − 1) + 𝑉))
269265, 267, 2683eqtr3d 2780 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (((𝐴 + (𝑉𝐷)) − 1) + 𝐷) = ((𝐴 − 1) + 𝑉))
270269oveq2d 7421 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑊 · (((𝐴 + (𝑉𝐷)) − 1) + 𝐷)) = (𝑊 · ((𝐴 − 1) + 𝑉)))
271260, 270eqtr3d 2774 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑊 · ((𝐴 + (𝑉𝐷)) − 1)) + (𝑊 · 𝐷)) = (𝑊 · ((𝐴 − 1) + 𝑉)))
272271oveq2d 7421 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐵 + ((𝑊 · ((𝐴 + (𝑉𝐷)) − 1)) + (𝑊 · 𝐷))) = (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))))
273259, 272eqtrd 2772 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝐵 + (𝑊 · ((𝐴 + (𝑉𝐷)) − 1))) + (𝑊 · 𝐷)) = (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))))
274251, 273eqtrid 2784 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑇 + (𝑊 · 𝐷)) = (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))))
275274oveq2d 7421 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝐸𝑖) + (𝑇 + (𝑊 · 𝐷))) = ((𝐸𝑖) + (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
276275adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝐸𝑖) + (𝑇 + (𝑊 · 𝐷))) = ((𝐸𝑖) + (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
27788adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℂ)
27880, 77, 277addassd 11232 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → (((𝐸𝑖) + 𝐵) + (𝑊 · ((𝐴 − 1) + 𝑉))) = ((𝐸𝑖) + (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
27980, 77addcomd 11412 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝐸𝑖) + 𝐵) = (𝐵 + (𝐸𝑖)))
280279oveq1d 7420 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → (((𝐸𝑖) + 𝐵) + (𝑊 · ((𝐴 − 1) + 𝑉))) = ((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))))
281276, 278, 2803eqtr2d 2778 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝐸𝑖) + (𝑇 + (𝑊 · 𝐷))) = ((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))))
282246, 250, 2813eqtrd 2776 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑇 + (𝑃𝑖)) = ((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉))))
283282, 245oveq12d 7423 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) = (((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)((𝐸𝑖) + (𝑊 · 𝐷))))
284 cnvimass 6077 . . . . . . . . . . . . . . . . . . 19 (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}) ⊆ dom 𝐺
285284, 14fssdm 6734 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}) ⊆ (1...𝑊))
286285adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}) ⊆ (1...𝑊))
287 vdwapid1 16904 . . . . . . . . . . . . . . . . . . 19 ((𝐾 ∈ ℕ ∧ (𝐵 + (𝐸𝑖)) ∈ ℕ ∧ (𝐸𝑖) ∈ ℕ) → (𝐵 + (𝐸𝑖)) ∈ ((𝐵 + (𝐸𝑖))(AP‘𝐾)(𝐸𝑖)))
288160, 162, 79, 287syl3anc 1371 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐵 + (𝐸𝑖)) ∈ ((𝐵 + (𝐸𝑖))(AP‘𝐾)(𝐸𝑖)))
289153, 288sseldd 3982 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐵 + (𝐸𝑖)) ∈ (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}))
290286, 289sseldd 3982 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐵 + (𝐸𝑖)) ∈ (1...𝑊))
291 fvoveq1 7428 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝐵 + (𝐸𝑖)) → (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))) = (𝐻‘((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))))
292 eqid 2732 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))))) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
293 fvex 6901 . . . . . . . . . . . . . . . . 17 (𝐻‘((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))) ∈ V
294291, 292, 293fvmpt 6995 . . . . . . . . . . . . . . . 16 ((𝐵 + (𝐸𝑖)) ∈ (1...𝑊) → ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))‘(𝐵 + (𝐸𝑖))) = (𝐻‘((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))))
295290, 294syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))‘(𝐵 + (𝐸𝑖))) = (𝐻‘((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))))
296 vdwapid1 16904 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐾 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → 𝐴 ∈ (𝐴(AP‘𝐾)𝐷))
29710, 41, 42, 296syl3anc 1371 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐴 ∈ (𝐴(AP‘𝐾)𝐷))
29843, 297sseldd 3982 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐴 ∈ (𝐹 “ {𝐺}))
299 fniniseg 7058 . . . . . . . . . . . . . . . . . . . . 21 (𝐹 Fn (1...𝑉) → (𝐴 ∈ (𝐹 “ {𝐺}) ↔ (𝐴 ∈ (1...𝑉) ∧ (𝐹𝐴) = 𝐺)))
300146, 299syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝐴 ∈ (𝐹 “ {𝐺}) ↔ (𝐴 ∈ (1...𝑉) ∧ (𝐹𝐴) = 𝐺)))
301298, 300mpbid 231 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐴 ∈ (1...𝑉) ∧ (𝐹𝐴) = 𝐺))
302301simprd 496 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐹𝐴) = 𝐺)
303301simpld 495 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐴 ∈ (1...𝑉))
304 oveq1 7412 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = 𝐴 → (𝑥 − 1) = (𝐴 − 1))
305304oveq1d 7420 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝐴 → ((𝑥 − 1) + 𝑉) = ((𝐴 − 1) + 𝑉))
306305oveq2d 7421 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝐴 → (𝑊 · ((𝑥 − 1) + 𝑉)) = (𝑊 · ((𝐴 − 1) + 𝑉)))
307306oveq2d 7421 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝐴 → (𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))) = (𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))))
308307fveq2d 6892 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝐴 → (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉)))) = (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
309308mpteq2dv 5249 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝐴 → (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))))))
310190mptex 7221 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))))) ∈ V
311309, 40, 310fvmpt 6995 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ (1...𝑉) → (𝐹𝐴) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))))))
312303, 311syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐹𝐴) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))))))
313302, 312eqtr3d 2774 . . . . . . . . . . . . . . . . 17 (𝜑𝐺 = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉))))))
314313fveq1d 6890 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐺‘(𝐵 + (𝐸𝑖))) = ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))‘(𝐵 + (𝐸𝑖))))
315314adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐺‘(𝐵 + (𝐸𝑖))) = ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))‘(𝐵 + (𝐸𝑖))))
316282fveq2d 6892 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻‘(𝑇 + (𝑃𝑖))) = (𝐻‘((𝐵 + (𝐸𝑖)) + (𝑊 · ((𝐴 − 1) + 𝑉)))))
317295, 315, 3163eqtr4rd 2783 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻‘(𝑇 + (𝑃𝑖))) = (𝐺‘(𝐵 + (𝐸𝑖))))
318317sneqd 4639 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑀)) → {(𝐻‘(𝑇 + (𝑃𝑖)))} = {(𝐺‘(𝐵 + (𝐸𝑖)))})
319318imaeq2d 6057 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))}) = (𝐻 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}))
320217, 283, 3193sstr4d 4028 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))}))
321320ex 413 . . . . . . . . . 10 (𝜑 → (𝑖 ∈ (1...𝑀) → ((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))})))
322252adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝐵 ∈ ℂ)
32388adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℂ)
324322, 323, 98addassd 11232 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) = (𝐵 + ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑚 · (𝑊 · 𝐷)))))
325127oveq2d 7421 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))) = (𝐵 + ((𝑊 · ((𝐴 − 1) + 𝑉)) + (𝑚 · (𝑊 · 𝐷)))))
326324, 325eqtr4d 2775 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) = (𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))))
32738adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝑉 ∈ ℕ)
32812adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝑊 ∈ ℕ)
329 eluzfz1 13504 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑀 ∈ (ℤ‘1) → 1 ∈ (1...𝑀))
33052, 329syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → 1 ∈ (1...𝑀))
331330ne0d 4334 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (1...𝑀) ≠ ∅)
332 elfzuz3 13494 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐵 + (𝐸𝑖)) ∈ (1...𝑊) → 𝑊 ∈ (ℤ‘(𝐵 + (𝐸𝑖))))
333290, 332syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑊 ∈ (ℤ‘(𝐵 + (𝐸𝑖))))
33416nnzd 12581 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑𝐵 ∈ ℤ)
335 uzid 12833 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐵 ∈ ℤ → 𝐵 ∈ (ℤ𝐵))
336334, 335syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝐵 ∈ (ℤ𝐵))
337336adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑖 ∈ (1...𝑀)) → 𝐵 ∈ (ℤ𝐵))
33879nnnn0d 12528 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐸𝑖) ∈ ℕ0)
339 uzaddcl 12884 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐵 ∈ (ℤ𝐵) ∧ (𝐸𝑖) ∈ ℕ0) → (𝐵 + (𝐸𝑖)) ∈ (ℤ𝐵))
340337, 338, 339syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐵 + (𝐸𝑖)) ∈ (ℤ𝐵))
341 uztrn 12836 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑊 ∈ (ℤ‘(𝐵 + (𝐸𝑖))) ∧ (𝐵 + (𝐸𝑖)) ∈ (ℤ𝐵)) → 𝑊 ∈ (ℤ𝐵))
342333, 340, 341syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑊 ∈ (ℤ𝐵))
343 eluzle 12831 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑊 ∈ (ℤ𝐵) → 𝐵𝑊)
344342, 343syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑖 ∈ (1...𝑀)) → 𝐵𝑊)
345344ralrimiva 3146 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ∀𝑖 ∈ (1...𝑀)𝐵𝑊)
346 r19.2z 4493 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...𝑀) ≠ ∅ ∧ ∀𝑖 ∈ (1...𝑀)𝐵𝑊) → ∃𝑖 ∈ (1...𝑀)𝐵𝑊)
347331, 345, 346syl2anc 584 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ∃𝑖 ∈ (1...𝑀)𝐵𝑊)
348 idd 24 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (1...𝑀) → (𝐵𝑊𝐵𝑊))
349348rexlimiv 3148 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑖 ∈ (1...𝑀)𝐵𝑊𝐵𝑊)
350347, 349syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐵𝑊)
35112nnzd 12581 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑊 ∈ ℤ)
352 fznn 13565 . . . . . . . . . . . . . . . . . . . . . 22 (𝑊 ∈ ℤ → (𝐵 ∈ (1...𝑊) ↔ (𝐵 ∈ ℕ ∧ 𝐵𝑊)))
353351, 352syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐵 ∈ (1...𝑊) ↔ (𝐵 ∈ ℕ ∧ 𝐵𝑊)))
35416, 350, 353mpbir2and 711 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐵 ∈ (1...𝑊))
355354adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → 𝐵 ∈ (1...𝑊))
356327, 328, 151, 355vdwlem3 16912 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉))) ∈ (1...(𝑊 · (2 · 𝑉))))
357326, 356eqeltrd 2833 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) ∈ (1...(𝑊 · (2 · 𝑉))))
358 fvoveq1 7428 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝐵 → (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))) = (𝐻‘(𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))
359 fvex 6901 . . . . . . . . . . . . . . . . . . . 20 (𝐻‘(𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))) ∈ V
360358, 178, 359fvmpt 6995 . . . . . . . . . . . . . . . . . . 19 (𝐵 ∈ (1...𝑊) → ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))‘𝐵) = (𝐻‘(𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))
361355, 360syl 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)))))
362194fveq1d 6890 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐺𝐵) = ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))‘𝐵))
363326fveq2d 6892 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐻‘((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷)))) = (𝐻‘(𝐵 + (𝑊 · (((𝐴 + (𝑚 · 𝐷)) − 1) + 𝑉)))))
364361, 362, 3633eqtr4rd 2783 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝐻‘((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷)))) = (𝐺𝐵))
365357, 364jca 512 . . . . . . . . . . . . . . . 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)))
366 eleq1 2821 . . . . . . . . . . . . . . . . 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)))))
367 fveqeq2 6897 . . . . . . . . . . . . . . . . 17 (𝑧 = ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) → ((𝐻𝑧) = (𝐺𝐵) ↔ (𝐻‘((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷)))) = (𝐺𝐵)))
368366, 367anbi12d 631 . . . . . . . . . . . . . . . 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))))
369365, 368syl5ibrcom 246 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ (0...(𝐾 − 1))) → (𝑧 = ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) → (𝑧 ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻𝑧) = (𝐺𝐵))))
370369rexlimdva 3155 . . . . . . . . . . . . . 14 (𝜑 → (∃𝑚 ∈ (0...(𝐾 − 1))𝑧 = ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) + (𝑚 · (𝑊 · 𝐷))) → (𝑧 ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻𝑧) = (𝐺𝐵))))
37116, 87nnaddcld 12260 . . . . . . . . . . . . . . 15 (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ ℕ)
372 vdwapval 16902 . . . . . . . . . . . . . . 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)))))
373139, 371, 65, 372syl3anc 1371 . . . . . . . . . . . . . 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)))))
374 fniniseg 7058 . . . . . . . . . . . . . . 15 (𝐻 Fn (1...(𝑊 · (2 · 𝑉))) → (𝑧 ∈ (𝐻 “ {(𝐺𝐵)}) ↔ (𝑧 ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻𝑧) = (𝐺𝐵))))
375212, 374syl 17 . . . . . . . . . . . . . 14 (𝜑 → (𝑧 ∈ (𝐻 “ {(𝐺𝐵)}) ↔ (𝑧 ∈ (1...(𝑊 · (2 · 𝑉))) ∧ (𝐻𝑧) = (𝐺𝐵))))
376370, 373, 3753imtr4d 293 . . . . . . . . . . . . 13 (𝜑 → (𝑧 ∈ ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)(𝑊 · 𝐷)) → 𝑧 ∈ (𝐻 “ {(𝐺𝐵)})))
377376ssrdv 3987 . . . . . . . . . . . 12 (𝜑 → ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)(𝑊 · 𝐷)) ⊆ (𝐻 “ {(𝐺𝐵)}))
37818peano2nnd 12225 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑀 + 1) ∈ ℕ)
379378, 51eleqtrdi 2843 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑀 + 1) ∈ (ℤ‘1))
380 eluzfz2 13505 . . . . . . . . . . . . . . . . 17 ((𝑀 + 1) ∈ (ℤ‘1) → (𝑀 + 1) ∈ (1...(𝑀 + 1)))
381 iftrue 4533 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑀 + 1) → if(𝑗 = (𝑀 + 1), 0, (𝐸𝑗)) = 0)
382381oveq1d 7420 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑀 + 1) → (if(𝑗 = (𝑀 + 1), 0, (𝐸𝑗)) + (𝑊 · 𝐷)) = (0 + (𝑊 · 𝐷)))
383 ovex 7438 . . . . . . . . . . . . . . . . . 18 (0 + (𝑊 · 𝐷)) ∈ V
384382, 46, 383fvmpt 6995 . . . . . . . . . . . . . . . . 17 ((𝑀 + 1) ∈ (1...(𝑀 + 1)) → (𝑃‘(𝑀 + 1)) = (0 + (𝑊 · 𝐷)))
385379, 380, 3843syl 18 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑃‘(𝑀 + 1)) = (0 + (𝑊 · 𝐷)))
386101addlidd 11411 . . . . . . . . . . . . . . . 16 (𝜑 → (0 + (𝑊 · 𝐷)) = (𝑊 · 𝐷))
387385, 386eqtrd 2772 . . . . . . . . . . . . . . 15 (𝜑 → (𝑃‘(𝑀 + 1)) = (𝑊 · 𝐷))
388387oveq2d 7421 . . . . . . . . . . . . . 14 (𝜑 → (𝑇 + (𝑃‘(𝑀 + 1))) = (𝑇 + (𝑊 · 𝐷)))
389388, 274eqtrd 2772 . . . . . . . . . . . . 13 (𝜑 → (𝑇 + (𝑃‘(𝑀 + 1))) = (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))))
390389, 387oveq12d 7423 . . . . . . . . . . . 12 (𝜑 → ((𝑇 + (𝑃‘(𝑀 + 1)))(AP‘𝐾)(𝑃‘(𝑀 + 1))) = ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))(AP‘𝐾)(𝑊 · 𝐷)))
391 fvoveq1 7428 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐵 → (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))) = (𝐻‘(𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
392 fvex 6901 . . . . . . . . . . . . . . . . 17 (𝐻‘(𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))) ∈ V
393391, 292, 392fvmpt 6995 . . . . . . . . . . . . . . . 16 (𝐵 ∈ (1...𝑊) → ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))‘𝐵) = (𝐻‘(𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
394354, 393syl 17 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))‘𝐵) = (𝐻‘(𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
395313fveq1d 6890 . . . . . . . . . . . . . . 15 (𝜑 → (𝐺𝐵) = ((𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝐴 − 1) + 𝑉)))))‘𝐵))
396389fveq2d 6892 . . . . . . . . . . . . . . 15 (𝜑 → (𝐻‘(𝑇 + (𝑃‘(𝑀 + 1)))) = (𝐻‘(𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
397394, 395, 3963eqtr4rd 2783 . . . . . . . . . . . . . 14 (𝜑 → (𝐻‘(𝑇 + (𝑃‘(𝑀 + 1)))) = (𝐺𝐵))
398397sneqd 4639 . . . . . . . . . . . . 13 (𝜑 → {(𝐻‘(𝑇 + (𝑃‘(𝑀 + 1))))} = {(𝐺𝐵)})
399398imaeq2d 6057 . . . . . . . . . . . 12 (𝜑 → (𝐻 “ {(𝐻‘(𝑇 + (𝑃‘(𝑀 + 1))))}) = (𝐻 “ {(𝐺𝐵)}))
400377, 390, 3993sstr4d 4028 . . . . . . . . . . 11 (𝜑 → ((𝑇 + (𝑃‘(𝑀 + 1)))(AP‘𝐾)(𝑃‘(𝑀 + 1))) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃‘(𝑀 + 1))))}))
401 fveq2 6888 . . . . . . . . . . . . . 14 (𝑖 = (𝑀 + 1) → (𝑃𝑖) = (𝑃‘(𝑀 + 1)))
402401oveq2d 7421 . . . . . . . . . . . . 13 (𝑖 = (𝑀 + 1) → (𝑇 + (𝑃𝑖)) = (𝑇 + (𝑃‘(𝑀 + 1))))
403402, 401oveq12d 7423 . . . . . . . . . . . 12 (𝑖 = (𝑀 + 1) → ((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) = ((𝑇 + (𝑃‘(𝑀 + 1)))(AP‘𝐾)(𝑃‘(𝑀 + 1))))
404402fveq2d 6892 . . . . . . . . . . . . . 14 (𝑖 = (𝑀 + 1) → (𝐻‘(𝑇 + (𝑃𝑖))) = (𝐻‘(𝑇 + (𝑃‘(𝑀 + 1)))))
405404sneqd 4639 . . . . . . . . . . . . 13 (𝑖 = (𝑀 + 1) → {(𝐻‘(𝑇 + (𝑃𝑖)))} = {(𝐻‘(𝑇 + (𝑃‘(𝑀 + 1))))})
406405imaeq2d 6057 . . . . . . . . . . . 12 (𝑖 = (𝑀 + 1) → (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))}) = (𝐻 “ {(𝐻‘(𝑇 + (𝑃‘(𝑀 + 1))))}))
407403, 406sseq12d 4014 . . . . . . . . . . 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))))})))
408400, 407syl5ibrcom 246 . . . . . . . . . 10 (𝜑 → (𝑖 = (𝑀 + 1) → ((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))})))
409321, 408jaod 857 . . . . . . . . 9 (𝜑 → ((𝑖 ∈ (1...𝑀) ∨ 𝑖 = (𝑀 + 1)) → ((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))})))
41075, 409sylbid 239 . . . . . . . 8 (𝜑 → (𝑖 ∈ (1...(𝑀 + 1)) → ((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))})))
411410ralrimiv 3145 . . . . . . 7 (𝜑 → ∀𝑖 ∈ (1...(𝑀 + 1))((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))}))
412411adantr 481 . . . . . 6 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → ∀𝑖 ∈ (1...(𝑀 + 1))((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))}))
413220rexeqdv 3326 . . . . . . . . . . . 12 (𝜑 → (∃𝑖 ∈ (1...(𝑀 + 1))𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ↔ ∃𝑖 ∈ ((1...𝑀) ∪ {(𝑀 + 1)})𝑥 = (𝐻‘(𝑇 + (𝑃𝑖)))))
414 rexun 4189 . . . . . . . . . . . . 13 (∃𝑖 ∈ ((1...𝑀) ∪ {(𝑀 + 1)})𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ↔ (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ∨ ∃𝑖 ∈ {(𝑀 + 1)}𝑥 = (𝐻‘(𝑇 + (𝑃𝑖)))))
415317eqeq2d 2743 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ↔ 𝑥 = (𝐺‘(𝐵 + (𝐸𝑖)))))
416415rexbidva 3176 . . . . . . . . . . . . . 14 (𝜑 → (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ↔ ∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸𝑖)))))
417 ovex 7438 . . . . . . . . . . . . . . . 16 (𝑀 + 1) ∈ V
418404eqeq2d 2743 . . . . . . . . . . . . . . . 16 (𝑖 = (𝑀 + 1) → (𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ↔ 𝑥 = (𝐻‘(𝑇 + (𝑃‘(𝑀 + 1))))))
419417, 418rexsn 4685 . . . . . . . . . . . . . . 15 (∃𝑖 ∈ {(𝑀 + 1)}𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ↔ 𝑥 = (𝐻‘(𝑇 + (𝑃‘(𝑀 + 1)))))
420397eqeq2d 2743 . . . . . . . . . . . . . . 15 (𝜑 → (𝑥 = (𝐻‘(𝑇 + (𝑃‘(𝑀 + 1)))) ↔ 𝑥 = (𝐺𝐵)))
421419, 420bitrid 282 . . . . . . . . . . . . . 14 (𝜑 → (∃𝑖 ∈ {(𝑀 + 1)}𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ↔ 𝑥 = (𝐺𝐵)))
422416, 421orbi12d 917 . . . . . . . . . . . . 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))))
423414, 422bitrid 282 . . . . . . . . . . . 12 (𝜑 → (∃𝑖 ∈ ((1...𝑀) ∪ {(𝑀 + 1)})𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ↔ (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸𝑖))) ∨ 𝑥 = (𝐺𝐵))))
424413, 423bitrd 278 . . . . . . . . . . 11 (𝜑 → (∃𝑖 ∈ (1...(𝑀 + 1))𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ↔ (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸𝑖))) ∨ 𝑥 = (𝐺𝐵))))
425424adantr 481 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → (∃𝑖 ∈ (1...(𝑀 + 1))𝑥 = (𝐻‘(𝑇 + (𝑃𝑖))) ↔ (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸𝑖))) ∨ 𝑥 = (𝐺𝐵))))
426425abbidv 2801 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → {𝑥 ∣ ∃𝑖 ∈ (1...(𝑀 + 1))𝑥 = (𝐻‘(𝑇 + (𝑃𝑖)))} = {𝑥 ∣ (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸𝑖))) ∨ 𝑥 = (𝐺𝐵))})
427 eqid 2732 . . . . . . . . . 10 (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃𝑖)))) = (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃𝑖))))
428427rnmpt 5952 . . . . . . . . 9 ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃𝑖)))) = {𝑥 ∣ ∃𝑖 ∈ (1...(𝑀 + 1))𝑥 = (𝐻‘(𝑇 + (𝑃𝑖)))}
4292rnmpt 5952 . . . . . . . . . . 11 ran 𝐽 = {𝑥 ∣ ∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸𝑖)))}
430 df-sn 4628 . . . . . . . . . . 11 {(𝐺𝐵)} = {𝑥𝑥 = (𝐺𝐵)}
431429, 430uneq12i 4160 . . . . . . . . . 10 (ran 𝐽 ∪ {(𝐺𝐵)}) = ({𝑥 ∣ ∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸𝑖)))} ∪ {𝑥𝑥 = (𝐺𝐵)})
432 unab 4297 . . . . . . . . . 10 ({𝑥 ∣ ∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸𝑖)))} ∪ {𝑥𝑥 = (𝐺𝐵)}) = {𝑥 ∣ (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸𝑖))) ∨ 𝑥 = (𝐺𝐵))}
433431, 432eqtri 2760 . . . . . . . . 9 (ran 𝐽 ∪ {(𝐺𝐵)}) = {𝑥 ∣ (∃𝑖 ∈ (1...𝑀)𝑥 = (𝐺‘(𝐵 + (𝐸𝑖))) ∨ 𝑥 = (𝐺𝐵))}
434426, 428, 4333eqtr4g 2797 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃𝑖)))) = (ran 𝐽 ∪ {(𝐺𝐵)}))
435434fveq2d 6892 . . . . . . 7 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → (♯‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃𝑖))))) = (♯‘(ran 𝐽 ∪ {(𝐺𝐵)})))
436 fzfi 13933 . . . . . . . . . 10 (1...𝑀) ∈ Fin
437 dffn4 6808 . . . . . . . . . . 11 (𝐽 Fn (1...𝑀) ↔ 𝐽:(1...𝑀)–onto→ran 𝐽)
4383, 437mpbi 229 . . . . . . . . . 10 𝐽:(1...𝑀)–onto→ran 𝐽
439 fofi 9334 . . . . . . . . . 10 (((1...𝑀) ∈ Fin ∧ 𝐽:(1...𝑀)–onto→ran 𝐽) → ran 𝐽 ∈ Fin)
440436, 438, 439mp2an 690 . . . . . . . . 9 ran 𝐽 ∈ Fin
441440a1i 11 . . . . . . . 8 (𝜑 → ran 𝐽 ∈ Fin)
442 fvex 6901 . . . . . . . . 9 (𝐺𝐵) ∈ V
443 hashunsng 14348 . . . . . . . . 9 ((𝐺𝐵) ∈ V → ((ran 𝐽 ∈ Fin ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → (♯‘(ran 𝐽 ∪ {(𝐺𝐵)})) = ((♯‘ran 𝐽) + 1)))
444442, 443ax-mp 5 . . . . . . . 8 ((ran 𝐽 ∈ Fin ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → (♯‘(ran 𝐽 ∪ {(𝐺𝐵)})) = ((♯‘ran 𝐽) + 1))
445441, 444sylan 580 . . . . . . 7 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → (♯‘(ran 𝐽 ∪ {(𝐺𝐵)})) = ((♯‘ran 𝐽) + 1))
44644adantr 481 . . . . . . . 8 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → (♯‘ran 𝐽) = 𝑀)
447446oveq1d 7420 . . . . . . 7 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → ((♯‘ran 𝐽) + 1) = (𝑀 + 1))
448435, 445, 4473eqtrd 2776 . . . . . 6 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → (♯‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃𝑖))))) = (𝑀 + 1))
449412, 448jca 512 . . . . 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)))
450 oveq1 7412 . . . . . . . . . 10 (𝑎 = 𝑇 → (𝑎 + (𝑑𝑖)) = (𝑇 + (𝑑𝑖)))
451450oveq1d 7420 . . . . . . . . 9 (𝑎 = 𝑇 → ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) = ((𝑇 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)))
452 fvoveq1 7428 . . . . . . . . . . 11 (𝑎 = 𝑇 → (𝐻‘(𝑎 + (𝑑𝑖))) = (𝐻‘(𝑇 + (𝑑𝑖))))
453452sneqd 4639 . . . . . . . . . 10 (𝑎 = 𝑇 → {(𝐻‘(𝑎 + (𝑑𝑖)))} = {(𝐻‘(𝑇 + (𝑑𝑖)))})
454453imaeq2d 6057 . . . . . . . . 9 (𝑎 = 𝑇 → (𝐻 “ {(𝐻‘(𝑎 + (𝑑𝑖)))}) = (𝐻 “ {(𝐻‘(𝑇 + (𝑑𝑖)))}))
455451, 454sseq12d 4014 . . . . . . . 8 (𝑎 = 𝑇 → (((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑎 + (𝑑𝑖)))}) ↔ ((𝑇 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑑𝑖)))})))
456455ralbidv 3177 . . . . . . 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)))})))
457452mpteq2dv 5249 . . . . . . . . 9 (𝑎 = 𝑇 → (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑎 + (𝑑𝑖)))) = (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑𝑖)))))
458457rneqd 5935 . . . . . . . 8 (𝑎 = 𝑇 → ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑎 + (𝑑𝑖)))) = ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑𝑖)))))
459458fveqeq2d 6896 . . . . . . 7 (𝑎 = 𝑇 → ((♯‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑎 + (𝑑𝑖))))) = (𝑀 + 1) ↔ (♯‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑𝑖))))) = (𝑀 + 1)))
460456, 459anbi12d 631 . . . . . 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))))
461 fveq1 6887 . . . . . . . . . . 11 (𝑑 = 𝑃 → (𝑑𝑖) = (𝑃𝑖))
462461oveq2d 7421 . . . . . . . . . 10 (𝑑 = 𝑃 → (𝑇 + (𝑑𝑖)) = (𝑇 + (𝑃𝑖)))
463462, 461oveq12d 7423 . . . . . . . . 9 (𝑑 = 𝑃 → ((𝑇 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) = ((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)))
464462fveq2d 6892 . . . . . . . . . . 11 (𝑑 = 𝑃 → (𝐻‘(𝑇 + (𝑑𝑖))) = (𝐻‘(𝑇 + (𝑃𝑖))))
465464sneqd 4639 . . . . . . . . . 10 (𝑑 = 𝑃 → {(𝐻‘(𝑇 + (𝑑𝑖)))} = {(𝐻‘(𝑇 + (𝑃𝑖)))})
466465imaeq2d 6057 . . . . . . . . 9 (𝑑 = 𝑃 → (𝐻 “ {(𝐻‘(𝑇 + (𝑑𝑖)))}) = (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))}))
467463, 466sseq12d 4014 . . . . . . . 8 (𝑑 = 𝑃 → (((𝑇 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑑𝑖)))}) ↔ ((𝑇 + (𝑃𝑖))(AP‘𝐾)(𝑃𝑖)) ⊆ (𝐻 “ {(𝐻‘(𝑇 + (𝑃𝑖)))})))
468467ralbidv 3177 . . . . . . 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)))})))
469464mpteq2dv 5249 . . . . . . . . 9 (𝑑 = 𝑃 → (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑𝑖)))) = (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃𝑖)))))
470469rneqd 5935 . . . . . . . 8 (𝑑 = 𝑃 → ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑𝑖)))) = ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃𝑖)))))
471470fveqeq2d 6896 . . . . . . 7 (𝑑 = 𝑃 → ((♯‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑑𝑖))))) = (𝑀 + 1) ↔ (♯‘ran (𝑖 ∈ (1...(𝑀 + 1)) ↦ (𝐻‘(𝑇 + (𝑃𝑖))))) = (𝑀 + 1)))
472468, 471anbi12d 631 . . . . . 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))))
473460, 472rspc2ev 3623 . . . . 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)))
47448, 73, 449, 473syl3anc 1371 . . . 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)))
475 ovex 7438 . . . . 5 (1...(𝑊 · (2 · 𝑉))) ∈ V
47610adantr 481 . . . . . 6 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → 𝐾 ∈ ℕ)
477476nnnn0d 12528 . . . . 5 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → 𝐾 ∈ ℕ0)
47839adantr 481 . . . . 5 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → 𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)
47918adantr 481 . . . . . 6 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → 𝑀 ∈ ℕ)
480479peano2nnd 12225 . . . . 5 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → (𝑀 + 1) ∈ ℕ)
481 eqid 2732 . . . . 5 (1...(𝑀 + 1)) = (1...(𝑀 + 1))
482475, 477, 478, 480, 481vdwpc 16909 . . . 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))))
483474, 482mpbird 256 . . 3 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → ⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻)
484483orcd 871 . 2 ((𝜑 ∧ ¬ (𝐺𝐵) ∈ ran 𝐽) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺))
48537, 484pm2.61dan 811 1 (𝜑 → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wcel 2106  {cab 2709  wne 2940  wral 3061  wrex 3070  Vcvv 3474  cun 3945  wss 3947  c0 4321  ifcif 4527  {csn 4627  cop 4633   class class class wbr 5147  cmpt 5230  ccnv 5674  ran crn 5676  cima 5678   Fn wfn 6535  wf 6536  ontowfo 6538  cfv 6540  (class class class)co 7405  m cmap 8816  Fincfn 8935  cc 11104  cr 11105  0cc0 11106  1c1 11107   + caddc 11109   · cmul 11111   < clt 11244  cle 11245  cmin 11440  cn 12208  2c2 12263  0cn0 12468  cz 12554  cuz 12818  ...cfz 13480  chash 14286  APcvdwa 16894   MonoAP cvdwm 16895   PolyAP cvdwp 16896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-oadd 8466  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-dju 9892  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-n0 12469  df-z 12555  df-uz 12819  df-rp 12971  df-fz 13481  df-hash 14287  df-vdwap 16897  df-vdwmc 16898  df-vdwpc 16899
This theorem is referenced by:  vdwlem7  16916
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