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Theorem prodmodclem3 11356
 Description: Lemma for prodmodc 11359. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 11-Apr-2024.)
Hypotheses
Ref Expression
prodmo.1 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))
prodmo.2 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
prodmodc.3 𝐺 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (𝑓𝑗) / 𝑘𝐵, 1))
prodmodclem3.4 𝐻 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (𝐾𝑗) / 𝑘𝐵, 1))
prodmolem3.5 (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
prodmolem3.6 (𝜑𝑓:(1...𝑀)–1-1-onto𝐴)
prodmolem3.7 (𝜑𝐾:(1...𝑁)–1-1-onto𝐴)
Assertion
Ref Expression
prodmodclem3 (𝜑 → (seq1( · , 𝐺)‘𝑀) = (seq1( · , 𝐻)‘𝑁))
Distinct variable groups:   𝐴,𝑗,𝑘   𝐵,𝑗   𝑗,𝐺   𝑗,𝐾,𝑘   𝑗,𝑀   𝑓,𝑗,𝑘   𝜑,𝑘
Allowed substitution hints:   𝜑(𝑓,𝑗)   𝐴(𝑓)   𝐵(𝑓,𝑘)   𝐹(𝑓,𝑗,𝑘)   𝐺(𝑓,𝑘)   𝐻(𝑓,𝑗,𝑘)   𝐾(𝑓)   𝑀(𝑓,𝑘)   𝑁(𝑓,𝑗,𝑘)

Proof of Theorem prodmodclem3
Dummy variables 𝑖 𝑚 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulcl 7759 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑚 · 𝑦) ∈ ℂ)
21adantl 275 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑚 · 𝑦) ∈ ℂ)
3 mulcom 7761 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑚 · 𝑦) = (𝑦 · 𝑚))
43adantl 275 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑚 · 𝑦) = (𝑦 · 𝑚))
5 mulass 7763 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝑚 · 𝑦) · 𝑥) = (𝑚 · (𝑦 · 𝑥)))
65adantl 275 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → ((𝑚 · 𝑦) · 𝑥) = (𝑚 · (𝑦 · 𝑥)))
7 prodmolem3.5 . . . . 5 (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
87simpld 111 . . . 4 (𝜑𝑀 ∈ ℕ)
9 nnuz 9373 . . . 4 ℕ = (ℤ‘1)
108, 9eleqtrdi 2232 . . 3 (𝜑𝑀 ∈ (ℤ‘1))
11 prodmolem3.6 . . . . . 6 (𝜑𝑓:(1...𝑀)–1-1-onto𝐴)
12 f1ocnv 5380 . . . . . 6 (𝑓:(1...𝑀)–1-1-onto𝐴𝑓:𝐴1-1-onto→(1...𝑀))
1311, 12syl 14 . . . . 5 (𝜑𝑓:𝐴1-1-onto→(1...𝑀))
14 prodmolem3.7 . . . . 5 (𝜑𝐾:(1...𝑁)–1-1-onto𝐴)
15 f1oco 5390 . . . . 5 ((𝑓:𝐴1-1-onto→(1...𝑀) ∧ 𝐾:(1...𝑁)–1-1-onto𝐴) → (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
1613, 14, 15syl2anc 408 . . . 4 (𝜑 → (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
177ancomd 265 . . . . . . 7 (𝜑 → (𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ))
1817, 14, 11nnf1o 11157 . . . . . 6 (𝜑𝑀 = 𝑁)
1918oveq2d 5790 . . . . 5 (𝜑 → (1...𝑀) = (1...𝑁))
2019f1oeq2d 5363 . . . 4 (𝜑 → ((𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀) ↔ (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀)))
2116, 20mpbird 166 . . 3 (𝜑 → (𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀))
22 prodmodc.3 . . . . 5 𝐺 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (𝑓𝑗) / 𝑘𝐵, 1))
23 breq1 3932 . . . . . 6 (𝑗 = 𝑚 → (𝑗 ≤ (♯‘𝐴) ↔ 𝑚 ≤ (♯‘𝐴)))
24 fveq2 5421 . . . . . . 7 (𝑗 = 𝑚 → (𝑓𝑗) = (𝑓𝑚))
2524csbeq1d 3010 . . . . . 6 (𝑗 = 𝑚(𝑓𝑗) / 𝑘𝐵 = (𝑓𝑚) / 𝑘𝐵)
2623, 25ifbieq1d 3494 . . . . 5 (𝑗 = 𝑚 → if(𝑗 ≤ (♯‘𝐴), (𝑓𝑗) / 𝑘𝐵, 1) = if(𝑚 ≤ (♯‘𝐴), (𝑓𝑚) / 𝑘𝐵, 1))
27 elnnuz 9374 . . . . . . 7 (𝑚 ∈ ℕ ↔ 𝑚 ∈ (ℤ‘1))
2827biimpri 132 . . . . . 6 (𝑚 ∈ (ℤ‘1) → 𝑚 ∈ ℕ)
2928adantl 275 . . . . 5 ((𝜑𝑚 ∈ (ℤ‘1)) → 𝑚 ∈ ℕ)
30 f1of 5367 . . . . . . . . . 10 (𝑓:(1...𝑀)–1-1-onto𝐴𝑓:(1...𝑀)⟶𝐴)
3111, 30syl 14 . . . . . . . . 9 (𝜑𝑓:(1...𝑀)⟶𝐴)
3231ad2antrr 479 . . . . . . . 8 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 𝑓:(1...𝑀)⟶𝐴)
33 1zzd 9093 . . . . . . . . . 10 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 1 ∈ ℤ)
348nnzd 9184 . . . . . . . . . . 11 (𝜑𝑀 ∈ ℤ)
3534ad2antrr 479 . . . . . . . . . 10 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 𝑀 ∈ ℤ)
36 eluzelz 9347 . . . . . . . . . . 11 (𝑚 ∈ (ℤ‘1) → 𝑚 ∈ ℤ)
3736ad2antlr 480 . . . . . . . . . 10 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 𝑚 ∈ ℤ)
3833, 35, 373jca 1161 . . . . . . . . 9 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → (1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑚 ∈ ℤ))
39 eluzle 9350 . . . . . . . . . . 11 (𝑚 ∈ (ℤ‘1) → 1 ≤ 𝑚)
4039ad2antlr 480 . . . . . . . . . 10 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 1 ≤ 𝑚)
41 simpr 109 . . . . . . . . . . 11 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 𝑚 ≤ (♯‘𝐴))
428nnnn0d 9042 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ ℕ0)
43 hashfz1 10541 . . . . . . . . . . . . . 14 (𝑀 ∈ ℕ0 → (♯‘(1...𝑀)) = 𝑀)
4442, 43syl 14 . . . . . . . . . . . . 13 (𝜑 → (♯‘(1...𝑀)) = 𝑀)
45 1zzd 9093 . . . . . . . . . . . . . . 15 (𝜑 → 1 ∈ ℤ)
4645, 34fzfigd 10216 . . . . . . . . . . . . . 14 (𝜑 → (1...𝑀) ∈ Fin)
4746, 11fihasheqf1od 10548 . . . . . . . . . . . . 13 (𝜑 → (♯‘(1...𝑀)) = (♯‘𝐴))
4844, 47eqtr3d 2174 . . . . . . . . . . . 12 (𝜑𝑀 = (♯‘𝐴))
4948ad2antrr 479 . . . . . . . . . . 11 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 𝑀 = (♯‘𝐴))
5041, 49breqtrrd 3956 . . . . . . . . . 10 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 𝑚𝑀)
5140, 50jca 304 . . . . . . . . 9 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → (1 ≤ 𝑚𝑚𝑀))
52 elfz2 9809 . . . . . . . . 9 (𝑚 ∈ (1...𝑀) ↔ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑚 ∈ ℤ) ∧ (1 ≤ 𝑚𝑚𝑀)))
5338, 51, 52sylanbrc 413 . . . . . . . 8 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 𝑚 ∈ (1...𝑀))
5432, 53ffvelrnd 5556 . . . . . . 7 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → (𝑓𝑚) ∈ 𝐴)
55 prodmo.2 . . . . . . . . 9 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
5655ralrimiva 2505 . . . . . . . 8 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
5756ad2antrr 479 . . . . . . 7 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
58 nfcsb1v 3035 . . . . . . . . 9 𝑘(𝑓𝑚) / 𝑘𝐵
5958nfel1 2292 . . . . . . . 8 𝑘(𝑓𝑚) / 𝑘𝐵 ∈ ℂ
60 csbeq1a 3012 . . . . . . . . 9 (𝑘 = (𝑓𝑚) → 𝐵 = (𝑓𝑚) / 𝑘𝐵)
6160eleq1d 2208 . . . . . . . 8 (𝑘 = (𝑓𝑚) → (𝐵 ∈ ℂ ↔ (𝑓𝑚) / 𝑘𝐵 ∈ ℂ))
6259, 61rspc 2783 . . . . . . 7 ((𝑓𝑚) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝑓𝑚) / 𝑘𝐵 ∈ ℂ))
6354, 57, 62sylc 62 . . . . . 6 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → (𝑓𝑚) / 𝑘𝐵 ∈ ℂ)
64 1cnd 7794 . . . . . 6 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ ¬ 𝑚 ≤ (♯‘𝐴)) → 1 ∈ ℂ)
6529nnzd 9184 . . . . . . 7 ((𝜑𝑚 ∈ (ℤ‘1)) → 𝑚 ∈ ℤ)
6648, 34eqeltrrd 2217 . . . . . . . 8 (𝜑 → (♯‘𝐴) ∈ ℤ)
6766adantr 274 . . . . . . 7 ((𝜑𝑚 ∈ (ℤ‘1)) → (♯‘𝐴) ∈ ℤ)
68 zdcle 9139 . . . . . . 7 ((𝑚 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → DECID 𝑚 ≤ (♯‘𝐴))
6965, 67, 68syl2anc 408 . . . . . 6 ((𝜑𝑚 ∈ (ℤ‘1)) → DECID 𝑚 ≤ (♯‘𝐴))
7063, 64, 69ifcldadc 3501 . . . . 5 ((𝜑𝑚 ∈ (ℤ‘1)) → if(𝑚 ≤ (♯‘𝐴), (𝑓𝑚) / 𝑘𝐵, 1) ∈ ℂ)
7122, 26, 29, 70fvmptd3 5514 . . . 4 ((𝜑𝑚 ∈ (ℤ‘1)) → (𝐺𝑚) = if(𝑚 ≤ (♯‘𝐴), (𝑓𝑚) / 𝑘𝐵, 1))
7271, 70eqeltrd 2216 . . 3 ((𝜑𝑚 ∈ (ℤ‘1)) → (𝐺𝑚) ∈ ℂ)
73 prodmodclem3.4 . . . . 5 𝐻 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (𝐾𝑗) / 𝑘𝐵, 1))
74 fveq2 5421 . . . . . . 7 (𝑗 = 𝑚 → (𝐾𝑗) = (𝐾𝑚))
7574csbeq1d 3010 . . . . . 6 (𝑗 = 𝑚(𝐾𝑗) / 𝑘𝐵 = (𝐾𝑚) / 𝑘𝐵)
7623, 75ifbieq1d 3494 . . . . 5 (𝑗 = 𝑚 → if(𝑗 ≤ (♯‘𝐴), (𝐾𝑗) / 𝑘𝐵, 1) = if(𝑚 ≤ (♯‘𝐴), (𝐾𝑚) / 𝑘𝐵, 1))
7714ad2antrr 479 . . . . . . . . 9 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 𝐾:(1...𝑁)–1-1-onto𝐴)
78 f1of 5367 . . . . . . . . 9 (𝐾:(1...𝑁)–1-1-onto𝐴𝐾:(1...𝑁)⟶𝐴)
7977, 78syl 14 . . . . . . . 8 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 𝐾:(1...𝑁)⟶𝐴)
8019ad2antrr 479 . . . . . . . . 9 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → (1...𝑀) = (1...𝑁))
8153, 80eleqtrd 2218 . . . . . . . 8 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 𝑚 ∈ (1...𝑁))
8279, 81ffvelrnd 5556 . . . . . . 7 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → (𝐾𝑚) ∈ 𝐴)
83 nfcsb1v 3035 . . . . . . . . 9 𝑘(𝐾𝑚) / 𝑘𝐵
8483nfel1 2292 . . . . . . . 8 𝑘(𝐾𝑚) / 𝑘𝐵 ∈ ℂ
85 csbeq1a 3012 . . . . . . . . 9 (𝑘 = (𝐾𝑚) → 𝐵 = (𝐾𝑚) / 𝑘𝐵)
8685eleq1d 2208 . . . . . . . 8 (𝑘 = (𝐾𝑚) → (𝐵 ∈ ℂ ↔ (𝐾𝑚) / 𝑘𝐵 ∈ ℂ))
8784, 86rspc 2783 . . . . . . 7 ((𝐾𝑚) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝐾𝑚) / 𝑘𝐵 ∈ ℂ))
8882, 57, 87sylc 62 . . . . . 6 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → (𝐾𝑚) / 𝑘𝐵 ∈ ℂ)
8988, 64, 69ifcldadc 3501 . . . . 5 ((𝜑𝑚 ∈ (ℤ‘1)) → if(𝑚 ≤ (♯‘𝐴), (𝐾𝑚) / 𝑘𝐵, 1) ∈ ℂ)
9073, 76, 29, 89fvmptd3 5514 . . . 4 ((𝜑𝑚 ∈ (ℤ‘1)) → (𝐻𝑚) = if(𝑚 ≤ (♯‘𝐴), (𝐾𝑚) / 𝑘𝐵, 1))
9190, 89eqeltrd 2216 . . 3 ((𝜑𝑚 ∈ (ℤ‘1)) → (𝐻𝑚) ∈ ℂ)
9219f1oeq2d 5363 . . . . . . . . . 10 (𝜑 → (𝐾:(1...𝑀)–1-1-onto𝐴𝐾:(1...𝑁)–1-1-onto𝐴))
9314, 92mpbird 166 . . . . . . . . 9 (𝜑𝐾:(1...𝑀)–1-1-onto𝐴)
94 f1of 5367 . . . . . . . . 9 (𝐾:(1...𝑀)–1-1-onto𝐴𝐾:(1...𝑀)⟶𝐴)
9593, 94syl 14 . . . . . . . 8 (𝜑𝐾:(1...𝑀)⟶𝐴)
96 fvco3 5492 . . . . . . . 8 ((𝐾:(1...𝑀)⟶𝐴𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) = (𝑓‘(𝐾𝑖)))
9795, 96sylan 281 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) = (𝑓‘(𝐾𝑖)))
9897fveq2d 5425 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘((𝑓𝐾)‘𝑖)) = (𝑓‘(𝑓‘(𝐾𝑖))))
9911adantr 274 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑓:(1...𝑀)–1-1-onto𝐴)
10095ffvelrnda 5555 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) ∈ 𝐴)
101 f1ocnvfv2 5679 . . . . . . 7 ((𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (𝐾𝑖) ∈ 𝐴) → (𝑓‘(𝑓‘(𝐾𝑖))) = (𝐾𝑖))
10299, 100, 101syl2anc 408 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘(𝑓‘(𝐾𝑖))) = (𝐾𝑖))
10398, 102eqtrd 2172 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘((𝑓𝐾)‘𝑖)) = (𝐾𝑖))
104103csbeq1d 3010 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵 = (𝐾𝑖) / 𝑘𝐵)
105 breq1 3932 . . . . . . 7 (𝑗 = ((𝑓𝐾)‘𝑖) → (𝑗 ≤ (♯‘𝐴) ↔ ((𝑓𝐾)‘𝑖) ≤ (♯‘𝐴)))
106 fveq2 5421 . . . . . . . 8 (𝑗 = ((𝑓𝐾)‘𝑖) → (𝑓𝑗) = (𝑓‘((𝑓𝐾)‘𝑖)))
107106csbeq1d 3010 . . . . . . 7 (𝑗 = ((𝑓𝐾)‘𝑖) → (𝑓𝑗) / 𝑘𝐵 = (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵)
108105, 107ifbieq1d 3494 . . . . . 6 (𝑗 = ((𝑓𝐾)‘𝑖) → if(𝑗 ≤ (♯‘𝐴), (𝑓𝑗) / 𝑘𝐵, 1) = if(((𝑓𝐾)‘𝑖) ≤ (♯‘𝐴), (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵, 1))
109 f1of 5367 . . . . . . . . 9 ((𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀) → (𝑓𝐾):(1...𝑀)⟶(1...𝑀))
11021, 109syl 14 . . . . . . . 8 (𝜑 → (𝑓𝐾):(1...𝑀)⟶(1...𝑀))
111110ffvelrnda 5555 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) ∈ (1...𝑀))
112 elfznn 9846 . . . . . . 7 (((𝑓𝐾)‘𝑖) ∈ (1...𝑀) → ((𝑓𝐾)‘𝑖) ∈ ℕ)
113111, 112syl 14 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) ∈ ℕ)
114 elfzle2 9820 . . . . . . . . . 10 (((𝑓𝐾)‘𝑖) ∈ (1...𝑀) → ((𝑓𝐾)‘𝑖) ≤ 𝑀)
115111, 114syl 14 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) ≤ 𝑀)
11648adantr 274 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑀 = (♯‘𝐴))
117115, 116breqtrd 3954 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) ≤ (♯‘𝐴))
118117iftrued 3481 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → if(((𝑓𝐾)‘𝑖) ≤ (♯‘𝐴), (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵, 1) = (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵)
11956adantr 274 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑀)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
120 nfcsb1v 3035 . . . . . . . . . . 11 𝑘(𝐾𝑖) / 𝑘𝐵
121120nfel1 2292 . . . . . . . . . 10 𝑘(𝐾𝑖) / 𝑘𝐵 ∈ ℂ
122 csbeq1a 3012 . . . . . . . . . . 11 (𝑘 = (𝐾𝑖) → 𝐵 = (𝐾𝑖) / 𝑘𝐵)
123122eleq1d 2208 . . . . . . . . . 10 (𝑘 = (𝐾𝑖) → (𝐵 ∈ ℂ ↔ (𝐾𝑖) / 𝑘𝐵 ∈ ℂ))
124121, 123rspc 2783 . . . . . . . . 9 ((𝐾𝑖) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝐾𝑖) / 𝑘𝐵 ∈ ℂ))
125100, 119, 124sylc 62 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) / 𝑘𝐵 ∈ ℂ)
126104, 125eqeltrd 2216 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵 ∈ ℂ)
127118, 126eqeltrd 2216 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → if(((𝑓𝐾)‘𝑖) ≤ (♯‘𝐴), (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵, 1) ∈ ℂ)
12822, 108, 113, 127fvmptd3 5514 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐺‘((𝑓𝐾)‘𝑖)) = if(((𝑓𝐾)‘𝑖) ≤ (♯‘𝐴), (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵, 1))
129128, 118eqtrd 2172 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐺‘((𝑓𝐾)‘𝑖)) = (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵)
130 breq1 3932 . . . . . . 7 (𝑗 = 𝑖 → (𝑗 ≤ (♯‘𝐴) ↔ 𝑖 ≤ (♯‘𝐴)))
131 fveq2 5421 . . . . . . . 8 (𝑗 = 𝑖 → (𝐾𝑗) = (𝐾𝑖))
132131csbeq1d 3010 . . . . . . 7 (𝑗 = 𝑖(𝐾𝑗) / 𝑘𝐵 = (𝐾𝑖) / 𝑘𝐵)
133130, 132ifbieq1d 3494 . . . . . 6 (𝑗 = 𝑖 → if(𝑗 ≤ (♯‘𝐴), (𝐾𝑗) / 𝑘𝐵, 1) = if(𝑖 ≤ (♯‘𝐴), (𝐾𝑖) / 𝑘𝐵, 1))
134 elfznn 9846 . . . . . . 7 (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℕ)
135134adantl 275 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑖 ∈ ℕ)
136 elfzle2 9820 . . . . . . . . . 10 (𝑖 ∈ (1...𝑀) → 𝑖𝑀)
137136adantl 275 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑖𝑀)
138137, 116breqtrd 3954 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑖 ≤ (♯‘𝐴))
139138iftrued 3481 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → if(𝑖 ≤ (♯‘𝐴), (𝐾𝑖) / 𝑘𝐵, 1) = (𝐾𝑖) / 𝑘𝐵)
140139, 125eqeltrd 2216 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → if(𝑖 ≤ (♯‘𝐴), (𝐾𝑖) / 𝑘𝐵, 1) ∈ ℂ)
14173, 133, 135, 140fvmptd3 5514 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻𝑖) = if(𝑖 ≤ (♯‘𝐴), (𝐾𝑖) / 𝑘𝐵, 1))
142141, 139eqtrd 2172 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻𝑖) = (𝐾𝑖) / 𝑘𝐵)
143104, 129, 1423eqtr4rd 2183 . . 3 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻𝑖) = (𝐺‘((𝑓𝐾)‘𝑖)))
1442, 4, 6, 10, 21, 72, 91, 143seq3f1o 10289 . 2 (𝜑 → (seq1( · , 𝐻)‘𝑀) = (seq1( · , 𝐺)‘𝑀))
14518fveq2d 5425 . 2 (𝜑 → (seq1( · , 𝐻)‘𝑀) = (seq1( · , 𝐻)‘𝑁))
146144, 145eqtr3d 2174 1 (𝜑 → (seq1( · , 𝐺)‘𝑀) = (seq1( · , 𝐻)‘𝑁))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  DECID wdc 819   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  ∀wral 2416  ⦋csb 3003  ifcif 3474   class class class wbr 3929   ↦ cmpt 3989  ◡ccnv 4538   ∘ ccom 4543  ⟶wf 5119  –1-1-onto→wf1o 5122  ‘cfv 5123  (class class class)co 5774  ℂcc 7630  1c1 7633   · cmul 7637   ≤ cle 7813  ℕcn 8732  ℕ0cn0 8989  ℤcz 9066  ℤ≥cuz 9338  ...cfz 9802  seqcseq 10230  ♯chash 10533 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-0id 7740  ax-rnegex 7741  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748 This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-1o 6313  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-inn 8733  df-n0 8990  df-z 9067  df-uz 9339  df-fz 9803  df-fzo 9932  df-seqfrec 10231  df-ihash 10534 This theorem is referenced by:  prodmodclem2a  11357  prodmodc  11359
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