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Theorem prodmodclem3 11567
Description: Lemma for prodmodc 11570. (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 7929 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑚 · 𝑦) ∈ ℂ)
21adantl 277 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑚 · 𝑦) ∈ ℂ)
3 mulcom 7931 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑚 · 𝑦) = (𝑦 · 𝑚))
43adantl 277 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑚 · 𝑦) = (𝑦 · 𝑚))
5 mulass 7933 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝑚 · 𝑦) · 𝑥) = (𝑚 · (𝑦 · 𝑥)))
65adantl 277 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → ((𝑚 · 𝑦) · 𝑥) = (𝑚 · (𝑦 · 𝑥)))
7 prodmolem3.5 . . . . 5 (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
87simpld 112 . . . 4 (𝜑𝑀 ∈ ℕ)
9 nnuz 9552 . . . 4 ℕ = (ℤ‘1)
108, 9eleqtrdi 2270 . . 3 (𝜑𝑀 ∈ (ℤ‘1))
11 prodmolem3.6 . . . . . 6 (𝜑𝑓:(1...𝑀)–1-1-onto𝐴)
12 f1ocnv 5470 . . . . . 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 5480 . . . . 5 ((𝑓:𝐴1-1-onto→(1...𝑀) ∧ 𝐾:(1...𝑁)–1-1-onto𝐴) → (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
1613, 14, 15syl2anc 411 . . . 4 (𝜑 → (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
177ancomd 267 . . . . . . 7 (𝜑 → (𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ))
1817, 14, 11nnf1o 11368 . . . . . 6 (𝜑𝑀 = 𝑁)
1918oveq2d 5885 . . . . 5 (𝜑 → (1...𝑀) = (1...𝑁))
2019f1oeq2d 5453 . . . 4 (𝜑 → ((𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀) ↔ (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀)))
2116, 20mpbird 167 . . 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 4003 . . . . . 6 (𝑗 = 𝑚 → (𝑗 ≤ (♯‘𝐴) ↔ 𝑚 ≤ (♯‘𝐴)))
24 fveq2 5511 . . . . . . 7 (𝑗 = 𝑚 → (𝑓𝑗) = (𝑓𝑚))
2524csbeq1d 3064 . . . . . 6 (𝑗 = 𝑚(𝑓𝑗) / 𝑘𝐵 = (𝑓𝑚) / 𝑘𝐵)
2623, 25ifbieq1d 3556 . . . . 5 (𝑗 = 𝑚 → if(𝑗 ≤ (♯‘𝐴), (𝑓𝑗) / 𝑘𝐵, 1) = if(𝑚 ≤ (♯‘𝐴), (𝑓𝑚) / 𝑘𝐵, 1))
27 elnnuz 9553 . . . . . . 7 (𝑚 ∈ ℕ ↔ 𝑚 ∈ (ℤ‘1))
2827biimpri 133 . . . . . 6 (𝑚 ∈ (ℤ‘1) → 𝑚 ∈ ℕ)
2928adantl 277 . . . . 5 ((𝜑𝑚 ∈ (ℤ‘1)) → 𝑚 ∈ ℕ)
30 f1of 5457 . . . . . . . . . 10 (𝑓:(1...𝑀)–1-1-onto𝐴𝑓:(1...𝑀)⟶𝐴)
3111, 30syl 14 . . . . . . . . 9 (𝜑𝑓:(1...𝑀)⟶𝐴)
3231ad2antrr 488 . . . . . . . 8 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 𝑓:(1...𝑀)⟶𝐴)
33 1zzd 9269 . . . . . . . . . 10 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 1 ∈ ℤ)
348nnzd 9363 . . . . . . . . . . 11 (𝜑𝑀 ∈ ℤ)
3534ad2antrr 488 . . . . . . . . . 10 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 𝑀 ∈ ℤ)
36 eluzelz 9526 . . . . . . . . . . 11 (𝑚 ∈ (ℤ‘1) → 𝑚 ∈ ℤ)
3736ad2antlr 489 . . . . . . . . . 10 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 𝑚 ∈ ℤ)
3833, 35, 373jca 1177 . . . . . . . . 9 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → (1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑚 ∈ ℤ))
39 eluzle 9529 . . . . . . . . . . 11 (𝑚 ∈ (ℤ‘1) → 1 ≤ 𝑚)
4039ad2antlr 489 . . . . . . . . . 10 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 1 ≤ 𝑚)
41 simpr 110 . . . . . . . . . . 11 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 𝑚 ≤ (♯‘𝐴))
428nnnn0d 9218 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ ℕ0)
43 hashfz1 10747 . . . . . . . . . . . . . 14 (𝑀 ∈ ℕ0 → (♯‘(1...𝑀)) = 𝑀)
4442, 43syl 14 . . . . . . . . . . . . 13 (𝜑 → (♯‘(1...𝑀)) = 𝑀)
45 1zzd 9269 . . . . . . . . . . . . . . 15 (𝜑 → 1 ∈ ℤ)
4645, 34fzfigd 10417 . . . . . . . . . . . . . 14 (𝜑 → (1...𝑀) ∈ Fin)
4746, 11fihasheqf1od 10753 . . . . . . . . . . . . 13 (𝜑 → (♯‘(1...𝑀)) = (♯‘𝐴))
4844, 47eqtr3d 2212 . . . . . . . . . . . 12 (𝜑𝑀 = (♯‘𝐴))
4948ad2antrr 488 . . . . . . . . . . 11 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 𝑀 = (♯‘𝐴))
5041, 49breqtrrd 4028 . . . . . . . . . 10 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 𝑚𝑀)
5140, 50jca 306 . . . . . . . . 9 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → (1 ≤ 𝑚𝑚𝑀))
52 elfz2 10002 . . . . . . . . 9 (𝑚 ∈ (1...𝑀) ↔ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑚 ∈ ℤ) ∧ (1 ≤ 𝑚𝑚𝑀)))
5338, 51, 52sylanbrc 417 . . . . . . . 8 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 𝑚 ∈ (1...𝑀))
5432, 53ffvelcdmd 5648 . . . . . . 7 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → (𝑓𝑚) ∈ 𝐴)
55 prodmo.2 . . . . . . . . 9 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
5655ralrimiva 2550 . . . . . . . 8 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
5756ad2antrr 488 . . . . . . 7 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
58 nfcsb1v 3090 . . . . . . . . 9 𝑘(𝑓𝑚) / 𝑘𝐵
5958nfel1 2330 . . . . . . . 8 𝑘(𝑓𝑚) / 𝑘𝐵 ∈ ℂ
60 csbeq1a 3066 . . . . . . . . 9 (𝑘 = (𝑓𝑚) → 𝐵 = (𝑓𝑚) / 𝑘𝐵)
6160eleq1d 2246 . . . . . . . 8 (𝑘 = (𝑓𝑚) → (𝐵 ∈ ℂ ↔ (𝑓𝑚) / 𝑘𝐵 ∈ ℂ))
6259, 61rspc 2835 . . . . . . 7 ((𝑓𝑚) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝑓𝑚) / 𝑘𝐵 ∈ ℂ))
6354, 57, 62sylc 62 . . . . . 6 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → (𝑓𝑚) / 𝑘𝐵 ∈ ℂ)
64 1cnd 7964 . . . . . 6 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ ¬ 𝑚 ≤ (♯‘𝐴)) → 1 ∈ ℂ)
6529nnzd 9363 . . . . . . 7 ((𝜑𝑚 ∈ (ℤ‘1)) → 𝑚 ∈ ℤ)
6648, 34eqeltrrd 2255 . . . . . . . 8 (𝜑 → (♯‘𝐴) ∈ ℤ)
6766adantr 276 . . . . . . 7 ((𝜑𝑚 ∈ (ℤ‘1)) → (♯‘𝐴) ∈ ℤ)
68 zdcle 9318 . . . . . . 7 ((𝑚 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → DECID 𝑚 ≤ (♯‘𝐴))
6965, 67, 68syl2anc 411 . . . . . 6 ((𝜑𝑚 ∈ (ℤ‘1)) → DECID 𝑚 ≤ (♯‘𝐴))
7063, 64, 69ifcldadc 3563 . . . . 5 ((𝜑𝑚 ∈ (ℤ‘1)) → if(𝑚 ≤ (♯‘𝐴), (𝑓𝑚) / 𝑘𝐵, 1) ∈ ℂ)
7122, 26, 29, 70fvmptd3 5605 . . . 4 ((𝜑𝑚 ∈ (ℤ‘1)) → (𝐺𝑚) = if(𝑚 ≤ (♯‘𝐴), (𝑓𝑚) / 𝑘𝐵, 1))
7271, 70eqeltrd 2254 . . 3 ((𝜑𝑚 ∈ (ℤ‘1)) → (𝐺𝑚) ∈ ℂ)
73 prodmodclem3.4 . . . . 5 𝐻 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (𝐾𝑗) / 𝑘𝐵, 1))
74 fveq2 5511 . . . . . . 7 (𝑗 = 𝑚 → (𝐾𝑗) = (𝐾𝑚))
7574csbeq1d 3064 . . . . . 6 (𝑗 = 𝑚(𝐾𝑗) / 𝑘𝐵 = (𝐾𝑚) / 𝑘𝐵)
7623, 75ifbieq1d 3556 . . . . 5 (𝑗 = 𝑚 → if(𝑗 ≤ (♯‘𝐴), (𝐾𝑗) / 𝑘𝐵, 1) = if(𝑚 ≤ (♯‘𝐴), (𝐾𝑚) / 𝑘𝐵, 1))
7714ad2antrr 488 . . . . . . . . 9 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 𝐾:(1...𝑁)–1-1-onto𝐴)
78 f1of 5457 . . . . . . . . 9 (𝐾:(1...𝑁)–1-1-onto𝐴𝐾:(1...𝑁)⟶𝐴)
7977, 78syl 14 . . . . . . . 8 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 𝐾:(1...𝑁)⟶𝐴)
8019ad2antrr 488 . . . . . . . . 9 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → (1...𝑀) = (1...𝑁))
8153, 80eleqtrd 2256 . . . . . . . 8 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 𝑚 ∈ (1...𝑁))
8279, 81ffvelcdmd 5648 . . . . . . 7 (((𝜑𝑚 ∈ (ℤ‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → (𝐾𝑚) ∈ 𝐴)
83 nfcsb1v 3090 . . . . . . . . 9 𝑘(𝐾𝑚) / 𝑘𝐵
8483nfel1 2330 . . . . . . . 8 𝑘(𝐾𝑚) / 𝑘𝐵 ∈ ℂ
85 csbeq1a 3066 . . . . . . . . 9 (𝑘 = (𝐾𝑚) → 𝐵 = (𝐾𝑚) / 𝑘𝐵)
8685eleq1d 2246 . . . . . . . 8 (𝑘 = (𝐾𝑚) → (𝐵 ∈ ℂ ↔ (𝐾𝑚) / 𝑘𝐵 ∈ ℂ))
8784, 86rspc 2835 . . . . . . 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 3563 . . . . 5 ((𝜑𝑚 ∈ (ℤ‘1)) → if(𝑚 ≤ (♯‘𝐴), (𝐾𝑚) / 𝑘𝐵, 1) ∈ ℂ)
9073, 76, 29, 89fvmptd3 5605 . . . 4 ((𝜑𝑚 ∈ (ℤ‘1)) → (𝐻𝑚) = if(𝑚 ≤ (♯‘𝐴), (𝐾𝑚) / 𝑘𝐵, 1))
9190, 89eqeltrd 2254 . . 3 ((𝜑𝑚 ∈ (ℤ‘1)) → (𝐻𝑚) ∈ ℂ)
9219f1oeq2d 5453 . . . . . . . . . 10 (𝜑 → (𝐾:(1...𝑀)–1-1-onto𝐴𝐾:(1...𝑁)–1-1-onto𝐴))
9314, 92mpbird 167 . . . . . . . . 9 (𝜑𝐾:(1...𝑀)–1-1-onto𝐴)
94 f1of 5457 . . . . . . . . 9 (𝐾:(1...𝑀)–1-1-onto𝐴𝐾:(1...𝑀)⟶𝐴)
9593, 94syl 14 . . . . . . . 8 (𝜑𝐾:(1...𝑀)⟶𝐴)
96 fvco3 5583 . . . . . . . 8 ((𝐾:(1...𝑀)⟶𝐴𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) = (𝑓‘(𝐾𝑖)))
9795, 96sylan 283 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) = (𝑓‘(𝐾𝑖)))
9897fveq2d 5515 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘((𝑓𝐾)‘𝑖)) = (𝑓‘(𝑓‘(𝐾𝑖))))
9911adantr 276 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑓:(1...𝑀)–1-1-onto𝐴)
10095ffvelcdmda 5647 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) ∈ 𝐴)
101 f1ocnvfv2 5773 . . . . . . 7 ((𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (𝐾𝑖) ∈ 𝐴) → (𝑓‘(𝑓‘(𝐾𝑖))) = (𝐾𝑖))
10299, 100, 101syl2anc 411 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘(𝑓‘(𝐾𝑖))) = (𝐾𝑖))
10398, 102eqtrd 2210 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘((𝑓𝐾)‘𝑖)) = (𝐾𝑖))
104103csbeq1d 3064 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵 = (𝐾𝑖) / 𝑘𝐵)
105 breq1 4003 . . . . . . 7 (𝑗 = ((𝑓𝐾)‘𝑖) → (𝑗 ≤ (♯‘𝐴) ↔ ((𝑓𝐾)‘𝑖) ≤ (♯‘𝐴)))
106 fveq2 5511 . . . . . . . 8 (𝑗 = ((𝑓𝐾)‘𝑖) → (𝑓𝑗) = (𝑓‘((𝑓𝐾)‘𝑖)))
107106csbeq1d 3064 . . . . . . 7 (𝑗 = ((𝑓𝐾)‘𝑖) → (𝑓𝑗) / 𝑘𝐵 = (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵)
108105, 107ifbieq1d 3556 . . . . . 6 (𝑗 = ((𝑓𝐾)‘𝑖) → if(𝑗 ≤ (♯‘𝐴), (𝑓𝑗) / 𝑘𝐵, 1) = if(((𝑓𝐾)‘𝑖) ≤ (♯‘𝐴), (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵, 1))
109 f1of 5457 . . . . . . . . 9 ((𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀) → (𝑓𝐾):(1...𝑀)⟶(1...𝑀))
11021, 109syl 14 . . . . . . . 8 (𝜑 → (𝑓𝐾):(1...𝑀)⟶(1...𝑀))
111110ffvelcdmda 5647 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) ∈ (1...𝑀))
112 elfznn 10040 . . . . . . 7 (((𝑓𝐾)‘𝑖) ∈ (1...𝑀) → ((𝑓𝐾)‘𝑖) ∈ ℕ)
113111, 112syl 14 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) ∈ ℕ)
114 elfzle2 10014 . . . . . . . . . 10 (((𝑓𝐾)‘𝑖) ∈ (1...𝑀) → ((𝑓𝐾)‘𝑖) ≤ 𝑀)
115111, 114syl 14 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) ≤ 𝑀)
11648adantr 276 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑀 = (♯‘𝐴))
117115, 116breqtrd 4026 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) ≤ (♯‘𝐴))
118117iftrued 3541 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → if(((𝑓𝐾)‘𝑖) ≤ (♯‘𝐴), (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵, 1) = (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵)
11956adantr 276 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑀)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
120 nfcsb1v 3090 . . . . . . . . . . 11 𝑘(𝐾𝑖) / 𝑘𝐵
121120nfel1 2330 . . . . . . . . . 10 𝑘(𝐾𝑖) / 𝑘𝐵 ∈ ℂ
122 csbeq1a 3066 . . . . . . . . . . 11 (𝑘 = (𝐾𝑖) → 𝐵 = (𝐾𝑖) / 𝑘𝐵)
123122eleq1d 2246 . . . . . . . . . 10 (𝑘 = (𝐾𝑖) → (𝐵 ∈ ℂ ↔ (𝐾𝑖) / 𝑘𝐵 ∈ ℂ))
124121, 123rspc 2835 . . . . . . . . 9 ((𝐾𝑖) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝐾𝑖) / 𝑘𝐵 ∈ ℂ))
125100, 119, 124sylc 62 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) / 𝑘𝐵 ∈ ℂ)
126104, 125eqeltrd 2254 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵 ∈ ℂ)
127118, 126eqeltrd 2254 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → if(((𝑓𝐾)‘𝑖) ≤ (♯‘𝐴), (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵, 1) ∈ ℂ)
12822, 108, 113, 127fvmptd3 5605 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐺‘((𝑓𝐾)‘𝑖)) = if(((𝑓𝐾)‘𝑖) ≤ (♯‘𝐴), (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵, 1))
129128, 118eqtrd 2210 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐺‘((𝑓𝐾)‘𝑖)) = (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵)
130 breq1 4003 . . . . . . 7 (𝑗 = 𝑖 → (𝑗 ≤ (♯‘𝐴) ↔ 𝑖 ≤ (♯‘𝐴)))
131 fveq2 5511 . . . . . . . 8 (𝑗 = 𝑖 → (𝐾𝑗) = (𝐾𝑖))
132131csbeq1d 3064 . . . . . . 7 (𝑗 = 𝑖(𝐾𝑗) / 𝑘𝐵 = (𝐾𝑖) / 𝑘𝐵)
133130, 132ifbieq1d 3556 . . . . . 6 (𝑗 = 𝑖 → if(𝑗 ≤ (♯‘𝐴), (𝐾𝑗) / 𝑘𝐵, 1) = if(𝑖 ≤ (♯‘𝐴), (𝐾𝑖) / 𝑘𝐵, 1))
134 elfznn 10040 . . . . . . 7 (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℕ)
135134adantl 277 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑖 ∈ ℕ)
136 elfzle2 10014 . . . . . . . . . 10 (𝑖 ∈ (1...𝑀) → 𝑖𝑀)
137136adantl 277 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑖𝑀)
138137, 116breqtrd 4026 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑖 ≤ (♯‘𝐴))
139138iftrued 3541 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → if(𝑖 ≤ (♯‘𝐴), (𝐾𝑖) / 𝑘𝐵, 1) = (𝐾𝑖) / 𝑘𝐵)
140139, 125eqeltrd 2254 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → if(𝑖 ≤ (♯‘𝐴), (𝐾𝑖) / 𝑘𝐵, 1) ∈ ℂ)
14173, 133, 135, 140fvmptd3 5605 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻𝑖) = if(𝑖 ≤ (♯‘𝐴), (𝐾𝑖) / 𝑘𝐵, 1))
142141, 139eqtrd 2210 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻𝑖) = (𝐾𝑖) / 𝑘𝐵)
143104, 129, 1423eqtr4rd 2221 . . 3 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻𝑖) = (𝐺‘((𝑓𝐾)‘𝑖)))
1442, 4, 6, 10, 21, 72, 91, 143seq3f1o 10490 . 2 (𝜑 → (seq1( · , 𝐻)‘𝑀) = (seq1( · , 𝐺)‘𝑀))
14518fveq2d 5515 . 2 (𝜑 → (seq1( · , 𝐻)‘𝑀) = (seq1( · , 𝐻)‘𝑁))
146144, 145eqtr3d 2212 1 (𝜑 → (seq1( · , 𝐺)‘𝑀) = (seq1( · , 𝐻)‘𝑁))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  DECID wdc 834  w3a 978   = wceq 1353  wcel 2148  wral 2455  csb 3057  ifcif 3534   class class class wbr 4000  cmpt 4061  ccnv 4622  ccom 4627  wf 5208  1-1-ontowf1o 5211  cfv 5212  (class class class)co 5869  cc 7800  1c1 7803   · cmul 7807  cle 7983  cn 8908  0cn0 9165  cz 9242  cuz 9517  ...cfz 9995  seqcseq 10431  chash 10739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-1o 6411  df-er 6529  df-en 6735  df-dom 6736  df-fin 6737  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-inn 8909  df-n0 9166  df-z 9243  df-uz 9518  df-fz 9996  df-fzo 10129  df-seqfrec 10432  df-ihash 10740
This theorem is referenced by:  prodmodclem2a  11568  prodmodc  11570
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