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Theorem prodmolem2a 14503
 Description: Lemma for prodmo 14505. (Contributed by Scott Fenton, 4-Dec-2017.)
Hypotheses
Ref Expression
prodmo.1 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))
prodmo.2 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
prodmo.3 𝐺 = (𝑗 ∈ ℕ ↦ (𝑓𝑗) / 𝑘𝐵)
prodmolem2.4 𝐻 = (𝑗 ∈ ℕ ↦ (𝐾𝑗) / 𝑘𝐵)
prodmolem2.5 (𝜑𝑁 ∈ ℕ)
prodmolem2.6 (𝜑𝑀 ∈ ℤ)
prodmolem2.7 (𝜑𝐴 ⊆ (ℤ𝑀))
prodmolem2.8 (𝜑𝑓:(1...𝑁)–1-1-onto𝐴)
prodmolem2.9 (𝜑𝐾 Isom < , < ((1...(#‘𝐴)), 𝐴))
Assertion
Ref Expression
prodmolem2a (𝜑 → seq𝑀( · , 𝐹) ⇝ (seq1( · , 𝐺)‘𝑁))
Distinct variable groups:   𝐴,𝑘   𝑘,𝐹   𝜑,𝑘   𝐴,𝑗   𝐵,𝑗   𝑓,𝑗,𝑘   𝑗,𝐺   𝑗,𝑘,𝜑   𝑗,𝐾,𝑘   𝑗,𝑀,𝑘   𝑗,𝑁,𝑘
Allowed substitution hints:   𝜑(𝑓)   𝐴(𝑓)   𝐵(𝑓,𝑘)   𝐹(𝑓,𝑗)   𝐺(𝑓,𝑘)   𝐻(𝑓,𝑗,𝑘)   𝐾(𝑓)   𝑀(𝑓)   𝑁(𝑓)

Proof of Theorem prodmolem2a
Dummy variables 𝑛 𝑚 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prodmo.1 . . 3 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))
2 prodmo.2 . . 3 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
3 prodmolem2.7 . . . 4 (𝜑𝐴 ⊆ (ℤ𝑀))
4 prodmolem2.9 . . . . . . 7 (𝜑𝐾 Isom < , < ((1...(#‘𝐴)), 𝐴))
5 prodmolem2.8 . . . . . . . . . . . 12 (𝜑𝑓:(1...𝑁)–1-1-onto𝐴)
6 ovex 6577 . . . . . . . . . . . . 13 (1...𝑁) ∈ V
76f1oen 7862 . . . . . . . . . . . 12 (𝑓:(1...𝑁)–1-1-onto𝐴 → (1...𝑁) ≈ 𝐴)
85, 7syl 17 . . . . . . . . . . 11 (𝜑 → (1...𝑁) ≈ 𝐴)
9 fzfid 12634 . . . . . . . . . . . 12 (𝜑 → (1...𝑁) ∈ Fin)
108ensymd 7893 . . . . . . . . . . . . 13 (𝜑𝐴 ≈ (1...𝑁))
11 enfii 8062 . . . . . . . . . . . . 13 (((1...𝑁) ∈ Fin ∧ 𝐴 ≈ (1...𝑁)) → 𝐴 ∈ Fin)
129, 10, 11syl2anc 691 . . . . . . . . . . . 12 (𝜑𝐴 ∈ Fin)
13 hashen 12997 . . . . . . . . . . . 12 (((1...𝑁) ∈ Fin ∧ 𝐴 ∈ Fin) → ((#‘(1...𝑁)) = (#‘𝐴) ↔ (1...𝑁) ≈ 𝐴))
149, 12, 13syl2anc 691 . . . . . . . . . . 11 (𝜑 → ((#‘(1...𝑁)) = (#‘𝐴) ↔ (1...𝑁) ≈ 𝐴))
158, 14mpbird 246 . . . . . . . . . 10 (𝜑 → (#‘(1...𝑁)) = (#‘𝐴))
16 prodmolem2.5 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℕ)
1716nnnn0d 11228 . . . . . . . . . . 11 (𝜑𝑁 ∈ ℕ0)
18 hashfz1 12996 . . . . . . . . . . 11 (𝑁 ∈ ℕ0 → (#‘(1...𝑁)) = 𝑁)
1917, 18syl 17 . . . . . . . . . 10 (𝜑 → (#‘(1...𝑁)) = 𝑁)
2015, 19eqtr3d 2646 . . . . . . . . 9 (𝜑 → (#‘𝐴) = 𝑁)
2120oveq2d 6565 . . . . . . . 8 (𝜑 → (1...(#‘𝐴)) = (1...𝑁))
22 isoeq4 6470 . . . . . . . 8 ((1...(#‘𝐴)) = (1...𝑁) → (𝐾 Isom < , < ((1...(#‘𝐴)), 𝐴) ↔ 𝐾 Isom < , < ((1...𝑁), 𝐴)))
2321, 22syl 17 . . . . . . 7 (𝜑 → (𝐾 Isom < , < ((1...(#‘𝐴)), 𝐴) ↔ 𝐾 Isom < , < ((1...𝑁), 𝐴)))
244, 23mpbid 221 . . . . . 6 (𝜑𝐾 Isom < , < ((1...𝑁), 𝐴))
25 isof1o 6473 . . . . . 6 (𝐾 Isom < , < ((1...𝑁), 𝐴) → 𝐾:(1...𝑁)–1-1-onto𝐴)
26 f1of 6050 . . . . . 6 (𝐾:(1...𝑁)–1-1-onto𝐴𝐾:(1...𝑁)⟶𝐴)
2724, 25, 263syl 18 . . . . 5 (𝜑𝐾:(1...𝑁)⟶𝐴)
28 nnuz 11599 . . . . . . 7 ℕ = (ℤ‘1)
2916, 28syl6eleq 2698 . . . . . 6 (𝜑𝑁 ∈ (ℤ‘1))
30 eluzfz2 12220 . . . . . 6 (𝑁 ∈ (ℤ‘1) → 𝑁 ∈ (1...𝑁))
3129, 30syl 17 . . . . 5 (𝜑𝑁 ∈ (1...𝑁))
3227, 31ffvelrnd 6268 . . . 4 (𝜑 → (𝐾𝑁) ∈ 𝐴)
333, 32sseldd 3569 . . 3 (𝜑 → (𝐾𝑁) ∈ (ℤ𝑀))
343sselda 3568 . . . . . 6 ((𝜑𝑗𝐴) → 𝑗 ∈ (ℤ𝑀))
3524, 25syl 17 . . . . . . . . 9 (𝜑𝐾:(1...𝑁)–1-1-onto𝐴)
36 f1ocnvfv2 6433 . . . . . . . . 9 ((𝐾:(1...𝑁)–1-1-onto𝐴𝑗𝐴) → (𝐾‘(𝐾𝑗)) = 𝑗)
3735, 36sylan 487 . . . . . . . 8 ((𝜑𝑗𝐴) → (𝐾‘(𝐾𝑗)) = 𝑗)
38 f1ocnv 6062 . . . . . . . . . . . 12 (𝐾:(1...𝑁)–1-1-onto𝐴𝐾:𝐴1-1-onto→(1...𝑁))
39 f1of 6050 . . . . . . . . . . . 12 (𝐾:𝐴1-1-onto→(1...𝑁) → 𝐾:𝐴⟶(1...𝑁))
4035, 38, 393syl 18 . . . . . . . . . . 11 (𝜑𝐾:𝐴⟶(1...𝑁))
4140ffvelrnda 6267 . . . . . . . . . 10 ((𝜑𝑗𝐴) → (𝐾𝑗) ∈ (1...𝑁))
42 elfzle2 12216 . . . . . . . . . 10 ((𝐾𝑗) ∈ (1...𝑁) → (𝐾𝑗) ≤ 𝑁)
4341, 42syl 17 . . . . . . . . 9 ((𝜑𝑗𝐴) → (𝐾𝑗) ≤ 𝑁)
4424adantr 480 . . . . . . . . . 10 ((𝜑𝑗𝐴) → 𝐾 Isom < , < ((1...𝑁), 𝐴))
45 fzssuz 12253 . . . . . . . . . . . . 13 (1...𝑁) ⊆ (ℤ‘1)
46 uzssz 11583 . . . . . . . . . . . . . 14 (ℤ‘1) ⊆ ℤ
47 zssre 11261 . . . . . . . . . . . . . 14 ℤ ⊆ ℝ
4846, 47sstri 3577 . . . . . . . . . . . . 13 (ℤ‘1) ⊆ ℝ
4945, 48sstri 3577 . . . . . . . . . . . 12 (1...𝑁) ⊆ ℝ
50 ressxr 9962 . . . . . . . . . . . 12 ℝ ⊆ ℝ*
5149, 50sstri 3577 . . . . . . . . . . 11 (1...𝑁) ⊆ ℝ*
5251a1i 11 . . . . . . . . . 10 ((𝜑𝑗𝐴) → (1...𝑁) ⊆ ℝ*)
53 uzssz 11583 . . . . . . . . . . . . . 14 (ℤ𝑀) ⊆ ℤ
5453, 47sstri 3577 . . . . . . . . . . . . 13 (ℤ𝑀) ⊆ ℝ
5554, 50sstri 3577 . . . . . . . . . . . 12 (ℤ𝑀) ⊆ ℝ*
563, 55syl6ss 3580 . . . . . . . . . . 11 (𝜑𝐴 ⊆ ℝ*)
5756adantr 480 . . . . . . . . . 10 ((𝜑𝑗𝐴) → 𝐴 ⊆ ℝ*)
5831adantr 480 . . . . . . . . . 10 ((𝜑𝑗𝐴) → 𝑁 ∈ (1...𝑁))
59 leisorel 13101 . . . . . . . . . 10 ((𝐾 Isom < , < ((1...𝑁), 𝐴) ∧ ((1...𝑁) ⊆ ℝ*𝐴 ⊆ ℝ*) ∧ ((𝐾𝑗) ∈ (1...𝑁) ∧ 𝑁 ∈ (1...𝑁))) → ((𝐾𝑗) ≤ 𝑁 ↔ (𝐾‘(𝐾𝑗)) ≤ (𝐾𝑁)))
6044, 52, 57, 41, 58, 59syl122anc 1327 . . . . . . . . 9 ((𝜑𝑗𝐴) → ((𝐾𝑗) ≤ 𝑁 ↔ (𝐾‘(𝐾𝑗)) ≤ (𝐾𝑁)))
6143, 60mpbid 221 . . . . . . . 8 ((𝜑𝑗𝐴) → (𝐾‘(𝐾𝑗)) ≤ (𝐾𝑁))
6237, 61eqbrtrrd 4607 . . . . . . 7 ((𝜑𝑗𝐴) → 𝑗 ≤ (𝐾𝑁))
633, 53syl6ss 3580 . . . . . . . . 9 (𝜑𝐴 ⊆ ℤ)
6463sselda 3568 . . . . . . . 8 ((𝜑𝑗𝐴) → 𝑗 ∈ ℤ)
65 eluzelz 11573 . . . . . . . . . 10 ((𝐾𝑁) ∈ (ℤ𝑀) → (𝐾𝑁) ∈ ℤ)
6633, 65syl 17 . . . . . . . . 9 (𝜑 → (𝐾𝑁) ∈ ℤ)
6766adantr 480 . . . . . . . 8 ((𝜑𝑗𝐴) → (𝐾𝑁) ∈ ℤ)
68 eluz 11577 . . . . . . . 8 ((𝑗 ∈ ℤ ∧ (𝐾𝑁) ∈ ℤ) → ((𝐾𝑁) ∈ (ℤ𝑗) ↔ 𝑗 ≤ (𝐾𝑁)))
6964, 67, 68syl2anc 691 . . . . . . 7 ((𝜑𝑗𝐴) → ((𝐾𝑁) ∈ (ℤ𝑗) ↔ 𝑗 ≤ (𝐾𝑁)))
7062, 69mpbird 246 . . . . . 6 ((𝜑𝑗𝐴) → (𝐾𝑁) ∈ (ℤ𝑗))
71 elfzuzb 12207 . . . . . 6 (𝑗 ∈ (𝑀...(𝐾𝑁)) ↔ (𝑗 ∈ (ℤ𝑀) ∧ (𝐾𝑁) ∈ (ℤ𝑗)))
7234, 70, 71sylanbrc 695 . . . . 5 ((𝜑𝑗𝐴) → 𝑗 ∈ (𝑀...(𝐾𝑁)))
7372ex 449 . . . 4 (𝜑 → (𝑗𝐴𝑗 ∈ (𝑀...(𝐾𝑁))))
7473ssrdv 3574 . . 3 (𝜑𝐴 ⊆ (𝑀...(𝐾𝑁)))
751, 2, 33, 74fprodcvg 14499 . 2 (𝜑 → seq𝑀( · , 𝐹) ⇝ (seq𝑀( · , 𝐹)‘(𝐾𝑁)))
76 mulid2 9917 . . . . 5 (𝑚 ∈ ℂ → (1 · 𝑚) = 𝑚)
7776adantl 481 . . . 4 ((𝜑𝑚 ∈ ℂ) → (1 · 𝑚) = 𝑚)
78 mulid1 9916 . . . . 5 (𝑚 ∈ ℂ → (𝑚 · 1) = 𝑚)
7978adantl 481 . . . 4 ((𝜑𝑚 ∈ ℂ) → (𝑚 · 1) = 𝑚)
80 mulcl 9899 . . . . 5 ((𝑚 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑚 · 𝑥) ∈ ℂ)
8180adantl 481 . . . 4 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑚 · 𝑥) ∈ ℂ)
82 1cnd 9935 . . . 4 (𝜑 → 1 ∈ ℂ)
8331, 21eleqtrrd 2691 . . . 4 (𝜑𝑁 ∈ (1...(#‘𝐴)))
84 iftrue 4042 . . . . . . . . . . 11 (𝑘𝐴 → if(𝑘𝐴, 𝐵, 1) = 𝐵)
8584adantl 481 . . . . . . . . . 10 ((𝜑𝑘𝐴) → if(𝑘𝐴, 𝐵, 1) = 𝐵)
8685, 2eqeltrd 2688 . . . . . . . . 9 ((𝜑𝑘𝐴) → if(𝑘𝐴, 𝐵, 1) ∈ ℂ)
8786ex 449 . . . . . . . 8 (𝜑 → (𝑘𝐴 → if(𝑘𝐴, 𝐵, 1) ∈ ℂ))
88 iffalse 4045 . . . . . . . . 9 𝑘𝐴 → if(𝑘𝐴, 𝐵, 1) = 1)
89 ax-1cn 9873 . . . . . . . . 9 1 ∈ ℂ
9088, 89syl6eqel 2696 . . . . . . . 8 𝑘𝐴 → if(𝑘𝐴, 𝐵, 1) ∈ ℂ)
9187, 90pm2.61d1 170 . . . . . . 7 (𝜑 → if(𝑘𝐴, 𝐵, 1) ∈ ℂ)
9291adantr 480 . . . . . 6 ((𝜑𝑘 ∈ ℤ) → if(𝑘𝐴, 𝐵, 1) ∈ ℂ)
9392, 1fmptd 6292 . . . . 5 (𝜑𝐹:ℤ⟶ℂ)
94 elfzelz 12213 . . . . 5 (𝑚 ∈ (𝑀...(𝐾‘(#‘𝐴))) → 𝑚 ∈ ℤ)
95 ffvelrn 6265 . . . . 5 ((𝐹:ℤ⟶ℂ ∧ 𝑚 ∈ ℤ) → (𝐹𝑚) ∈ ℂ)
9693, 94, 95syl2an 493 . . . 4 ((𝜑𝑚 ∈ (𝑀...(𝐾‘(#‘𝐴)))) → (𝐹𝑚) ∈ ℂ)
97 fveq2 6103 . . . . . . 7 (𝑘 = 𝑚 → (𝐹𝑘) = (𝐹𝑚))
9897eqeq1d 2612 . . . . . 6 (𝑘 = 𝑚 → ((𝐹𝑘) = 1 ↔ (𝐹𝑚) = 1))
99 fzssuz 12253 . . . . . . . . . 10 (𝑀...(𝐾‘(#‘𝐴))) ⊆ (ℤ𝑀)
10099, 53sstri 3577 . . . . . . . . 9 (𝑀...(𝐾‘(#‘𝐴))) ⊆ ℤ
101 eldifi 3694 . . . . . . . . 9 (𝑘 ∈ ((𝑀...(𝐾‘(#‘𝐴))) ∖ 𝐴) → 𝑘 ∈ (𝑀...(𝐾‘(#‘𝐴))))
102100, 101sseldi 3566 . . . . . . . 8 (𝑘 ∈ ((𝑀...(𝐾‘(#‘𝐴))) ∖ 𝐴) → 𝑘 ∈ ℤ)
103 eldifn 3695 . . . . . . . . . 10 (𝑘 ∈ ((𝑀...(𝐾‘(#‘𝐴))) ∖ 𝐴) → ¬ 𝑘𝐴)
104103, 88syl 17 . . . . . . . . 9 (𝑘 ∈ ((𝑀...(𝐾‘(#‘𝐴))) ∖ 𝐴) → if(𝑘𝐴, 𝐵, 1) = 1)
105104, 89syl6eqel 2696 . . . . . . . 8 (𝑘 ∈ ((𝑀...(𝐾‘(#‘𝐴))) ∖ 𝐴) → if(𝑘𝐴, 𝐵, 1) ∈ ℂ)
1061fvmpt2 6200 . . . . . . . 8 ((𝑘 ∈ ℤ ∧ if(𝑘𝐴, 𝐵, 1) ∈ ℂ) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 1))
107102, 105, 106syl2anc 691 . . . . . . 7 (𝑘 ∈ ((𝑀...(𝐾‘(#‘𝐴))) ∖ 𝐴) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 1))
108107, 104eqtrd 2644 . . . . . 6 (𝑘 ∈ ((𝑀...(𝐾‘(#‘𝐴))) ∖ 𝐴) → (𝐹𝑘) = 1)
10998, 108vtoclga 3245 . . . . 5 (𝑚 ∈ ((𝑀...(𝐾‘(#‘𝐴))) ∖ 𝐴) → (𝐹𝑚) = 1)
110109adantl 481 . . . 4 ((𝜑𝑚 ∈ ((𝑀...(𝐾‘(#‘𝐴))) ∖ 𝐴)) → (𝐹𝑚) = 1)
111 isof1o 6473 . . . . . . . 8 (𝐾 Isom < , < ((1...(#‘𝐴)), 𝐴) → 𝐾:(1...(#‘𝐴))–1-1-onto𝐴)
112 f1of 6050 . . . . . . . 8 (𝐾:(1...(#‘𝐴))–1-1-onto𝐴𝐾:(1...(#‘𝐴))⟶𝐴)
1134, 111, 1123syl 18 . . . . . . 7 (𝜑𝐾:(1...(#‘𝐴))⟶𝐴)
114113ffvelrnda 6267 . . . . . 6 ((𝜑𝑥 ∈ (1...(#‘𝐴))) → (𝐾𝑥) ∈ 𝐴)
115114iftrued 4044 . . . . 5 ((𝜑𝑥 ∈ (1...(#‘𝐴))) → if((𝐾𝑥) ∈ 𝐴, (𝐾𝑥) / 𝑘𝐵, 1) = (𝐾𝑥) / 𝑘𝐵)
11663adantr 480 . . . . . . 7 ((𝜑𝑥 ∈ (1...(#‘𝐴))) → 𝐴 ⊆ ℤ)
117116, 114sseldd 3569 . . . . . 6 ((𝜑𝑥 ∈ (1...(#‘𝐴))) → (𝐾𝑥) ∈ ℤ)
118 nfv 1830 . . . . . . . . 9 𝑘𝜑
119 nfv 1830 . . . . . . . . . . 11 𝑘(𝐾𝑥) ∈ 𝐴
120 nfcsb1v 3515 . . . . . . . . . . 11 𝑘(𝐾𝑥) / 𝑘𝐵
121 nfcv 2751 . . . . . . . . . . 11 𝑘1
122119, 120, 121nfif 4065 . . . . . . . . . 10 𝑘if((𝐾𝑥) ∈ 𝐴, (𝐾𝑥) / 𝑘𝐵, 1)
123122nfel1 2765 . . . . . . . . 9 𝑘if((𝐾𝑥) ∈ 𝐴, (𝐾𝑥) / 𝑘𝐵, 1) ∈ ℂ
124118, 123nfim 1813 . . . . . . . 8 𝑘(𝜑 → if((𝐾𝑥) ∈ 𝐴, (𝐾𝑥) / 𝑘𝐵, 1) ∈ ℂ)
125 fvex 6113 . . . . . . . 8 (𝐾𝑥) ∈ V
126 eleq1 2676 . . . . . . . . . . 11 (𝑘 = (𝐾𝑥) → (𝑘𝐴 ↔ (𝐾𝑥) ∈ 𝐴))
127 csbeq1a 3508 . . . . . . . . . . 11 (𝑘 = (𝐾𝑥) → 𝐵 = (𝐾𝑥) / 𝑘𝐵)
128126, 127ifbieq1d 4059 . . . . . . . . . 10 (𝑘 = (𝐾𝑥) → if(𝑘𝐴, 𝐵, 1) = if((𝐾𝑥) ∈ 𝐴, (𝐾𝑥) / 𝑘𝐵, 1))
129128eleq1d 2672 . . . . . . . . 9 (𝑘 = (𝐾𝑥) → (if(𝑘𝐴, 𝐵, 1) ∈ ℂ ↔ if((𝐾𝑥) ∈ 𝐴, (𝐾𝑥) / 𝑘𝐵, 1) ∈ ℂ))
130129imbi2d 329 . . . . . . . 8 (𝑘 = (𝐾𝑥) → ((𝜑 → if(𝑘𝐴, 𝐵, 1) ∈ ℂ) ↔ (𝜑 → if((𝐾𝑥) ∈ 𝐴, (𝐾𝑥) / 𝑘𝐵, 1) ∈ ℂ)))
131124, 125, 130, 91vtoclf 3231 . . . . . . 7 (𝜑 → if((𝐾𝑥) ∈ 𝐴, (𝐾𝑥) / 𝑘𝐵, 1) ∈ ℂ)
132131adantr 480 . . . . . 6 ((𝜑𝑥 ∈ (1...(#‘𝐴))) → if((𝐾𝑥) ∈ 𝐴, (𝐾𝑥) / 𝑘𝐵, 1) ∈ ℂ)
133 eleq1 2676 . . . . . . . 8 (𝑛 = (𝐾𝑥) → (𝑛𝐴 ↔ (𝐾𝑥) ∈ 𝐴))
134 csbeq1 3502 . . . . . . . 8 (𝑛 = (𝐾𝑥) → 𝑛 / 𝑘𝐵 = (𝐾𝑥) / 𝑘𝐵)
135133, 134ifbieq1d 4059 . . . . . . 7 (𝑛 = (𝐾𝑥) → if(𝑛𝐴, 𝑛 / 𝑘𝐵, 1) = if((𝐾𝑥) ∈ 𝐴, (𝐾𝑥) / 𝑘𝐵, 1))
136 nfcv 2751 . . . . . . . . 9 𝑛if(𝑘𝐴, 𝐵, 1)
137 nfv 1830 . . . . . . . . . 10 𝑘 𝑛𝐴
138 nfcsb1v 3515 . . . . . . . . . 10 𝑘𝑛 / 𝑘𝐵
139137, 138, 121nfif 4065 . . . . . . . . 9 𝑘if(𝑛𝐴, 𝑛 / 𝑘𝐵, 1)
140 eleq1 2676 . . . . . . . . . 10 (𝑘 = 𝑛 → (𝑘𝐴𝑛𝐴))
141 csbeq1a 3508 . . . . . . . . . 10 (𝑘 = 𝑛𝐵 = 𝑛 / 𝑘𝐵)
142140, 141ifbieq1d 4059 . . . . . . . . 9 (𝑘 = 𝑛 → if(𝑘𝐴, 𝐵, 1) = if(𝑛𝐴, 𝑛 / 𝑘𝐵, 1))
143136, 139, 142cbvmpt 4677 . . . . . . . 8 (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1)) = (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 1))
1441, 143eqtri 2632 . . . . . . 7 𝐹 = (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 1))
145135, 144fvmptg 6189 . . . . . 6 (((𝐾𝑥) ∈ ℤ ∧ if((𝐾𝑥) ∈ 𝐴, (𝐾𝑥) / 𝑘𝐵, 1) ∈ ℂ) → (𝐹‘(𝐾𝑥)) = if((𝐾𝑥) ∈ 𝐴, (𝐾𝑥) / 𝑘𝐵, 1))
146117, 132, 145syl2anc 691 . . . . 5 ((𝜑𝑥 ∈ (1...(#‘𝐴))) → (𝐹‘(𝐾𝑥)) = if((𝐾𝑥) ∈ 𝐴, (𝐾𝑥) / 𝑘𝐵, 1))
147 elfznn 12241 . . . . . . 7 (𝑥 ∈ (1...(#‘𝐴)) → 𝑥 ∈ ℕ)
148147adantl 481 . . . . . 6 ((𝜑𝑥 ∈ (1...(#‘𝐴))) → 𝑥 ∈ ℕ)
149115, 132eqeltrrd 2689 . . . . . 6 ((𝜑𝑥 ∈ (1...(#‘𝐴))) → (𝐾𝑥) / 𝑘𝐵 ∈ ℂ)
150 fveq2 6103 . . . . . . . 8 (𝑗 = 𝑥 → (𝐾𝑗) = (𝐾𝑥))
151150csbeq1d 3506 . . . . . . 7 (𝑗 = 𝑥(𝐾𝑗) / 𝑘𝐵 = (𝐾𝑥) / 𝑘𝐵)
152 prodmolem2.4 . . . . . . 7 𝐻 = (𝑗 ∈ ℕ ↦ (𝐾𝑗) / 𝑘𝐵)
153151, 152fvmptg 6189 . . . . . 6 ((𝑥 ∈ ℕ ∧ (𝐾𝑥) / 𝑘𝐵 ∈ ℂ) → (𝐻𝑥) = (𝐾𝑥) / 𝑘𝐵)
154148, 149, 153syl2anc 691 . . . . 5 ((𝜑𝑥 ∈ (1...(#‘𝐴))) → (𝐻𝑥) = (𝐾𝑥) / 𝑘𝐵)
155115, 146, 1543eqtr4rd 2655 . . . 4 ((𝜑𝑥 ∈ (1...(#‘𝐴))) → (𝐻𝑥) = (𝐹‘(𝐾𝑥)))
15677, 79, 81, 82, 4, 83, 3, 96, 110, 155seqcoll 13105 . . 3 (𝜑 → (seq𝑀( · , 𝐹)‘(𝐾𝑁)) = (seq1( · , 𝐻)‘𝑁))
157 prodmo.3 . . . 4 𝐺 = (𝑗 ∈ ℕ ↦ (𝑓𝑗) / 𝑘𝐵)
15816, 16jca 553 . . . 4 (𝜑 → (𝑁 ∈ ℕ ∧ 𝑁 ∈ ℕ))
1591, 2, 157, 152, 158, 5, 35prodmolem3 14502 . . 3 (𝜑 → (seq1( · , 𝐺)‘𝑁) = (seq1( · , 𝐻)‘𝑁))
160156, 159eqtr4d 2647 . 2 (𝜑 → (seq𝑀( · , 𝐹)‘(𝐾𝑁)) = (seq1( · , 𝐺)‘𝑁))
16175, 160breqtrd 4609 1 (𝜑 → seq𝑀( · , 𝐹) ⇝ (seq1( · , 𝐺)‘𝑁))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ⦋csb 3499   ∖ cdif 3537   ⊆ wss 3540  ifcif 4036   class class class wbr 4583   ↦ cmpt 4643  ◡ccnv 5037  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804   Isom wiso 5805  (class class class)co 6549   ≈ cen 7838  Fincfn 7841  ℂcc 9813  ℝcr 9814  1c1 9816   · cmul 9820  ℝ*cxr 9952   < clt 9953   ≤ cle 9954  ℕcn 10897  ℕ0cn0 11169  ℤcz 11254  ℤ≥cuz 11563  ...cfz 12197  seqcseq 12663  #chash 12979   ⇝ cli 14063 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067 This theorem is referenced by:  prodmolem2  14504  zprod  14506
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