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Theorem summodclem3 11163
Description: Lemma for summodc 11166. (Contributed by Mario Carneiro, 29-Mar-2014.) (Revised by Jim Kingdon, 9-Apr-2023.)
Hypotheses
Ref Expression
isummo.1 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))
isummo.2 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
isummolem3.5 (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
isummolem3.6 (𝜑𝑓:(1...𝑀)–1-1-onto𝐴)
isummolem3.7 (𝜑𝐾:(1...𝑁)–1-1-onto𝐴)
isummolem3.g 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0))
isummolem3.4 𝐻 = (𝑛 ∈ ℕ ↦ if(𝑛𝑁, (𝐾𝑛) / 𝑘𝐵, 0))
Assertion
Ref Expression
summodclem3 (𝜑 → (seq1( + , 𝐺)‘𝑀) = (seq1( + , 𝐻)‘𝑁))
Distinct variable groups:   𝑘,𝑛,𝐴   𝑛,𝐹   𝑘,𝑁,𝑛   𝜑,𝑘,𝑛   𝑘,𝑀,𝑛   𝐵,𝑛   𝑘,𝐾,𝑛   𝑓,𝑘,𝑛
Allowed substitution hints:   𝜑(𝑓)   𝐴(𝑓)   𝐵(𝑓,𝑘)   𝐹(𝑓,𝑘)   𝐺(𝑓,𝑘,𝑛)   𝐻(𝑓,𝑘,𝑛)   𝐾(𝑓)   𝑀(𝑓)   𝑁(𝑓)

Proof of Theorem summodclem3
Dummy variables 𝑖 𝑗 𝑚 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addcl 7759 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 + 𝑗) ∈ ℂ)
21adantl 275 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 + 𝑗) ∈ ℂ)
3 addcom 7913 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 + 𝑗) = (𝑗 + 𝑚))
43adantl 275 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 + 𝑗) = (𝑗 + 𝑚))
5 addass 7764 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑚 + 𝑗) + 𝑦) = (𝑚 + (𝑗 + 𝑦)))
65adantl 275 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → ((𝑚 + 𝑗) + 𝑦) = (𝑚 + (𝑗 + 𝑦)))
7 isummolem3.5 . . . . 5 (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
87simpld 111 . . . 4 (𝜑𝑀 ∈ ℕ)
9 nnuz 9375 . . . 4 ℕ = (ℤ‘1)
108, 9eleqtrdi 2232 . . 3 (𝜑𝑀 ∈ (ℤ‘1))
11 isummolem3.6 . . . . . 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 isummolem3.7 . . . . 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))
177, 11, 14nnf1o 11159 . . . . . . 7 (𝜑𝑁 = 𝑀)
1817eqcomd 2145 . . . . . 6 (𝜑𝑀 = 𝑁)
1918oveq2d 5790 . . . . 5 (𝜑 → (1...𝑀) = (1...𝑁))
20 f1oeq2 5357 . . . . 5 ((1...𝑀) = (1...𝑁) → ((𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀) ↔ (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀)))
2119, 20syl 14 . . . 4 (𝜑 → ((𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀) ↔ (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀)))
2216, 21mpbird 166 . . 3 (𝜑 → (𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀))
23 elnnuz 9376 . . . 4 (𝑚 ∈ ℕ ↔ 𝑚 ∈ (ℤ‘1))
24 isummolem3.g . . . . . . 7 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0))
25 breq1 3932 . . . . . . . 8 (𝑛 = 𝑚 → (𝑛𝑀𝑚𝑀))
26 fveq2 5421 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑓𝑛) = (𝑓𝑚))
2726csbeq1d 3010 . . . . . . . 8 (𝑛 = 𝑚(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑚) / 𝑘𝐵)
2825, 27ifbieq1d 3494 . . . . . . 7 (𝑛 = 𝑚 → if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0) = if(𝑚𝑀, (𝑓𝑚) / 𝑘𝐵, 0))
29 simplr 519 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → 𝑚 ∈ ℕ)
30 elfzle2 9822 . . . . . . . . . 10 (𝑚 ∈ (1...𝑀) → 𝑚𝑀)
3130adantl 275 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → 𝑚𝑀)
3231iftrued 3481 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → if(𝑚𝑀, (𝑓𝑚) / 𝑘𝐵, 0) = (𝑓𝑚) / 𝑘𝐵)
33 f1of 5367 . . . . . . . . . . . 12 (𝑓:(1...𝑀)–1-1-onto𝐴𝑓:(1...𝑀)⟶𝐴)
3411, 33syl 14 . . . . . . . . . . 11 (𝜑𝑓:(1...𝑀)⟶𝐴)
3534ffvelrnda 5555 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...𝑀)) → (𝑓𝑚) ∈ 𝐴)
36 isummo.2 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
3736ralrimiva 2505 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
3837adantr 274 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...𝑀)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
39 nfcsb1v 3035 . . . . . . . . . . . 12 𝑘(𝑓𝑚) / 𝑘𝐵
4039nfel1 2292 . . . . . . . . . . 11 𝑘(𝑓𝑚) / 𝑘𝐵 ∈ ℂ
41 csbeq1a 3012 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑚) → 𝐵 = (𝑓𝑚) / 𝑘𝐵)
4241eleq1d 2208 . . . . . . . . . . 11 (𝑘 = (𝑓𝑚) → (𝐵 ∈ ℂ ↔ (𝑓𝑚) / 𝑘𝐵 ∈ ℂ))
4340, 42rspc 2783 . . . . . . . . . 10 ((𝑓𝑚) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝑓𝑚) / 𝑘𝐵 ∈ ℂ))
4435, 38, 43sylc 62 . . . . . . . . 9 ((𝜑𝑚 ∈ (1...𝑀)) → (𝑓𝑚) / 𝑘𝐵 ∈ ℂ)
4544adantlr 468 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → (𝑓𝑚) / 𝑘𝐵 ∈ ℂ)
4632, 45eqeltrd 2216 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → if(𝑚𝑀, (𝑓𝑚) / 𝑘𝐵, 0) ∈ ℂ)
4724, 28, 29, 46fvmptd3 5514 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → (𝐺𝑚) = if(𝑚𝑀, (𝑓𝑚) / 𝑘𝐵, 0))
4847, 46eqeltrd 2216 . . . . 5 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → (𝐺𝑚) ∈ ℂ)
49 simplr 519 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → 𝑚 ∈ ℕ)
508ad2antrr 479 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → 𝑀 ∈ ℕ)
5150nnzd 9186 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → 𝑀 ∈ ℤ)
52 eluzp1l 9364 . . . . . . . . . . 11 ((𝑀 ∈ ℤ ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → 𝑀 < 𝑚)
5351, 52sylancom 416 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → 𝑀 < 𝑚)
5449nnzd 9186 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → 𝑚 ∈ ℤ)
55 zltnle 9114 . . . . . . . . . . 11 ((𝑀 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑀 < 𝑚 ↔ ¬ 𝑚𝑀))
5651, 54, 55syl2anc 408 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → (𝑀 < 𝑚 ↔ ¬ 𝑚𝑀))
5753, 56mpbid 146 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → ¬ 𝑚𝑀)
5857iffalsed 3484 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → if(𝑚𝑀, (𝑓𝑚) / 𝑘𝐵, 0) = 0)
59 0cn 7772 . . . . . . . 8 0 ∈ ℂ
6058, 59eqeltrdi 2230 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → if(𝑚𝑀, (𝑓𝑚) / 𝑘𝐵, 0) ∈ ℂ)
6124, 28, 49, 60fvmptd3 5514 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → (𝐺𝑚) = if(𝑚𝑀, (𝑓𝑚) / 𝑘𝐵, 0))
6261, 60eqeltrd 2216 . . . . 5 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → (𝐺𝑚) ∈ ℂ)
63 nnsplit 9928 . . . . . . . . 9 (𝑀 ∈ ℕ → ℕ = ((1...𝑀) ∪ (ℤ‘(𝑀 + 1))))
648, 63syl 14 . . . . . . . 8 (𝜑 → ℕ = ((1...𝑀) ∪ (ℤ‘(𝑀 + 1))))
6564eleq2d 2209 . . . . . . 7 (𝜑 → (𝑚 ∈ ℕ ↔ 𝑚 ∈ ((1...𝑀) ∪ (ℤ‘(𝑀 + 1)))))
6665biimpa 294 . . . . . 6 ((𝜑𝑚 ∈ ℕ) → 𝑚 ∈ ((1...𝑀) ∪ (ℤ‘(𝑀 + 1))))
67 elun 3217 . . . . . 6 (𝑚 ∈ ((1...𝑀) ∪ (ℤ‘(𝑀 + 1))) ↔ (𝑚 ∈ (1...𝑀) ∨ 𝑚 ∈ (ℤ‘(𝑀 + 1))))
6866, 67sylib 121 . . . . 5 ((𝜑𝑚 ∈ ℕ) → (𝑚 ∈ (1...𝑀) ∨ 𝑚 ∈ (ℤ‘(𝑀 + 1))))
6948, 62, 68mpjaodan 787 . . . 4 ((𝜑𝑚 ∈ ℕ) → (𝐺𝑚) ∈ ℂ)
7023, 69sylan2br 286 . . 3 ((𝜑𝑚 ∈ (ℤ‘1)) → (𝐺𝑚) ∈ ℂ)
7117oveq2d 5790 . . . . . . . . 9 (𝜑 → (1...𝑁) = (1...𝑀))
7271eleq2d 2209 . . . . . . . 8 (𝜑 → (𝑚 ∈ (1...𝑁) ↔ 𝑚 ∈ (1...𝑀)))
7372adantr 274 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (𝑚 ∈ (1...𝑁) ↔ 𝑚 ∈ (1...𝑀)))
7473pm5.32i 449 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) ↔ ((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)))
75 isummolem3.4 . . . . . . . 8 𝐻 = (𝑛 ∈ ℕ ↦ if(𝑛𝑁, (𝐾𝑛) / 𝑘𝐵, 0))
76 breq1 3932 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑛𝑁𝑚𝑁))
77 fveq2 5421 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝐾𝑛) = (𝐾𝑚))
7877csbeq1d 3010 . . . . . . . . 9 (𝑛 = 𝑚(𝐾𝑛) / 𝑘𝐵 = (𝐾𝑚) / 𝑘𝐵)
7976, 78ifbieq1d 3494 . . . . . . . 8 (𝑛 = 𝑚 → if(𝑛𝑁, (𝐾𝑛) / 𝑘𝐵, 0) = if(𝑚𝑁, (𝐾𝑚) / 𝑘𝐵, 0))
80 simplr 519 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → 𝑚 ∈ ℕ)
81 elfzle2 9822 . . . . . . . . . . 11 (𝑚 ∈ (1...𝑁) → 𝑚𝑁)
8281adantl 275 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → 𝑚𝑁)
8382iftrued 3481 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → if(𝑚𝑁, (𝐾𝑚) / 𝑘𝐵, 0) = (𝐾𝑚) / 𝑘𝐵)
84 f1of 5367 . . . . . . . . . . . . 13 (𝐾:(1...𝑁)–1-1-onto𝐴𝐾:(1...𝑁)⟶𝐴)
8514, 84syl 14 . . . . . . . . . . . 12 (𝜑𝐾:(1...𝑁)⟶𝐴)
8685ffvelrnda 5555 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (1...𝑁)) → (𝐾𝑚) ∈ 𝐴)
8737adantr 274 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (1...𝑁)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
88 nfcsb1v 3035 . . . . . . . . . . . . 13 𝑘(𝐾𝑚) / 𝑘𝐵
8988nfel1 2292 . . . . . . . . . . . 12 𝑘(𝐾𝑚) / 𝑘𝐵 ∈ ℂ
90 csbeq1a 3012 . . . . . . . . . . . . 13 (𝑘 = (𝐾𝑚) → 𝐵 = (𝐾𝑚) / 𝑘𝐵)
9190eleq1d 2208 . . . . . . . . . . . 12 (𝑘 = (𝐾𝑚) → (𝐵 ∈ ℂ ↔ (𝐾𝑚) / 𝑘𝐵 ∈ ℂ))
9289, 91rspc 2783 . . . . . . . . . . 11 ((𝐾𝑚) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝐾𝑚) / 𝑘𝐵 ∈ ℂ))
9386, 87, 92sylc 62 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...𝑁)) → (𝐾𝑚) / 𝑘𝐵 ∈ ℂ)
9493adantlr 468 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (𝐾𝑚) / 𝑘𝐵 ∈ ℂ)
9583, 94eqeltrd 2216 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → if(𝑚𝑁, (𝐾𝑚) / 𝑘𝐵, 0) ∈ ℂ)
9675, 79, 80, 95fvmptd3 5514 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (𝐻𝑚) = if(𝑚𝑁, (𝐾𝑚) / 𝑘𝐵, 0))
9796, 95eqeltrd 2216 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (𝐻𝑚) ∈ ℂ)
9874, 97sylbir 134 . . . . 5 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → (𝐻𝑚) ∈ ℂ)
9917breq2d 3941 . . . . . . . . . . . 12 (𝜑 → (𝑚𝑁𝑚𝑀))
10099notbid 656 . . . . . . . . . . 11 (𝜑 → (¬ 𝑚𝑁 ↔ ¬ 𝑚𝑀))
101100ad2antrr 479 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → (¬ 𝑚𝑁 ↔ ¬ 𝑚𝑀))
10257, 101mpbird 166 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → ¬ 𝑚𝑁)
103102iffalsed 3484 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → if(𝑚𝑁, (𝐾𝑚) / 𝑘𝐵, 0) = 0)
104103, 59eqeltrdi 2230 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → if(𝑚𝑁, (𝐾𝑚) / 𝑘𝐵, 0) ∈ ℂ)
10575, 79, 49, 104fvmptd3 5514 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → (𝐻𝑚) = if(𝑚𝑁, (𝐾𝑚) / 𝑘𝐵, 0))
106105, 104eqeltrd 2216 . . . . 5 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → (𝐻𝑚) ∈ ℂ)
10798, 106, 68mpjaodan 787 . . . 4 ((𝜑𝑚 ∈ ℕ) → (𝐻𝑚) ∈ ℂ)
10823, 107sylan2br 286 . . 3 ((𝜑𝑚 ∈ (ℤ‘1)) → (𝐻𝑚) ∈ ℂ)
109 f1oeq2 5357 . . . . . . . . . . 11 ((1...𝑀) = (1...𝑁) → (𝐾:(1...𝑀)–1-1-onto𝐴𝐾:(1...𝑁)–1-1-onto𝐴))
11019, 109syl 14 . . . . . . . . . 10 (𝜑 → (𝐾:(1...𝑀)–1-1-onto𝐴𝐾:(1...𝑁)–1-1-onto𝐴))
11114, 110mpbird 166 . . . . . . . . 9 (𝜑𝐾:(1...𝑀)–1-1-onto𝐴)
112 f1of 5367 . . . . . . . . 9 (𝐾:(1...𝑀)–1-1-onto𝐴𝐾:(1...𝑀)⟶𝐴)
113111, 112syl 14 . . . . . . . 8 (𝜑𝐾:(1...𝑀)⟶𝐴)
114 fvco3 5492 . . . . . . . 8 ((𝐾:(1...𝑀)⟶𝐴𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) = (𝑓‘(𝐾𝑖)))
115113, 114sylan 281 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) = (𝑓‘(𝐾𝑖)))
116115fveq2d 5425 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘((𝑓𝐾)‘𝑖)) = (𝑓‘(𝑓‘(𝐾𝑖))))
11711adantr 274 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑓:(1...𝑀)–1-1-onto𝐴)
118113ffvelrnda 5555 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) ∈ 𝐴)
119 f1ocnvfv2 5679 . . . . . . 7 ((𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (𝐾𝑖) ∈ 𝐴) → (𝑓‘(𝑓‘(𝐾𝑖))) = (𝐾𝑖))
120117, 118, 119syl2anc 408 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘(𝑓‘(𝐾𝑖))) = (𝐾𝑖))
121116, 120eqtr2d 2173 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) = (𝑓‘((𝑓𝐾)‘𝑖)))
122121csbeq1d 3010 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) / 𝑘𝐵 = (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵)
123 elfznn 9848 . . . . . 6 (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℕ)
124 elfzle2 9822 . . . . . . . . . 10 (𝑖 ∈ (1...𝑀) → 𝑖𝑀)
125124adantl 275 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑖𝑀)
12618breq2d 3941 . . . . . . . . . 10 (𝜑 → (𝑖𝑀𝑖𝑁))
127126adantr 274 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑖𝑀𝑖𝑁))
128125, 127mpbid 146 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑖𝑁)
129128iftrued 3481 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → if(𝑖𝑁, (𝐾𝑖) / 𝑘𝐵, 0) = (𝐾𝑖) / 𝑘𝐵)
13037adantr 274 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
131 nfcsb1v 3035 . . . . . . . . . 10 𝑘(𝐾𝑖) / 𝑘𝐵
132131nfel1 2292 . . . . . . . . 9 𝑘(𝐾𝑖) / 𝑘𝐵 ∈ ℂ
133 csbeq1a 3012 . . . . . . . . . 10 (𝑘 = (𝐾𝑖) → 𝐵 = (𝐾𝑖) / 𝑘𝐵)
134133eleq1d 2208 . . . . . . . . 9 (𝑘 = (𝐾𝑖) → (𝐵 ∈ ℂ ↔ (𝐾𝑖) / 𝑘𝐵 ∈ ℂ))
135132, 134rspc 2783 . . . . . . . 8 ((𝐾𝑖) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝐾𝑖) / 𝑘𝐵 ∈ ℂ))
136118, 130, 135sylc 62 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) / 𝑘𝐵 ∈ ℂ)
137129, 136eqeltrd 2216 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → if(𝑖𝑁, (𝐾𝑖) / 𝑘𝐵, 0) ∈ ℂ)
138 breq1 3932 . . . . . . . 8 (𝑛 = 𝑖 → (𝑛𝑁𝑖𝑁))
139 fveq2 5421 . . . . . . . . 9 (𝑛 = 𝑖 → (𝐾𝑛) = (𝐾𝑖))
140139csbeq1d 3010 . . . . . . . 8 (𝑛 = 𝑖(𝐾𝑛) / 𝑘𝐵 = (𝐾𝑖) / 𝑘𝐵)
141138, 140ifbieq1d 3494 . . . . . . 7 (𝑛 = 𝑖 → if(𝑛𝑁, (𝐾𝑛) / 𝑘𝐵, 0) = if(𝑖𝑁, (𝐾𝑖) / 𝑘𝐵, 0))
142141, 75fvmptg 5497 . . . . . 6 ((𝑖 ∈ ℕ ∧ if(𝑖𝑁, (𝐾𝑖) / 𝑘𝐵, 0) ∈ ℂ) → (𝐻𝑖) = if(𝑖𝑁, (𝐾𝑖) / 𝑘𝐵, 0))
143123, 137, 142syl2an2 583 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻𝑖) = if(𝑖𝑁, (𝐾𝑖) / 𝑘𝐵, 0))
144143, 129eqtrd 2172 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻𝑖) = (𝐾𝑖) / 𝑘𝐵)
145 breq1 3932 . . . . . . 7 (𝑛 = ((𝑓𝐾)‘𝑖) → (𝑛𝑀 ↔ ((𝑓𝐾)‘𝑖) ≤ 𝑀))
146 fveq2 5421 . . . . . . . 8 (𝑛 = ((𝑓𝐾)‘𝑖) → (𝑓𝑛) = (𝑓‘((𝑓𝐾)‘𝑖)))
147146csbeq1d 3010 . . . . . . 7 (𝑛 = ((𝑓𝐾)‘𝑖) → (𝑓𝑛) / 𝑘𝐵 = (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵)
148145, 147ifbieq1d 3494 . . . . . 6 (𝑛 = ((𝑓𝐾)‘𝑖) → if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0) = if(((𝑓𝐾)‘𝑖) ≤ 𝑀, (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵, 0))
149 f1of 5367 . . . . . . . . 9 ((𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀) → (𝑓𝐾):(1...𝑀)⟶(1...𝑀))
15022, 149syl 14 . . . . . . . 8 (𝜑 → (𝑓𝐾):(1...𝑀)⟶(1...𝑀))
151150ffvelrnda 5555 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) ∈ (1...𝑀))
152 elfznn 9848 . . . . . . 7 (((𝑓𝐾)‘𝑖) ∈ (1...𝑀) → ((𝑓𝐾)‘𝑖) ∈ ℕ)
153151, 152syl 14 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) ∈ ℕ)
154 elfzle2 9822 . . . . . . . . . 10 (((𝑓𝐾)‘𝑖) ∈ (1...𝑀) → ((𝑓𝐾)‘𝑖) ≤ 𝑀)
155151, 154syl 14 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) ≤ 𝑀)
156155iftrued 3481 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → if(((𝑓𝐾)‘𝑖) ≤ 𝑀, (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵, 0) = (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵)
157156, 122eqtr4d 2175 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → if(((𝑓𝐾)‘𝑖) ≤ 𝑀, (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵, 0) = (𝐾𝑖) / 𝑘𝐵)
158157, 136eqeltrd 2216 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → if(((𝑓𝐾)‘𝑖) ≤ 𝑀, (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵, 0) ∈ ℂ)
15924, 148, 153, 158fvmptd3 5514 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐺‘((𝑓𝐾)‘𝑖)) = if(((𝑓𝐾)‘𝑖) ≤ 𝑀, (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵, 0))
160159, 156eqtrd 2172 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐺‘((𝑓𝐾)‘𝑖)) = (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵)
161122, 144, 1603eqtr4d 2182 . . 3 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻𝑖) = (𝐺‘((𝑓𝐾)‘𝑖)))
1622, 4, 6, 10, 22, 70, 108, 161seq3f1o 10291 . 2 (𝜑 → (seq1( + , 𝐻)‘𝑀) = (seq1( + , 𝐺)‘𝑀))
16318fveq2d 5425 . 2 (𝜑 → (seq1( + , 𝐻)‘𝑀) = (seq1( + , 𝐻)‘𝑁))
164162, 163eqtr3d 2174 1 (𝜑 → (seq1( + , 𝐺)‘𝑀) = (seq1( + , 𝐻)‘𝑁))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 697  w3a 962   = wceq 1331  wcel 1480  wral 2416  csb 3003  cun 3069  ifcif 3474   class class class wbr 3929  cmpt 3989  ccnv 4538  ccom 4543  wf 5119  1-1-ontowf1o 5122  cfv 5123  (class class class)co 5774  cc 7632  0cc0 7634  1c1 7635   + caddc 7637   < clt 7814  cle 7815  cn 8734  cz 9068  cuz 9340  ...cfz 9804  seqcseq 10232
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 7725  ax-resscn 7726  ax-1cn 7727  ax-1re 7728  ax-icn 7729  ax-addcl 7730  ax-addrcl 7731  ax-mulcl 7732  ax-addcom 7734  ax-addass 7736  ax-distr 7738  ax-i2m1 7739  ax-0lt1 7740  ax-0id 7742  ax-rnegex 7743  ax-cnre 7745  ax-pre-ltirr 7746  ax-pre-ltwlin 7747  ax-pre-lttrn 7748  ax-pre-apti 7749  ax-pre-ltadd 7750
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 7816  df-mnf 7817  df-xr 7818  df-ltxr 7819  df-le 7820  df-sub 7949  df-neg 7950  df-inn 8735  df-n0 8992  df-z 9069  df-uz 9341  df-fz 9805  df-fzo 9934  df-seqfrec 10233  df-ihash 10536
This theorem is referenced by:  summodclem2a  11164  summodc  11166
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