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Theorem summodclem3 11372
Description: Lemma for summodc 11375. (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 7927 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 + 𝑗) ∈ ℂ)
21adantl 277 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 + 𝑗) ∈ ℂ)
3 addcom 8084 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 + 𝑗) = (𝑗 + 𝑚))
43adantl 277 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 + 𝑗) = (𝑗 + 𝑚))
5 addass 7932 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑚 + 𝑗) + 𝑦) = (𝑚 + (𝑗 + 𝑦)))
65adantl 277 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → ((𝑚 + 𝑗) + 𝑦) = (𝑚 + (𝑗 + 𝑦)))
7 isummolem3.5 . . . . 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 isummolem3.6 . . . . . 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 isummolem3.7 . . . . 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))
177, 11, 14nnf1o 11368 . . . . . . 7 (𝜑𝑁 = 𝑀)
1817eqcomd 2183 . . . . . 6 (𝜑𝑀 = 𝑁)
1918oveq2d 5885 . . . . 5 (𝜑 → (1...𝑀) = (1...𝑁))
20 f1oeq2 5446 . . . . 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 167 . . 3 (𝜑 → (𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀))
23 elnnuz 9553 . . . 4 (𝑚 ∈ ℕ ↔ 𝑚 ∈ (ℤ‘1))
24 isummolem3.g . . . . . . 7 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0))
25 breq1 4003 . . . . . . . 8 (𝑛 = 𝑚 → (𝑛𝑀𝑚𝑀))
26 fveq2 5511 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑓𝑛) = (𝑓𝑚))
2726csbeq1d 3064 . . . . . . . 8 (𝑛 = 𝑚(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑚) / 𝑘𝐵)
2825, 27ifbieq1d 3556 . . . . . . 7 (𝑛 = 𝑚 → if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0) = if(𝑚𝑀, (𝑓𝑚) / 𝑘𝐵, 0))
29 simplr 528 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → 𝑚 ∈ ℕ)
30 elfzle2 10014 . . . . . . . . . 10 (𝑚 ∈ (1...𝑀) → 𝑚𝑀)
3130adantl 277 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → 𝑚𝑀)
3231iftrued 3541 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → if(𝑚𝑀, (𝑓𝑚) / 𝑘𝐵, 0) = (𝑓𝑚) / 𝑘𝐵)
33 f1of 5457 . . . . . . . . . . . 12 (𝑓:(1...𝑀)–1-1-onto𝐴𝑓:(1...𝑀)⟶𝐴)
3411, 33syl 14 . . . . . . . . . . 11 (𝜑𝑓:(1...𝑀)⟶𝐴)
3534ffvelcdmda 5647 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...𝑀)) → (𝑓𝑚) ∈ 𝐴)
36 isummo.2 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
3736ralrimiva 2550 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
3837adantr 276 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...𝑀)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
39 nfcsb1v 3090 . . . . . . . . . . . 12 𝑘(𝑓𝑚) / 𝑘𝐵
4039nfel1 2330 . . . . . . . . . . 11 𝑘(𝑓𝑚) / 𝑘𝐵 ∈ ℂ
41 csbeq1a 3066 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑚) → 𝐵 = (𝑓𝑚) / 𝑘𝐵)
4241eleq1d 2246 . . . . . . . . . . 11 (𝑘 = (𝑓𝑚) → (𝐵 ∈ ℂ ↔ (𝑓𝑚) / 𝑘𝐵 ∈ ℂ))
4340, 42rspc 2835 . . . . . . . . . 10 ((𝑓𝑚) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝑓𝑚) / 𝑘𝐵 ∈ ℂ))
4435, 38, 43sylc 62 . . . . . . . . 9 ((𝜑𝑚 ∈ (1...𝑀)) → (𝑓𝑚) / 𝑘𝐵 ∈ ℂ)
4544adantlr 477 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → (𝑓𝑚) / 𝑘𝐵 ∈ ℂ)
4632, 45eqeltrd 2254 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → if(𝑚𝑀, (𝑓𝑚) / 𝑘𝐵, 0) ∈ ℂ)
4724, 28, 29, 46fvmptd3 5605 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → (𝐺𝑚) = if(𝑚𝑀, (𝑓𝑚) / 𝑘𝐵, 0))
4847, 46eqeltrd 2254 . . . . 5 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → (𝐺𝑚) ∈ ℂ)
49 simplr 528 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → 𝑚 ∈ ℕ)
508ad2antrr 488 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → 𝑀 ∈ ℕ)
5150nnzd 9363 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → 𝑀 ∈ ℤ)
52 eluzp1l 9541 . . . . . . . . . . 11 ((𝑀 ∈ ℤ ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → 𝑀 < 𝑚)
5351, 52sylancom 420 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → 𝑀 < 𝑚)
5449nnzd 9363 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → 𝑚 ∈ ℤ)
55 zltnle 9288 . . . . . . . . . . 11 ((𝑀 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑀 < 𝑚 ↔ ¬ 𝑚𝑀))
5651, 54, 55syl2anc 411 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → (𝑀 < 𝑚 ↔ ¬ 𝑚𝑀))
5753, 56mpbid 147 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → ¬ 𝑚𝑀)
5857iffalsed 3544 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → if(𝑚𝑀, (𝑓𝑚) / 𝑘𝐵, 0) = 0)
59 0cn 7940 . . . . . . . 8 0 ∈ ℂ
6058, 59eqeltrdi 2268 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → if(𝑚𝑀, (𝑓𝑚) / 𝑘𝐵, 0) ∈ ℂ)
6124, 28, 49, 60fvmptd3 5605 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → (𝐺𝑚) = if(𝑚𝑀, (𝑓𝑚) / 𝑘𝐵, 0))
6261, 60eqeltrd 2254 . . . . 5 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → (𝐺𝑚) ∈ ℂ)
63 nnsplit 10123 . . . . . . . . 9 (𝑀 ∈ ℕ → ℕ = ((1...𝑀) ∪ (ℤ‘(𝑀 + 1))))
648, 63syl 14 . . . . . . . 8 (𝜑 → ℕ = ((1...𝑀) ∪ (ℤ‘(𝑀 + 1))))
6564eleq2d 2247 . . . . . . 7 (𝜑 → (𝑚 ∈ ℕ ↔ 𝑚 ∈ ((1...𝑀) ∪ (ℤ‘(𝑀 + 1)))))
6665biimpa 296 . . . . . 6 ((𝜑𝑚 ∈ ℕ) → 𝑚 ∈ ((1...𝑀) ∪ (ℤ‘(𝑀 + 1))))
67 elun 3276 . . . . . 6 (𝑚 ∈ ((1...𝑀) ∪ (ℤ‘(𝑀 + 1))) ↔ (𝑚 ∈ (1...𝑀) ∨ 𝑚 ∈ (ℤ‘(𝑀 + 1))))
6866, 67sylib 122 . . . . 5 ((𝜑𝑚 ∈ ℕ) → (𝑚 ∈ (1...𝑀) ∨ 𝑚 ∈ (ℤ‘(𝑀 + 1))))
6948, 62, 68mpjaodan 798 . . . 4 ((𝜑𝑚 ∈ ℕ) → (𝐺𝑚) ∈ ℂ)
7023, 69sylan2br 288 . . 3 ((𝜑𝑚 ∈ (ℤ‘1)) → (𝐺𝑚) ∈ ℂ)
7117oveq2d 5885 . . . . . . . . 9 (𝜑 → (1...𝑁) = (1...𝑀))
7271eleq2d 2247 . . . . . . . 8 (𝜑 → (𝑚 ∈ (1...𝑁) ↔ 𝑚 ∈ (1...𝑀)))
7372adantr 276 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (𝑚 ∈ (1...𝑁) ↔ 𝑚 ∈ (1...𝑀)))
7473pm5.32i 454 . . . . . 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 4003 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑛𝑁𝑚𝑁))
77 fveq2 5511 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝐾𝑛) = (𝐾𝑚))
7877csbeq1d 3064 . . . . . . . . 9 (𝑛 = 𝑚(𝐾𝑛) / 𝑘𝐵 = (𝐾𝑚) / 𝑘𝐵)
7976, 78ifbieq1d 3556 . . . . . . . 8 (𝑛 = 𝑚 → if(𝑛𝑁, (𝐾𝑛) / 𝑘𝐵, 0) = if(𝑚𝑁, (𝐾𝑚) / 𝑘𝐵, 0))
80 simplr 528 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → 𝑚 ∈ ℕ)
81 elfzle2 10014 . . . . . . . . . . 11 (𝑚 ∈ (1...𝑁) → 𝑚𝑁)
8281adantl 277 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → 𝑚𝑁)
8382iftrued 3541 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → if(𝑚𝑁, (𝐾𝑚) / 𝑘𝐵, 0) = (𝐾𝑚) / 𝑘𝐵)
84 f1of 5457 . . . . . . . . . . . . 13 (𝐾:(1...𝑁)–1-1-onto𝐴𝐾:(1...𝑁)⟶𝐴)
8514, 84syl 14 . . . . . . . . . . . 12 (𝜑𝐾:(1...𝑁)⟶𝐴)
8685ffvelcdmda 5647 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (1...𝑁)) → (𝐾𝑚) ∈ 𝐴)
8737adantr 276 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (1...𝑁)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
88 nfcsb1v 3090 . . . . . . . . . . . . 13 𝑘(𝐾𝑚) / 𝑘𝐵
8988nfel1 2330 . . . . . . . . . . . 12 𝑘(𝐾𝑚) / 𝑘𝐵 ∈ ℂ
90 csbeq1a 3066 . . . . . . . . . . . . 13 (𝑘 = (𝐾𝑚) → 𝐵 = (𝐾𝑚) / 𝑘𝐵)
9190eleq1d 2246 . . . . . . . . . . . 12 (𝑘 = (𝐾𝑚) → (𝐵 ∈ ℂ ↔ (𝐾𝑚) / 𝑘𝐵 ∈ ℂ))
9289, 91rspc 2835 . . . . . . . . . . 11 ((𝐾𝑚) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝐾𝑚) / 𝑘𝐵 ∈ ℂ))
9386, 87, 92sylc 62 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...𝑁)) → (𝐾𝑚) / 𝑘𝐵 ∈ ℂ)
9493adantlr 477 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (𝐾𝑚) / 𝑘𝐵 ∈ ℂ)
9583, 94eqeltrd 2254 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → if(𝑚𝑁, (𝐾𝑚) / 𝑘𝐵, 0) ∈ ℂ)
9675, 79, 80, 95fvmptd3 5605 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (𝐻𝑚) = if(𝑚𝑁, (𝐾𝑚) / 𝑘𝐵, 0))
9796, 95eqeltrd 2254 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (𝐻𝑚) ∈ ℂ)
9874, 97sylbir 135 . . . . 5 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → (𝐻𝑚) ∈ ℂ)
9917breq2d 4012 . . . . . . . . . . . 12 (𝜑 → (𝑚𝑁𝑚𝑀))
10099notbid 667 . . . . . . . . . . 11 (𝜑 → (¬ 𝑚𝑁 ↔ ¬ 𝑚𝑀))
101100ad2antrr 488 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → (¬ 𝑚𝑁 ↔ ¬ 𝑚𝑀))
10257, 101mpbird 167 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → ¬ 𝑚𝑁)
103102iffalsed 3544 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → if(𝑚𝑁, (𝐾𝑚) / 𝑘𝐵, 0) = 0)
104103, 59eqeltrdi 2268 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → if(𝑚𝑁, (𝐾𝑚) / 𝑘𝐵, 0) ∈ ℂ)
10575, 79, 49, 104fvmptd3 5605 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → (𝐻𝑚) = if(𝑚𝑁, (𝐾𝑚) / 𝑘𝐵, 0))
106105, 104eqeltrd 2254 . . . . 5 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → (𝐻𝑚) ∈ ℂ)
10798, 106, 68mpjaodan 798 . . . 4 ((𝜑𝑚 ∈ ℕ) → (𝐻𝑚) ∈ ℂ)
10823, 107sylan2br 288 . . 3 ((𝜑𝑚 ∈ (ℤ‘1)) → (𝐻𝑚) ∈ ℂ)
109 f1oeq2 5446 . . . . . . . . . . 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 167 . . . . . . . . 9 (𝜑𝐾:(1...𝑀)–1-1-onto𝐴)
112 f1of 5457 . . . . . . . . 9 (𝐾:(1...𝑀)–1-1-onto𝐴𝐾:(1...𝑀)⟶𝐴)
113111, 112syl 14 . . . . . . . 8 (𝜑𝐾:(1...𝑀)⟶𝐴)
114 fvco3 5583 . . . . . . . 8 ((𝐾:(1...𝑀)⟶𝐴𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) = (𝑓‘(𝐾𝑖)))
115113, 114sylan 283 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) = (𝑓‘(𝐾𝑖)))
116115fveq2d 5515 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘((𝑓𝐾)‘𝑖)) = (𝑓‘(𝑓‘(𝐾𝑖))))
11711adantr 276 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑓:(1...𝑀)–1-1-onto𝐴)
118113ffvelcdmda 5647 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) ∈ 𝐴)
119 f1ocnvfv2 5773 . . . . . . 7 ((𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (𝐾𝑖) ∈ 𝐴) → (𝑓‘(𝑓‘(𝐾𝑖))) = (𝐾𝑖))
120117, 118, 119syl2anc 411 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘(𝑓‘(𝐾𝑖))) = (𝐾𝑖))
121116, 120eqtr2d 2211 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) = (𝑓‘((𝑓𝐾)‘𝑖)))
122121csbeq1d 3064 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) / 𝑘𝐵 = (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵)
123 elfznn 10040 . . . . . 6 (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℕ)
124 elfzle2 10014 . . . . . . . . . 10 (𝑖 ∈ (1...𝑀) → 𝑖𝑀)
125124adantl 277 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑖𝑀)
12618breq2d 4012 . . . . . . . . . 10 (𝜑 → (𝑖𝑀𝑖𝑁))
127126adantr 276 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑖𝑀𝑖𝑁))
128125, 127mpbid 147 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑖𝑁)
129128iftrued 3541 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → if(𝑖𝑁, (𝐾𝑖) / 𝑘𝐵, 0) = (𝐾𝑖) / 𝑘𝐵)
13037adantr 276 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
131 nfcsb1v 3090 . . . . . . . . . 10 𝑘(𝐾𝑖) / 𝑘𝐵
132131nfel1 2330 . . . . . . . . 9 𝑘(𝐾𝑖) / 𝑘𝐵 ∈ ℂ
133 csbeq1a 3066 . . . . . . . . . 10 (𝑘 = (𝐾𝑖) → 𝐵 = (𝐾𝑖) / 𝑘𝐵)
134133eleq1d 2246 . . . . . . . . 9 (𝑘 = (𝐾𝑖) → (𝐵 ∈ ℂ ↔ (𝐾𝑖) / 𝑘𝐵 ∈ ℂ))
135132, 134rspc 2835 . . . . . . . 8 ((𝐾𝑖) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝐾𝑖) / 𝑘𝐵 ∈ ℂ))
136118, 130, 135sylc 62 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) / 𝑘𝐵 ∈ ℂ)
137129, 136eqeltrd 2254 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → if(𝑖𝑁, (𝐾𝑖) / 𝑘𝐵, 0) ∈ ℂ)
138 breq1 4003 . . . . . . . 8 (𝑛 = 𝑖 → (𝑛𝑁𝑖𝑁))
139 fveq2 5511 . . . . . . . . 9 (𝑛 = 𝑖 → (𝐾𝑛) = (𝐾𝑖))
140139csbeq1d 3064 . . . . . . . 8 (𝑛 = 𝑖(𝐾𝑛) / 𝑘𝐵 = (𝐾𝑖) / 𝑘𝐵)
141138, 140ifbieq1d 3556 . . . . . . 7 (𝑛 = 𝑖 → if(𝑛𝑁, (𝐾𝑛) / 𝑘𝐵, 0) = if(𝑖𝑁, (𝐾𝑖) / 𝑘𝐵, 0))
142141, 75fvmptg 5588 . . . . . 6 ((𝑖 ∈ ℕ ∧ if(𝑖𝑁, (𝐾𝑖) / 𝑘𝐵, 0) ∈ ℂ) → (𝐻𝑖) = if(𝑖𝑁, (𝐾𝑖) / 𝑘𝐵, 0))
143123, 137, 142syl2an2 594 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻𝑖) = if(𝑖𝑁, (𝐾𝑖) / 𝑘𝐵, 0))
144143, 129eqtrd 2210 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻𝑖) = (𝐾𝑖) / 𝑘𝐵)
145 breq1 4003 . . . . . . 7 (𝑛 = ((𝑓𝐾)‘𝑖) → (𝑛𝑀 ↔ ((𝑓𝐾)‘𝑖) ≤ 𝑀))
146 fveq2 5511 . . . . . . . 8 (𝑛 = ((𝑓𝐾)‘𝑖) → (𝑓𝑛) = (𝑓‘((𝑓𝐾)‘𝑖)))
147146csbeq1d 3064 . . . . . . 7 (𝑛 = ((𝑓𝐾)‘𝑖) → (𝑓𝑛) / 𝑘𝐵 = (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵)
148145, 147ifbieq1d 3556 . . . . . 6 (𝑛 = ((𝑓𝐾)‘𝑖) → if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0) = if(((𝑓𝐾)‘𝑖) ≤ 𝑀, (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵, 0))
149 f1of 5457 . . . . . . . . 9 ((𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀) → (𝑓𝐾):(1...𝑀)⟶(1...𝑀))
15022, 149syl 14 . . . . . . . 8 (𝜑 → (𝑓𝐾):(1...𝑀)⟶(1...𝑀))
151150ffvelcdmda 5647 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) ∈ (1...𝑀))
152 elfznn 10040 . . . . . . 7 (((𝑓𝐾)‘𝑖) ∈ (1...𝑀) → ((𝑓𝐾)‘𝑖) ∈ ℕ)
153151, 152syl 14 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) ∈ ℕ)
154 elfzle2 10014 . . . . . . . . . 10 (((𝑓𝐾)‘𝑖) ∈ (1...𝑀) → ((𝑓𝐾)‘𝑖) ≤ 𝑀)
155151, 154syl 14 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) ≤ 𝑀)
156155iftrued 3541 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → if(((𝑓𝐾)‘𝑖) ≤ 𝑀, (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵, 0) = (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵)
157156, 122eqtr4d 2213 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → if(((𝑓𝐾)‘𝑖) ≤ 𝑀, (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵, 0) = (𝐾𝑖) / 𝑘𝐵)
158157, 136eqeltrd 2254 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → if(((𝑓𝐾)‘𝑖) ≤ 𝑀, (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵, 0) ∈ ℂ)
15924, 148, 153, 158fvmptd3 5605 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐺‘((𝑓𝐾)‘𝑖)) = if(((𝑓𝐾)‘𝑖) ≤ 𝑀, (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵, 0))
160159, 156eqtrd 2210 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐺‘((𝑓𝐾)‘𝑖)) = (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵)
161122, 144, 1603eqtr4d 2220 . . 3 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻𝑖) = (𝐺‘((𝑓𝐾)‘𝑖)))
1622, 4, 6, 10, 22, 70, 108, 161seq3f1o 10490 . 2 (𝜑 → (seq1( + , 𝐻)‘𝑀) = (seq1( + , 𝐺)‘𝑀))
16318fveq2d 5515 . 2 (𝜑 → (seq1( + , 𝐻)‘𝑀) = (seq1( + , 𝐻)‘𝑁))
164162, 163eqtr3d 2212 1 (𝜑 → (seq1( + , 𝐺)‘𝑀) = (seq1( + , 𝐻)‘𝑁))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  w3a 978   = wceq 1353  wcel 2148  wral 2455  csb 3057  cun 3127  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  0cc0 7802  1c1 7803   + caddc 7805   < clt 7982  cle 7983  cn 8908  cz 9242  cuz 9517  ...cfz 9995  seqcseq 10431
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-addass 7904  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:  summodclem2a  11373  summodc  11375
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