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Theorem summodclem3 10988
 Description: Lemma for summodc 10991. (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 7617 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 + 𝑗) ∈ ℂ)
21adantl 273 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 + 𝑗) ∈ ℂ)
3 addcom 7770 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 + 𝑗) = (𝑗 + 𝑚))
43adantl 273 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 + 𝑗) = (𝑗 + 𝑚))
5 addass 7622 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑚 + 𝑗) + 𝑦) = (𝑚 + (𝑗 + 𝑦)))
65adantl 273 . . 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 9211 . . . 4 ℕ = (ℤ‘1)
108, 9syl6eleq 2192 . . 3 (𝜑𝑀 ∈ (ℤ‘1))
11 isummolem3.6 . . . . . 6 (𝜑𝑓:(1...𝑀)–1-1-onto𝐴)
12 f1ocnv 5314 . . . . . 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 5324 . . . . 5 ((𝑓:𝐴1-1-onto→(1...𝑀) ∧ 𝐾:(1...𝑁)–1-1-onto𝐴) → (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
1613, 14, 15syl2anc 406 . . . 4 (𝜑 → (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
17 isummo.1 . . . . . . . 8 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))
18 isummo.2 . . . . . . . 8 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
1917, 18, 7, 11, 14isummolemnm 10987 . . . . . . 7 (𝜑𝑁 = 𝑀)
2019eqcomd 2105 . . . . . 6 (𝜑𝑀 = 𝑁)
2120oveq2d 5722 . . . . 5 (𝜑 → (1...𝑀) = (1...𝑁))
22 f1oeq2 5293 . . . . 5 ((1...𝑀) = (1...𝑁) → ((𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀) ↔ (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀)))
2321, 22syl 14 . . . 4 (𝜑 → ((𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀) ↔ (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀)))
2416, 23mpbird 166 . . 3 (𝜑 → (𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀))
25 elnnuz 9212 . . . 4 (𝑚 ∈ ℕ ↔ 𝑚 ∈ (ℤ‘1))
26 isummolem3.g . . . . . . 7 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0))
27 breq1 3878 . . . . . . . 8 (𝑛 = 𝑚 → (𝑛𝑀𝑚𝑀))
28 fveq2 5353 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑓𝑛) = (𝑓𝑚))
2928csbeq1d 2961 . . . . . . . 8 (𝑛 = 𝑚(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑚) / 𝑘𝐵)
3027, 29ifbieq1d 3441 . . . . . . 7 (𝑛 = 𝑚 → if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0) = if(𝑚𝑀, (𝑓𝑚) / 𝑘𝐵, 0))
31 simplr 500 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → 𝑚 ∈ ℕ)
32 elfzle2 9649 . . . . . . . . . 10 (𝑚 ∈ (1...𝑀) → 𝑚𝑀)
3332adantl 273 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → 𝑚𝑀)
3433iftrued 3428 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → if(𝑚𝑀, (𝑓𝑚) / 𝑘𝐵, 0) = (𝑓𝑚) / 𝑘𝐵)
35 f1of 5301 . . . . . . . . . . . 12 (𝑓:(1...𝑀)–1-1-onto𝐴𝑓:(1...𝑀)⟶𝐴)
3611, 35syl 14 . . . . . . . . . . 11 (𝜑𝑓:(1...𝑀)⟶𝐴)
3736ffvelrnda 5487 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...𝑀)) → (𝑓𝑚) ∈ 𝐴)
3818ralrimiva 2464 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
3938adantr 272 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...𝑀)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
40 nfcsb1v 2985 . . . . . . . . . . . 12 𝑘(𝑓𝑚) / 𝑘𝐵
4140nfel1 2251 . . . . . . . . . . 11 𝑘(𝑓𝑚) / 𝑘𝐵 ∈ ℂ
42 csbeq1a 2963 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑚) → 𝐵 = (𝑓𝑚) / 𝑘𝐵)
4342eleq1d 2168 . . . . . . . . . . 11 (𝑘 = (𝑓𝑚) → (𝐵 ∈ ℂ ↔ (𝑓𝑚) / 𝑘𝐵 ∈ ℂ))
4441, 43rspc 2738 . . . . . . . . . 10 ((𝑓𝑚) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝑓𝑚) / 𝑘𝐵 ∈ ℂ))
4537, 39, 44sylc 62 . . . . . . . . 9 ((𝜑𝑚 ∈ (1...𝑀)) → (𝑓𝑚) / 𝑘𝐵 ∈ ℂ)
4645adantlr 464 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → (𝑓𝑚) / 𝑘𝐵 ∈ ℂ)
4734, 46eqeltrd 2176 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → if(𝑚𝑀, (𝑓𝑚) / 𝑘𝐵, 0) ∈ ℂ)
4826, 30, 31, 47fvmptd3 5446 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → (𝐺𝑚) = if(𝑚𝑀, (𝑓𝑚) / 𝑘𝐵, 0))
4948, 47eqeltrd 2176 . . . . 5 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → (𝐺𝑚) ∈ ℂ)
50 simplr 500 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → 𝑚 ∈ ℕ)
518ad2antrr 475 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → 𝑀 ∈ ℕ)
5251nnzd 9024 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → 𝑀 ∈ ℤ)
53 eluzp1l 9200 . . . . . . . . . . 11 ((𝑀 ∈ ℤ ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → 𝑀 < 𝑚)
5452, 53sylancom 414 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → 𝑀 < 𝑚)
5550nnzd 9024 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → 𝑚 ∈ ℤ)
56 zltnle 8952 . . . . . . . . . . 11 ((𝑀 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑀 < 𝑚 ↔ ¬ 𝑚𝑀))
5752, 55, 56syl2anc 406 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → (𝑀 < 𝑚 ↔ ¬ 𝑚𝑀))
5854, 57mpbid 146 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → ¬ 𝑚𝑀)
5958iffalsed 3431 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → if(𝑚𝑀, (𝑓𝑚) / 𝑘𝐵, 0) = 0)
60 0cn 7630 . . . . . . . 8 0 ∈ ℂ
6159, 60syl6eqel 2190 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → if(𝑚𝑀, (𝑓𝑚) / 𝑘𝐵, 0) ∈ ℂ)
6226, 30, 50, 61fvmptd3 5446 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → (𝐺𝑚) = if(𝑚𝑀, (𝑓𝑚) / 𝑘𝐵, 0))
6362, 61eqeltrd 2176 . . . . 5 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → (𝐺𝑚) ∈ ℂ)
64 nnsplit 9755 . . . . . . . . 9 (𝑀 ∈ ℕ → ℕ = ((1...𝑀) ∪ (ℤ‘(𝑀 + 1))))
658, 64syl 14 . . . . . . . 8 (𝜑 → ℕ = ((1...𝑀) ∪ (ℤ‘(𝑀 + 1))))
6665eleq2d 2169 . . . . . . 7 (𝜑 → (𝑚 ∈ ℕ ↔ 𝑚 ∈ ((1...𝑀) ∪ (ℤ‘(𝑀 + 1)))))
6766biimpa 292 . . . . . 6 ((𝜑𝑚 ∈ ℕ) → 𝑚 ∈ ((1...𝑀) ∪ (ℤ‘(𝑀 + 1))))
68 elun 3164 . . . . . 6 (𝑚 ∈ ((1...𝑀) ∪ (ℤ‘(𝑀 + 1))) ↔ (𝑚 ∈ (1...𝑀) ∨ 𝑚 ∈ (ℤ‘(𝑀 + 1))))
6967, 68sylib 121 . . . . 5 ((𝜑𝑚 ∈ ℕ) → (𝑚 ∈ (1...𝑀) ∨ 𝑚 ∈ (ℤ‘(𝑀 + 1))))
7049, 63, 69mpjaodan 753 . . . 4 ((𝜑𝑚 ∈ ℕ) → (𝐺𝑚) ∈ ℂ)
7125, 70sylan2br 284 . . 3 ((𝜑𝑚 ∈ (ℤ‘1)) → (𝐺𝑚) ∈ ℂ)
7219oveq2d 5722 . . . . . . . . 9 (𝜑 → (1...𝑁) = (1...𝑀))
7372eleq2d 2169 . . . . . . . 8 (𝜑 → (𝑚 ∈ (1...𝑁) ↔ 𝑚 ∈ (1...𝑀)))
7473adantr 272 . . . . . . 7 ((𝜑𝑚 ∈ ℕ) → (𝑚 ∈ (1...𝑁) ↔ 𝑚 ∈ (1...𝑀)))
7574pm5.32i 445 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) ↔ ((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)))
76 isummolem3.4 . . . . . . . 8 𝐻 = (𝑛 ∈ ℕ ↦ if(𝑛𝑁, (𝐾𝑛) / 𝑘𝐵, 0))
77 breq1 3878 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑛𝑁𝑚𝑁))
78 fveq2 5353 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝐾𝑛) = (𝐾𝑚))
7978csbeq1d 2961 . . . . . . . . 9 (𝑛 = 𝑚(𝐾𝑛) / 𝑘𝐵 = (𝐾𝑚) / 𝑘𝐵)
8077, 79ifbieq1d 3441 . . . . . . . 8 (𝑛 = 𝑚 → if(𝑛𝑁, (𝐾𝑛) / 𝑘𝐵, 0) = if(𝑚𝑁, (𝐾𝑚) / 𝑘𝐵, 0))
81 simplr 500 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → 𝑚 ∈ ℕ)
82 elfzle2 9649 . . . . . . . . . . 11 (𝑚 ∈ (1...𝑁) → 𝑚𝑁)
8382adantl 273 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → 𝑚𝑁)
8483iftrued 3428 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → if(𝑚𝑁, (𝐾𝑚) / 𝑘𝐵, 0) = (𝐾𝑚) / 𝑘𝐵)
85 f1of 5301 . . . . . . . . . . . . 13 (𝐾:(1...𝑁)–1-1-onto𝐴𝐾:(1...𝑁)⟶𝐴)
8614, 85syl 14 . . . . . . . . . . . 12 (𝜑𝐾:(1...𝑁)⟶𝐴)
8786ffvelrnda 5487 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (1...𝑁)) → (𝐾𝑚) ∈ 𝐴)
8838adantr 272 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (1...𝑁)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
89 nfcsb1v 2985 . . . . . . . . . . . . 13 𝑘(𝐾𝑚) / 𝑘𝐵
9089nfel1 2251 . . . . . . . . . . . 12 𝑘(𝐾𝑚) / 𝑘𝐵 ∈ ℂ
91 csbeq1a 2963 . . . . . . . . . . . . 13 (𝑘 = (𝐾𝑚) → 𝐵 = (𝐾𝑚) / 𝑘𝐵)
9291eleq1d 2168 . . . . . . . . . . . 12 (𝑘 = (𝐾𝑚) → (𝐵 ∈ ℂ ↔ (𝐾𝑚) / 𝑘𝐵 ∈ ℂ))
9390, 92rspc 2738 . . . . . . . . . . 11 ((𝐾𝑚) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝐾𝑚) / 𝑘𝐵 ∈ ℂ))
9487, 88, 93sylc 62 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1...𝑁)) → (𝐾𝑚) / 𝑘𝐵 ∈ ℂ)
9594adantlr 464 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (𝐾𝑚) / 𝑘𝐵 ∈ ℂ)
9684, 95eqeltrd 2176 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → if(𝑚𝑁, (𝐾𝑚) / 𝑘𝐵, 0) ∈ ℂ)
9776, 80, 81, 96fvmptd3 5446 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (𝐻𝑚) = if(𝑚𝑁, (𝐾𝑚) / 𝑘𝐵, 0))
9897, 96eqeltrd 2176 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (𝐻𝑚) ∈ ℂ)
9975, 98sylbir 134 . . . . 5 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → (𝐻𝑚) ∈ ℂ)
10019breq2d 3887 . . . . . . . . . . . 12 (𝜑 → (𝑚𝑁𝑚𝑀))
101100notbid 633 . . . . . . . . . . 11 (𝜑 → (¬ 𝑚𝑁 ↔ ¬ 𝑚𝑀))
102101ad2antrr 475 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → (¬ 𝑚𝑁 ↔ ¬ 𝑚𝑀))
10358, 102mpbird 166 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → ¬ 𝑚𝑁)
104103iffalsed 3431 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → if(𝑚𝑁, (𝐾𝑚) / 𝑘𝐵, 0) = 0)
105104, 60syl6eqel 2190 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → if(𝑚𝑁, (𝐾𝑚) / 𝑘𝐵, 0) ∈ ℂ)
10676, 80, 50, 105fvmptd3 5446 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → (𝐻𝑚) = if(𝑚𝑁, (𝐾𝑚) / 𝑘𝐵, 0))
107106, 105eqeltrd 2176 . . . . 5 (((𝜑𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ‘(𝑀 + 1))) → (𝐻𝑚) ∈ ℂ)
10899, 107, 69mpjaodan 753 . . . 4 ((𝜑𝑚 ∈ ℕ) → (𝐻𝑚) ∈ ℂ)
10925, 108sylan2br 284 . . 3 ((𝜑𝑚 ∈ (ℤ‘1)) → (𝐻𝑚) ∈ ℂ)
110 f1oeq2 5293 . . . . . . . . . . 11 ((1...𝑀) = (1...𝑁) → (𝐾:(1...𝑀)–1-1-onto𝐴𝐾:(1...𝑁)–1-1-onto𝐴))
11121, 110syl 14 . . . . . . . . . 10 (𝜑 → (𝐾:(1...𝑀)–1-1-onto𝐴𝐾:(1...𝑁)–1-1-onto𝐴))
11214, 111mpbird 166 . . . . . . . . 9 (𝜑𝐾:(1...𝑀)–1-1-onto𝐴)
113 f1of 5301 . . . . . . . . 9 (𝐾:(1...𝑀)–1-1-onto𝐴𝐾:(1...𝑀)⟶𝐴)
114112, 113syl 14 . . . . . . . 8 (𝜑𝐾:(1...𝑀)⟶𝐴)
115 fvco3 5424 . . . . . . . 8 ((𝐾:(1...𝑀)⟶𝐴𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) = (𝑓‘(𝐾𝑖)))
116114, 115sylan 279 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) = (𝑓‘(𝐾𝑖)))
117116fveq2d 5357 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘((𝑓𝐾)‘𝑖)) = (𝑓‘(𝑓‘(𝐾𝑖))))
11811adantr 272 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑓:(1...𝑀)–1-1-onto𝐴)
119114ffvelrnda 5487 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) ∈ 𝐴)
120 f1ocnvfv2 5611 . . . . . . 7 ((𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (𝐾𝑖) ∈ 𝐴) → (𝑓‘(𝑓‘(𝐾𝑖))) = (𝐾𝑖))
121118, 119, 120syl2anc 406 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘(𝑓‘(𝐾𝑖))) = (𝐾𝑖))
122117, 121eqtr2d 2133 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) = (𝑓‘((𝑓𝐾)‘𝑖)))
123122csbeq1d 2961 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) / 𝑘𝐵 = (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵)
124 elfznn 9675 . . . . . 6 (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℕ)
125 elfzle2 9649 . . . . . . . . . 10 (𝑖 ∈ (1...𝑀) → 𝑖𝑀)
126125adantl 273 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑖𝑀)
12720breq2d 3887 . . . . . . . . . 10 (𝜑 → (𝑖𝑀𝑖𝑁))
128127adantr 272 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑖𝑀𝑖𝑁))
129126, 128mpbid 146 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑖𝑁)
130129iftrued 3428 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → if(𝑖𝑁, (𝐾𝑖) / 𝑘𝐵, 0) = (𝐾𝑖) / 𝑘𝐵)
13138adantr 272 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
132 nfcsb1v 2985 . . . . . . . . . 10 𝑘(𝐾𝑖) / 𝑘𝐵
133132nfel1 2251 . . . . . . . . 9 𝑘(𝐾𝑖) / 𝑘𝐵 ∈ ℂ
134 csbeq1a 2963 . . . . . . . . . 10 (𝑘 = (𝐾𝑖) → 𝐵 = (𝐾𝑖) / 𝑘𝐵)
135134eleq1d 2168 . . . . . . . . 9 (𝑘 = (𝐾𝑖) → (𝐵 ∈ ℂ ↔ (𝐾𝑖) / 𝑘𝐵 ∈ ℂ))
136133, 135rspc 2738 . . . . . . . 8 ((𝐾𝑖) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝐾𝑖) / 𝑘𝐵 ∈ ℂ))
137119, 131, 136sylc 62 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) / 𝑘𝐵 ∈ ℂ)
138130, 137eqeltrd 2176 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → if(𝑖𝑁, (𝐾𝑖) / 𝑘𝐵, 0) ∈ ℂ)
139 breq1 3878 . . . . . . . 8 (𝑛 = 𝑖 → (𝑛𝑁𝑖𝑁))
140 fveq2 5353 . . . . . . . . 9 (𝑛 = 𝑖 → (𝐾𝑛) = (𝐾𝑖))
141140csbeq1d 2961 . . . . . . . 8 (𝑛 = 𝑖(𝐾𝑛) / 𝑘𝐵 = (𝐾𝑖) / 𝑘𝐵)
142139, 141ifbieq1d 3441 . . . . . . 7 (𝑛 = 𝑖 → if(𝑛𝑁, (𝐾𝑛) / 𝑘𝐵, 0) = if(𝑖𝑁, (𝐾𝑖) / 𝑘𝐵, 0))
143142, 76fvmptg 5429 . . . . . 6 ((𝑖 ∈ ℕ ∧ if(𝑖𝑁, (𝐾𝑖) / 𝑘𝐵, 0) ∈ ℂ) → (𝐻𝑖) = if(𝑖𝑁, (𝐾𝑖) / 𝑘𝐵, 0))
144124, 138, 143syl2an2 564 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻𝑖) = if(𝑖𝑁, (𝐾𝑖) / 𝑘𝐵, 0))
145144, 130eqtrd 2132 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻𝑖) = (𝐾𝑖) / 𝑘𝐵)
146 breq1 3878 . . . . . . 7 (𝑛 = ((𝑓𝐾)‘𝑖) → (𝑛𝑀 ↔ ((𝑓𝐾)‘𝑖) ≤ 𝑀))
147 fveq2 5353 . . . . . . . 8 (𝑛 = ((𝑓𝐾)‘𝑖) → (𝑓𝑛) = (𝑓‘((𝑓𝐾)‘𝑖)))
148147csbeq1d 2961 . . . . . . 7 (𝑛 = ((𝑓𝐾)‘𝑖) → (𝑓𝑛) / 𝑘𝐵 = (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵)
149146, 148ifbieq1d 3441 . . . . . 6 (𝑛 = ((𝑓𝐾)‘𝑖) → if(𝑛𝑀, (𝑓𝑛) / 𝑘𝐵, 0) = if(((𝑓𝐾)‘𝑖) ≤ 𝑀, (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵, 0))
150 f1of 5301 . . . . . . . . 9 ((𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀) → (𝑓𝐾):(1...𝑀)⟶(1...𝑀))
15124, 150syl 14 . . . . . . . 8 (𝜑 → (𝑓𝐾):(1...𝑀)⟶(1...𝑀))
152151ffvelrnda 5487 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) ∈ (1...𝑀))
153 elfznn 9675 . . . . . . 7 (((𝑓𝐾)‘𝑖) ∈ (1...𝑀) → ((𝑓𝐾)‘𝑖) ∈ ℕ)
154152, 153syl 14 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) ∈ ℕ)
155 elfzle2 9649 . . . . . . . . . 10 (((𝑓𝐾)‘𝑖) ∈ (1...𝑀) → ((𝑓𝐾)‘𝑖) ≤ 𝑀)
156152, 155syl 14 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) ≤ 𝑀)
157156iftrued 3428 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → if(((𝑓𝐾)‘𝑖) ≤ 𝑀, (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵, 0) = (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵)
158157, 123eqtr4d 2135 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → if(((𝑓𝐾)‘𝑖) ≤ 𝑀, (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵, 0) = (𝐾𝑖) / 𝑘𝐵)
159158, 137eqeltrd 2176 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → if(((𝑓𝐾)‘𝑖) ≤ 𝑀, (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵, 0) ∈ ℂ)
16026, 149, 154, 159fvmptd3 5446 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐺‘((𝑓𝐾)‘𝑖)) = if(((𝑓𝐾)‘𝑖) ≤ 𝑀, (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵, 0))
161160, 157eqtrd 2132 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐺‘((𝑓𝐾)‘𝑖)) = (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵)
162123, 145, 1613eqtr4d 2142 . . 3 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻𝑖) = (𝐺‘((𝑓𝐾)‘𝑖)))
1632, 4, 6, 10, 24, 71, 109, 162seq3f1o 10118 . 2 (𝜑 → (seq1( + , 𝐻)‘𝑀) = (seq1( + , 𝐺)‘𝑀))
16420fveq2d 5357 . 2 (𝜑 → (seq1( + , 𝐻)‘𝑀) = (seq1( + , 𝐻)‘𝑁))
165163, 164eqtr3d 2134 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 670   ∧ w3a 930   = wceq 1299   ∈ wcel 1448  ∀wral 2375  ⦋csb 2955   ∪ cun 3019  ifcif 3421   class class class wbr 3875   ↦ cmpt 3929  ◡ccnv 4476   ∘ ccom 4481  ⟶wf 5055  –1-1-onto→wf1o 5058  ‘cfv 5059  (class class class)co 5706  ℂcc 7498  0cc0 7500  1c1 7501   + caddc 7503   < clt 7672   ≤ cle 7673  ℕcn 8578  ℤcz 8906  ℤ≥cuz 9176  ...cfz 9631  seqcseq 10059 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-addcom 7595  ax-addass 7597  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-0id 7603  ax-rnegex 7604  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611 This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-1o 6243  df-er 6359  df-en 6565  df-dom 6566  df-fin 6567  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-inn 8579  df-n0 8830  df-z 8907  df-uz 9177  df-fz 9632  df-fzo 9761  df-seqfrec 10060  df-ihash 10363 This theorem is referenced by:  summodclem2a  10989  summodc  10991
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