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Mirrors > Home > MPE Home > Th. List > sumeq2sdv | Structured version Visualization version GIF version |
Description: Equality deduction for sum. (Contributed by NM, 3-Jan-2006.) (Proof shortened by Glauco Siliprandi, 5-Apr-2020.) Avoid axioms. (Revised by GG, 14-Aug-2025.) |
Ref | Expression |
---|---|
sumeq2sdv.1 | ⊢ (𝜑 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
sumeq2sdv | ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sumeq2sdv.1 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = 𝐶) | |
2 | 1 | csbeq2dv 3914 | . . . . . . . . . 10 ⊢ (𝜑 → ⦋𝑛 / 𝑘⦌𝐵 = ⦋𝑛 / 𝑘⦌𝐶) |
3 | 2 | ifeq1d 4549 | . . . . . . . . 9 ⊢ (𝜑 → if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0) = if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0)) |
4 | 3 | mpteq2dv 5249 | . . . . . . . 8 ⊢ (𝜑 → (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) |
5 | 4 | seqeq3d 14046 | . . . . . . 7 ⊢ (𝜑 → seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) = seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0)))) |
6 | 5 | breq1d 5157 | . . . . . 6 ⊢ (𝜑 → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥 ↔ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥)) |
7 | 6 | anbi2d 630 | . . . . 5 ⊢ (𝜑 → ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥))) |
8 | 7 | rexbidv 3176 | . . . 4 ⊢ (𝜑 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥))) |
9 | 1 | csbeq2dv 3914 | . . . . . . . . . . 11 ⊢ (𝜑 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑛) / 𝑘⦌𝐶) |
10 | 9 | mpteq2dv 5249 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵) = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶)) |
11 | 10 | seqeq3d 14046 | . . . . . . . . 9 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)) = seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))) |
12 | 11 | fveq1d 6908 | . . . . . . . 8 ⊢ (𝜑 → (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚)) |
13 | 12 | eqeq2d 2745 | . . . . . . 7 ⊢ (𝜑 → (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚) ↔ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚))) |
14 | 13 | anbi2d 630 | . . . . . 6 ⊢ (𝜑 → ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚)))) |
15 | 14 | exbidv 1918 | . . . . 5 ⊢ (𝜑 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚)))) |
16 | 15 | rexbidv 3176 | . . . 4 ⊢ (𝜑 → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚)))) |
17 | 8, 16 | orbi12d 918 | . . 3 ⊢ (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚))))) |
18 | 17 | iotabidv 6546 | . 2 ⊢ (𝜑 → (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))) = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚))))) |
19 | df-sum 15719 | . 2 ⊢ Σ𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))) | |
20 | df-sum 15719 | . 2 ⊢ Σ𝑘 ∈ 𝐴 𝐶 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚)))) | |
21 | 18, 19, 20 | 3eqtr4g 2799 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1536 ∃wex 1775 ∈ wcel 2105 ∃wrex 3067 ⦋csb 3907 ⊆ wss 3962 ifcif 4530 class class class wbr 5147 ↦ cmpt 5230 ℩cio 6513 –1-1-onto→wf1o 6561 ‘cfv 6562 (class class class)co 7430 0cc0 11152 1c1 11153 + caddc 11155 ℕcn 12263 ℤcz 12610 ℤ≥cuz 12875 ...cfz 13543 seqcseq 14038 ⇝ cli 15516 Σcsu 15718 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-xp 5694 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-iota 6515 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-seq 14039 df-sum 15719 |
This theorem is referenced by: sumsplit 15800 fsumrlim 15843 hash2iun1dif1 15856 incexclem 15868 bpolylem 16080 bpolyval 16081 efval 16111 rpnnen2lem12 16257 pcfac 16932 ramcl 17062 cshwshashnsame 17137 fsumcn 24907 fsum2cn 24908 lebnumlem3 25008 rrxdsfival 25460 uniioombllem6 25636 itg1climres 25763 itgeq1f 25820 itgeq1fOLD 25821 itgeq1 25822 cbvitgv 25826 itgeq2 25827 dvmptfsum 26027 elplyr 26254 plyeq0lem 26263 plyadd 26270 plymul 26271 coeeu 26278 coelem 26279 coeeq 26280 coeidlem 26290 coeid 26291 coeid2 26292 plyco 26294 plycjlem 26330 aareccl 26382 taylply2 26423 taylply2OLD 26424 pserdvlem2 26486 pserdv 26487 abelthlem6 26494 abelthlem9 26498 logtayl 26716 leibpi 26999 basellem3 27140 dchrvmasum2if 27555 dchrvmaeq0 27562 rpvmasum2 27570 dchrisum0re 27571 brcgr 28929 axsegcon 28956 dipfval 30730 ipval 30731 fsumiunle 32835 itgeq12dv 34307 eulerpartleme 34344 eulerpartlemr 34355 eulerpartlemn 34362 reprsum 34606 reprsuc 34608 reprpmtf1o 34619 vtsval 34630 iprodgam 35721 fwddifnval 36144 sumeq12sdv 36199 itgeq12sdv 36201 cbvitgdavw 36263 cbvitgdavw2 36279 knoppndvlem6 36499 knoppf 36517 rrnmval 37814 fsumshftd 38933 fsumcnf 44958 mccl 45553 dvnmul 45898 dvmptfprod 45900 dvnprodlem1 45901 dvnprodlem3 45903 dvnprod 45904 stoweidlem17 45972 stoweidlem26 45981 stoweidlem30 45985 stoweidlem32 45987 dirkertrigeq 46056 dirkeritg 46057 fourierdlem83 46144 fourierdlem103 46164 etransclem11 46200 etransclem24 46213 etransclem26 46215 etransclem27 46216 etransclem28 46217 etransclem31 46220 etransclem35 46224 etransclem46 46235 etransclem47 46236 rrndistlt 46245 ioorrnopn 46260 sge0val 46321 hoiqssbllem2 46578 nnsum3primes4 47712 nnsum4primesodd 47720 nnsum4primesoddALTV 47721 nnsum4primesevenALTV 47725 nn0sumshdiglemB 48469 nn0sumshdiglem1 48470 aacllem 49031 |
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