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Mirrors > Home > MPE Home > Th. List > seqeq123d | Structured version Visualization version GIF version |
Description: Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.) |
Ref | Expression |
---|---|
seqeq123d.1 | ⊢ (𝜑 → 𝑀 = 𝑁) |
seqeq123d.2 | ⊢ (𝜑 → + = 𝑄) |
seqeq123d.3 | ⊢ (𝜑 → 𝐹 = 𝐺) |
Ref | Expression |
---|---|
seqeq123d | ⊢ (𝜑 → seq𝑀( + , 𝐹) = seq𝑁(𝑄, 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seqeq123d.1 | . . 3 ⊢ (𝜑 → 𝑀 = 𝑁) | |
2 | 1 | seqeq1d 13978 | . 2 ⊢ (𝜑 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹)) |
3 | seqeq123d.2 | . . 3 ⊢ (𝜑 → + = 𝑄) | |
4 | 3 | seqeq2d 13979 | . 2 ⊢ (𝜑 → seq𝑁( + , 𝐹) = seq𝑁(𝑄, 𝐹)) |
5 | seqeq123d.3 | . . 3 ⊢ (𝜑 → 𝐹 = 𝐺) | |
6 | 5 | seqeq3d 13980 | . 2 ⊢ (𝜑 → seq𝑁(𝑄, 𝐹) = seq𝑁(𝑄, 𝐺)) |
7 | 2, 4, 6 | 3eqtrd 2770 | 1 ⊢ (𝜑 → seq𝑀( + , 𝐹) = seq𝑁(𝑄, 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 seqcseq 13972 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-xp 5675 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-iota 6489 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-seq 13973 |
This theorem is referenced by: relexpsucnnr 14978 sseqval 33917 bj-finsumval0 36673 itcoval 47622 |
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