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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 13374 | . 2 ⊢ (𝜑 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹)) |
3 | seqeq123d.2 | . . 3 ⊢ (𝜑 → + = 𝑄) | |
4 | 3 | seqeq2d 13375 | . 2 ⊢ (𝜑 → seq𝑁( + , 𝐹) = seq𝑁(𝑄, 𝐹)) |
5 | seqeq123d.3 | . . 3 ⊢ (𝜑 → 𝐹 = 𝐺) | |
6 | 5 | seqeq3d 13376 | . 2 ⊢ (𝜑 → seq𝑁(𝑄, 𝐹) = seq𝑁(𝑄, 𝐺)) |
7 | 2, 4, 6 | 3eqtrd 2860 | 1 ⊢ (𝜑 → seq𝑀( + , 𝐹) = seq𝑁(𝑄, 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 seqcseq 13368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-xp 5560 df-cnv 5562 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-iota 6313 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-seq 13369 |
This theorem is referenced by: relexpsucnnr 14383 sseqval 31646 bj-finsumval0 34566 |
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