Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 12669 | . 2 ⊢ (𝜑 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹)) |
3 | seqeq123d.2 | . . 3 ⊢ (𝜑 → + = 𝑄) | |
4 | 3 | seqeq2d 12670 | . 2 ⊢ (𝜑 → seq𝑁( + , 𝐹) = seq𝑁(𝑄, 𝐹)) |
5 | seqeq123d.3 | . . 3 ⊢ (𝜑 → 𝐹 = 𝐺) | |
6 | 5 | seqeq3d 12671 | . 2 ⊢ (𝜑 → seq𝑁(𝑄, 𝐹) = seq𝑁(𝑄, 𝐺)) |
7 | 2, 4, 6 | 3eqtrd 2648 | 1 ⊢ (𝜑 → seq𝑀( + , 𝐹) = seq𝑁(𝑄, 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 seqcseq 12663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-iota 5768 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-seq 12664 |
This theorem is referenced by: relexpsucnnr 13613 sseqval 29777 bj-finsumval0 32324 |
Copyright terms: Public domain | W3C validator |