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Mirrors > Home > MPE Home > Th. List > seqeq2d | Structured version Visualization version GIF version |
Description: Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.) |
Ref | Expression |
---|---|
seqeqd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
seqeq2d | ⊢ (𝜑 → seq𝑀(𝐴, 𝐹) = seq𝑀(𝐵, 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seqeqd.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | seqeq2 14006 | . 2 ⊢ (𝐴 = 𝐵 → seq𝑀(𝐴, 𝐹) = seq𝑀(𝐵, 𝐹)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → seq𝑀(𝐴, 𝐹) = seq𝑀(𝐵, 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 seqcseq 14002 |
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 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-xp 5684 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-iota 6501 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-seq 14003 |
This theorem is referenced by: seqeq123d 14011 sadfval 16430 smufval 16455 gsumvalx 18639 gsumpropd 18641 gsumress 18645 mulgfval 19033 mulgfvalALT 19034 submmulg 19081 subgmulg 19103 dvnfval 25896 ressmulgnnd 41701 |
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