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Mirrors > Home > MPE Home > Th. List > seqeq1d | 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 |
---|---|
seqeq1d | ⊢ (𝜑 → seq𝐴( + , 𝐹) = seq𝐵( + , 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seqeqd.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | seqeq1 12666 | . 2 ⊢ (𝐴 = 𝐵 → seq𝐴( + , 𝐹) = seq𝐵( + , 𝐹)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → seq𝐴( + , 𝐹) = seq𝐵( + , 𝐹)) |
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-wrecs 7294 df-recs 7355 df-rdg 7393 df-seq 12664 |
This theorem is referenced by: seqeq123d 12672 seqf1olem2 12703 bcval5 12967 bcn2 12968 seqshft 13673 iserex 14235 isershft 14242 isercoll2 14247 isumsplit 14411 cvgrat 14454 ntrivcvg 14468 ntrivcvgtail 14471 fprodser 14518 eftlub 14678 gsumval2a 17102 gsumccat 17201 mulgnndir 17392 mulgnndirOLD 17393 geolim3 23898 fmul01lt1lem2 38652 stirlinglem7 38973 stirlinglem12 38978 |
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