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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 13724 | . 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 1539 seqcseq 13721 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-xp 5595 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-iota 6391 df-fv 6441 df-ov 7278 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-seq 13722 |
This theorem is referenced by: seqeq123d 13730 seqf1olem2 13763 bcval5 14032 bcn2 14033 seqshft 14796 iserex 15368 isershft 15375 isercoll2 15380 isumsplit 15552 cvgrat 15595 ntrivcvg 15609 ntrivcvgtail 15612 fprodser 15659 eftlub 15818 gsumval2a 18369 gsumsgrpccat 18478 gsumccatOLD 18479 mulgnndir 18732 geolim3 25499 fmul01lt1lem2 43126 stirlinglem7 43621 stirlinglem12 43626 |
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