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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 12804 | . 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 1483 seqcseq 12801 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-iota 5851 df-fv 5896 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-seq 12802 |
This theorem is referenced by: seqeq123d 12810 seqf1olem2 12841 bcval5 13105 bcn2 13106 seqshft 13825 iserex 14387 isershft 14394 isercoll2 14399 isumsplit 14572 cvgrat 14615 ntrivcvg 14629 ntrivcvgtail 14632 fprodser 14679 eftlub 14839 gsumval2a 17279 gsumccat 17378 mulgnndir 17569 mulgnndirOLD 17570 geolim3 24094 fmul01lt1lem2 39817 stirlinglem7 40297 stirlinglem12 40302 |
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