![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 14042 | . 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 1537 seqcseq 14039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-xp 5695 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-iota 6516 df-fv 6571 df-ov 7434 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-seq 14040 |
This theorem is referenced by: seqeq123d 14048 seqf1olem2 14080 bcval5 14354 bcn2 14355 seqshft 15121 iserex 15690 isershft 15697 isercoll2 15702 isumsplit 15873 cvgrat 15916 ntrivcvg 15930 ntrivcvgtail 15933 fprodser 15982 eftlub 16142 gsumval2a 18711 gsumsgrpccat 18866 mulgnndir 19134 geolim3 26396 fmul01lt1lem2 45541 stirlinglem7 46036 stirlinglem12 46041 |
Copyright terms: Public domain | W3C validator |