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Mirrors > Home > MPE Home > Th. List > oveq | Structured version Visualization version GIF version |
Description: Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.) |
Ref | Expression |
---|---|
oveq | ⊢ (𝐹 = 𝐺 → (𝐴𝐹𝐵) = (𝐴𝐺𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 6906 | . 2 ⊢ (𝐹 = 𝐺 → (𝐹‘〈𝐴, 𝐵〉) = (𝐺‘〈𝐴, 𝐵〉)) | |
2 | df-ov 7434 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
3 | df-ov 7434 | . 2 ⊢ (𝐴𝐺𝐵) = (𝐺‘〈𝐴, 𝐵〉) | |
4 | 1, 2, 3 | 3eqtr4g 2800 | 1 ⊢ (𝐹 = 𝐺 → (𝐴𝐹𝐵) = (𝐴𝐺𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 〈cop 4637 ‘cfv 6563 (class class class)co 7431 |
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-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-ss 3980 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 |
This theorem is referenced by: oveqi 7444 oveqd 7448 ifov 7534 ovmpodf 7589 ovmpodv2 7591 seqomeq12 8493 mapxpen 9182 seqeq2 14043 relexp0g 15058 relexpsucnnr 15061 cat1 18151 ismgm 18667 mgmsscl 18671 issgrp 18746 ismnddef 18762 grpissubg 19177 isga 19322 isrng 20172 islmod 20879 lmodfopne 20915 mamuval 22413 dmatel 22515 dmatmulcl 22522 scmate 22532 scmateALT 22534 mvmulval 22565 marrepval0 22583 marepvval0 22588 submaval0 22602 mdetleib 22609 mdetleib1 22613 mdet0pr 22614 mdetunilem1 22634 maduval 22660 minmar1val0 22669 cpmatel 22733 mat2pmatval 22746 cpm2mval 22772 decpmatval0 22786 pmatcollpw3lem 22805 mptcoe1matfsupp 22824 mp2pm2mplem4 22831 chpscmat 22864 ispsmet 24330 ismet 24349 isxmet 24350 ishtpy 25018 isphtpy 25027 addsval 28010 mulsval 28150 isgrpo 30526 gidval 30541 grpoinvfval 30551 isablo 30575 vciOLD 30590 isvclem 30606 isnvlem 30639 isphg 30846 ofceq 34078 cvmlift2lem13 35300 ismtyval 37787 isass 37833 isexid 37834 elghomlem1OLD 37872 iscom2 37982 iscllaw 48033 iscomlaw 48034 isasslaw 48036 dmatALTbasel 48248 isthinc 48821 |
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