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Mirrors > Home > MPE Home > Th. List > rhmsubclem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for rhmsubc 20585. (Contributed by AV, 2-Mar-2020.) |
Ref | Expression |
---|---|
rngcrescrhm.u | β’ (π β π β π) |
rngcrescrhm.c | β’ πΆ = (RngCatβπ) |
rngcrescrhm.r | β’ (π β π = (Ring β© π)) |
rngcrescrhm.h | β’ π» = ( RingHom βΎ (π Γ π )) |
Ref | Expression |
---|---|
rhmsubclem2 | β’ ((π β§ π β π β§ π β π ) β (ππ»π) = (π RingHom π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpi 5706 | . . . 4 β’ ((π β π β§ π β π ) β β¨π, πβ© β (π Γ π )) | |
2 | 1 | 3adant1 1127 | . . 3 β’ ((π β§ π β π β§ π β π ) β β¨π, πβ© β (π Γ π )) |
3 | 2 | fvresd 6905 | . 2 β’ ((π β§ π β π β§ π β π ) β (( RingHom βΎ (π Γ π ))ββ¨π, πβ©) = ( RingHom ββ¨π, πβ©)) |
4 | df-ov 7408 | . . 3 β’ (ππ»π) = (π»ββ¨π, πβ©) | |
5 | rngcrescrhm.h | . . . 4 β’ π» = ( RingHom βΎ (π Γ π )) | |
6 | 5 | fveq1i 6886 | . . 3 β’ (π»ββ¨π, πβ©) = (( RingHom βΎ (π Γ π ))ββ¨π, πβ©) |
7 | 4, 6 | eqtri 2754 | . 2 β’ (ππ»π) = (( RingHom βΎ (π Γ π ))ββ¨π, πβ©) |
8 | df-ov 7408 | . 2 β’ (π RingHom π) = ( RingHom ββ¨π, πβ©) | |
9 | 3, 7, 8 | 3eqtr4g 2791 | 1 β’ ((π β§ π β π β§ π β π ) β (ππ»π) = (π RingHom π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 β© cin 3942 β¨cop 4629 Γ cxp 5667 βΎ cres 5671 βcfv 6537 (class class class)co 7405 Ringcrg 20138 RingHom crh 20371 RngCatcrngc 20512 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-xp 5675 df-res 5681 df-iota 6489 df-fv 6545 df-ov 7408 |
This theorem is referenced by: rhmsubclem3 20583 rhmsubclem4 20584 |
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