Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > rhmsubclem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for rhmsubc 46106. (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 5668 | . . . 4 β’ ((π β π β§ π β π ) β β¨π, πβ© β (π Γ π )) | |
2 | 1 | 3adant1 1131 | . . 3 β’ ((π β§ π β π β§ π β π ) β β¨π, πβ© β (π Γ π )) |
3 | 2 | fvresd 6858 | . 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 7353 | . . 3 β’ (ππ»π) = (π»ββ¨π, πβ©) | |
5 | rngcrescrhm.h | . . . 4 β’ π» = ( RingHom βΎ (π Γ π )) | |
6 | 5 | fveq1i 6839 | . . 3 β’ (π»ββ¨π, πβ©) = (( RingHom βΎ (π Γ π ))ββ¨π, πβ©) |
7 | 4, 6 | eqtri 2766 | . 2 β’ (ππ»π) = (( RingHom βΎ (π Γ π ))ββ¨π, πβ©) |
8 | df-ov 7353 | . 2 β’ (π RingHom π) = ( RingHom ββ¨π, πβ©) | |
9 | 3, 7, 8 | 3eqtr4g 2803 | 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 1088 = wceq 1542 β wcel 2107 β© cin 3908 β¨cop 4591 Γ cxp 5629 βΎ cres 5633 βcfv 6492 (class class class)co 7350 Ringcrg 19888 RingHom crh 20066 RngCatcrngc 45973 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-xp 5637 df-res 5643 df-iota 6444 df-fv 6500 df-ov 7353 |
This theorem is referenced by: rhmsubclem3 46104 rhmsubclem4 46105 |
Copyright terms: Public domain | W3C validator |