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Mirrors > Home > MPE Home > Th. List > Mathboxes > rhmsubclem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for rhmsubc 46137. (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 5667 | . . . 4 β’ ((π β π β§ π β π ) β β¨π, πβ© β (π Γ π )) | |
2 | 1 | 3adant1 1130 | . . 3 β’ ((π β§ π β π β§ π β π ) β β¨π, πβ© β (π Γ π )) |
3 | 2 | fvresd 6857 | . 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 7352 | . . 3 β’ (ππ»π) = (π»ββ¨π, πβ©) | |
5 | rngcrescrhm.h | . . . 4 β’ π» = ( RingHom βΎ (π Γ π )) | |
6 | 5 | fveq1i 6838 | . . 3 β’ (π»ββ¨π, πβ©) = (( RingHom βΎ (π Γ π ))ββ¨π, πβ©) |
7 | 4, 6 | eqtri 2765 | . 2 β’ (ππ»π) = (( RingHom βΎ (π Γ π ))ββ¨π, πβ©) |
8 | df-ov 7352 | . 2 β’ (π RingHom π) = ( RingHom ββ¨π, πβ©) | |
9 | 3, 7, 8 | 3eqtr4g 2802 | 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 1087 = wceq 1541 β wcel 2106 β© cin 3907 β¨cop 4590 Γ cxp 5628 βΎ cres 5632 βcfv 6491 (class class class)co 7349 Ringcrg 19888 RingHom crh 20066 RngCatcrngc 46004 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-xp 5636 df-res 5642 df-iota 6443 df-fv 6499 df-ov 7352 |
This theorem is referenced by: rhmsubclem3 46135 rhmsubclem4 46136 |
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