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Mirrors > Home > MPE Home > Th. List > Mathboxes > rhmsubcALTVlem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for rhmsubcALTV 46496. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
rngcrescrhmALTV.u | β’ (π β π β π) |
rngcrescrhmALTV.c | β’ πΆ = (RngCatALTVβπ) |
rngcrescrhmALTV.r | β’ (π β π = (Ring β© π)) |
rngcrescrhmALTV.h | β’ π» = ( RingHom βΎ (π Γ π )) |
Ref | Expression |
---|---|
rhmsubcALTVlem2 | β’ ((π β§ π β π β§ π β π ) β (ππ»π) = (π RingHom π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpi 5674 | . . . 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 6866 | . 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 7364 | . . 3 β’ (ππ»π) = (π»ββ¨π, πβ©) | |
5 | rngcrescrhmALTV.h | . . . 4 β’ π» = ( RingHom βΎ (π Γ π )) | |
6 | 5 | fveq1i 6847 | . . 3 β’ (π»ββ¨π, πβ©) = (( RingHom βΎ (π Γ π ))ββ¨π, πβ©) |
7 | 4, 6 | eqtri 2761 | . 2 β’ (ππ»π) = (( RingHom βΎ (π Γ π ))ββ¨π, πβ©) |
8 | df-ov 7364 | . 2 β’ (π RingHom π) = ( RingHom ββ¨π, πβ©) | |
9 | 3, 7, 8 | 3eqtr4g 2798 | 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 3913 β¨cop 4596 Γ cxp 5635 βΎ cres 5639 βcfv 6500 (class class class)co 7361 Ringcrg 19972 RingHom crh 20153 RngCatALTVcrngcALTV 46346 |
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 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
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 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-xp 5643 df-res 5649 df-iota 6452 df-fv 6508 df-ov 7364 |
This theorem is referenced by: rhmsubcALTVlem3 46494 rhmsubcALTVlem4 46495 |
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