Theorem rhmsubcALTVlem2 42105

Description: Lemma 2 for rhmsubcALTV 42108. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)

Hypotheses

Ref	Expression
rngcrescrhmALTV.u	|- ( ph -> U e. V )
rngcrescrhmALTV.c	|- C = (RngCatALTV ` U )
rngcrescrhmALTV.r	|- ( ph -> R = ( Ring i^i U ) )
rngcrescrhmALTV.h	|- H = ( RingHom |` ( R X. R ) )

Assertion

Ref	Expression
rhmsubcALTVlem2	|- ( ( ph /\ X e. R /\ Y e. R ) -> ( X H Y ) = ( X RingHom Y ) )

Proof of Theorem rhmsubcALTVlem2

Step	Hyp	Ref	Expression
1		opelxpi 5148	. . . 4 |- ( ( X e. R /\ Y e. R ) -> <. X , Y >. e. ( R X. R ) )
2	1	3adant1 1079	. . 3 |- ( ( ph /\ X e. R /\ Y e. R ) -> <. X , Y >. e. ( R X. R ) )
3		fvres 6207	. . 3 |- ( <. X , Y >. e. ( R X. R ) -> ( ( RingHom |` ( R X. R ) ) ` <. X , Y >. ) = ( RingHom ` <. X , Y >. ) )
4	2, 3	syl 17	. 2 |- ( ( ph /\ X e. R /\ Y e. R ) -> ( ( RingHom |` ( R X. R ) ) ` <. X , Y >. ) = ( RingHom ` <. X , Y >. ) )
5		df-ov 6653	. . 3 |- ( X H Y ) = ( H ` <. X , Y >. )
6		rngcrescrhmALTV.h	. . . 4 |- H = ( RingHom |` ( R X. R ) )
7	6	fveq1i 6192	. . 3 |- ( H ` <. X , Y >. ) = ( ( RingHom |` ( R X. R ) ) ` <. X , Y >. )
8	5, 7	eqtri 2644	. 2 |- ( X H Y ) = ( ( RingHom |` ( R X. R ) ) ` <. X , Y >. )
9		df-ov 6653	. 2 |- ( X RingHom Y ) = ( RingHom ` <. X , Y >. )
10	4, 8, 9	3eqtr4g 2681	1 |- ( ( ph /\ X e. R /\ Y e. R ) -> ( X H Y ) = ( X RingHom Y ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   i^i cin 3573   <.cop 4183   X. cxp 5112   |` cres 5116   ` cfv 5888  (class class class)co 6650   Ringcrg 18547   RingHom crh 18712  RngCatALTVcrngcALTV 41958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-res 5126  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  rhmsubcALTVlem3  42106  rhmsubcALTVlem4  42107