Theorem isrnghm2d 41901

Description: Demonstration of non-unital ring homomorphism. (Contributed by AV, 23-Feb-2020.)

Hypotheses

Ref	Expression
isrnghmd.b	|- B = ( Base ` R )
isrnghmd.t	|- .x. = ( .r ` R )
isrnghmd.u	|- .X. = ( .r ` S )
isrnghmd.r	|- ( ph -> R e. Rng )
isrnghmd.s	|- ( ph -> S e. Rng )
isrnghmd.ht	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( F ` ( x .x. y ) ) = ( ( F ` x ) .X. ( F ` y ) ) )
isrnghm2d.f	|- ( ph -> F e. ( R GrpHom S ) )

Assertion

Ref	Expression
isrnghm2d	|- ( ph -> F e. ( R RngHomo S ) )

Distinct variable groups:   ph, x, y   x, B, y   x, F, y   x, R, y   x, S, y

Allowed substitution hints:   .x. ( x, y)   .X. ( x, y)

Proof of Theorem isrnghm2d

Step	Hyp	Ref	Expression
1		isrnghmd.r	. . 3 |- ( ph -> R e. Rng )
2		isrnghmd.s	. . 3 |- ( ph -> S e. Rng )
3	1, 2	jca 554	. 2 |- ( ph -> ( R e. Rng /\ S e. Rng ) )
4		isrnghm2d.f	. . 3 |- ( ph -> F e. ( R GrpHom S ) )
5		isrnghmd.ht	. . . 4 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( F ` ( x .x. y ) ) = ( ( F ` x ) .X. ( F ` y ) ) )
6	5	ralrimivva 2971	. . 3 |- ( ph -> A. x e. B A. y e. B ( F ` ( x .x. y ) ) = ( ( F ` x ) .X. ( F ` y ) ) )
7	4, 6	jca 554	. 2 |- ( ph -> ( F e. ( R GrpHom S ) /\ A. x e. B A. y e. B ( F ` ( x .x. y ) ) = ( ( F ` x ) .X. ( F ` y ) ) ) )
8		isrnghmd.b	. . 3 |- B = ( Base ` R )
9		isrnghmd.t	. . 3 |- .x. = ( .r ` R )
10		isrnghmd.u	. . 3 |- .X. = ( .r ` S )
11	8, 9, 10	isrnghm 41892	. 2 |- ( F e. ( R RngHomo S ) <-> ( ( R e. Rng /\ S e. Rng ) /\ ( F e. ( R GrpHom S ) /\ A. x e. B A. y e. B ( F ` ( x .x. y ) ) = ( ( F ` x ) .X. ( F ` y ) ) ) ) )
12	3, 7, 11	sylanbrc 698	1 |- ( ph -> F e. ( R RngHomo S ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   ` cfv 5888  (class class class)co 6650   Basecbs 15857   .rcmulr 15942   GrpHom cghm 17657  Rngcrng 41874   RngHomo crngh 41885

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-ghm 17658  df-abl 18196  df-rng0 41875  df-rnghomo 41887

This theorem is referenced by:  isrnghmd  41902