Theorem ress0g 13084

Description: 0g is unaffected by restriction. This is a bit more generic than submnd0 13085. (Contributed by Thierry Arnoux, 23-Oct-2017.)

Hypotheses

Ref	Expression
ress0g.s	|- S = ( R ↾s A )
ress0g.b	|- B = ( Base ` R )
ress0g.0	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
ress0g	|- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> .0. = ( 0g ` S ) )

Proof of Theorem ress0g

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ress0g.s	. . . 4 |- S = ( R ↾s A )
2	1	a1i 9	. . 3 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> S = ( R ↾s A ) )
3		ress0g.b	. . . 4 |- B = ( Base ` R )
4	3	a1i 9	. . 3 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> B = ( Base ` R ) )
5		simp1 999	. . 3 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> R e. Mnd )
6		simp3 1001	. . 3 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> A C_ B )
7	2, 4, 5, 6	ressbas2d 12746	. 2 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> A = ( Base ` S ) )
8		eqidd 2197	. . 3 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> ( +g ` R ) = ( +g ` R ) )
9		basfn 12736	. . . . . 6 |- Base Fn _V
10	5	elexd 2776	. . . . . 6 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> R e. _V )
11		funfvex 5575	. . . . . . 7 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
12	11	funfni 5358	. . . . . 6 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
13	9, 10, 12	sylancr 414	. . . . 5 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> ( Base ` R ) e. _V )
14	3, 13	eqeltrid 2283	. . . 4 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> B e. _V )
15	14, 6	ssexd 4173	. . 3 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> A e. _V )
16	2, 8, 15, 5	ressplusgd 12806	. 2 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> ( +g ` R ) = ( +g ` S ) )
17		simp2 1000	. 2 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> .0. e. A )
18		simpl1 1002	. . 3 |- ( ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) /\ x e. A ) -> R e. Mnd )
19	6	sselda 3183	. . 3 |- ( ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) /\ x e. A ) -> x e. B )
20		eqid 2196	. . . 4 |- ( +g ` R ) = ( +g ` R )
21		ress0g.0	. . . 4 |- .0. = ( 0g ` R )
22	3, 20, 21	mndlid 13076	. . 3 |- ( ( R e. Mnd /\ x e. B ) -> ( .0. ( +g ` R ) x ) = x )
23	18, 19, 22	syl2anc 411	. 2 |- ( ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) /\ x e. A ) -> ( .0. ( +g ` R ) x ) = x )
24	3, 20, 21	mndrid 13077	. . 3 |- ( ( R e. Mnd /\ x e. B ) -> ( x ( +g ` R ) .0. ) = x )
25	18, 19, 24	syl2anc 411	. 2 |- ( ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) /\ x e. A ) -> ( x ( +g ` R ) .0. ) = x )
26	7, 16, 17, 23, 25	grpidd 13026	1 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> .0. = ( 0g ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2167   _Vcvv 2763   C_ wss 3157   Fn wfn 5253   ` cfv 5258  (class class class)co 5922   Basecbs 12678   ↾_s cress 12679   +g cplusg 12755   0gc0g 12927   Mndcmnd 13057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058

This theorem is referenced by:  submnd0  13085  zring0  14156