Theorem ress0g 13673

Description: 0g is unaffected by restriction. This is a bit more generic than submnd0 13674. (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 1024	. . 3 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> R e. Mnd )
6		simp3 1026	. . 3 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> A C_ B )
7	2, 4, 5, 6	ressbas2d 13298	. 2 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> A = ( Base ` S ) )
8		eqidd 2235	. . 3 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> ( +g ` R ) = ( +g ` R ) )
9		basfn 13288	. . . . . 6 |- Base Fn _V
10	5	elexd 2829	. . . . . 6 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> R e. _V )
11		funfvex 5689	. . . . . . 7 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
12	11	funfni 5460	. . . . . 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 2321	. . . 4 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> B e. _V )
15	14, 6	ssexd 4252	. . 3 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> A e. _V )
16	2, 8, 15, 5	ressplusgd 13359	. 2 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> ( +g ` R ) = ( +g ` S ) )
17		simp2 1025	. 2 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> .0. e. A )
18		simpl1 1027	. . 3 |- ( ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) /\ x e. A ) -> R e. Mnd )
19	6	sselda 3240	. . 3 |- ( ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) /\ x e. A ) -> x e. B )
20		eqid 2234	. . . 4 |- ( +g ` R ) = ( +g ` R )
21		ress0g.0	. . . 4 |- .0. = ( 0g ` R )
22	3, 20, 21	mndlid 13665	. . 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 13666	. . 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 13613	1 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> .0. = ( 0g ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2205   _Vcvv 2815   C_ wss 3213   Fn wfn 5349   ` cfv 5354  (class class class)co 6052   Basecbs 13229   ↾_s cress 13230   +g cplusg 13307   0gc0g 13486   Mndcmnd 13646

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-pre-ltirr 8241  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-iota 5314  df-fun 5356  df-fn 5357  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-pnf 8312  df-mnf 8313  df-ltxr 8315  df-inn 9240  df-2 9298  df-ndx 13232  df-slot 13233  df-base 13235  df-sets 13236  df-iress 13237  df-plusg 13320  df-0g 13488  df-mgm 13586  df-sgrp 13632  df-mnd 13647

This theorem is referenced by:  submnd0  13674  zring0  14765