Theorem ress0g 13476

Description: 0g is unaffected by restriction. This is a bit more generic than submnd0 13477. (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 1021	. . 3 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> R e. Mnd )
6		simp3 1023	. . 3 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> A C_ B )
7	2, 4, 5, 6	ressbas2d 13101	. 2 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> A = ( Base ` S ) )
8		eqidd 2230	. . 3 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> ( +g ` R ) = ( +g ` R ) )
9		basfn 13091	. . . . . 6 |- Base Fn _V
10	5	elexd 2813	. . . . . 6 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> R e. _V )
11		funfvex 5644	. . . . . . 7 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
12	11	funfni 5423	. . . . . 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 2316	. . . 4 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> B e. _V )
15	14, 6	ssexd 4224	. . 3 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> A e. _V )
16	2, 8, 15, 5	ressplusgd 13162	. 2 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> ( +g ` R ) = ( +g ` S ) )
17		simp2 1022	. 2 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> .0. e. A )
18		simpl1 1024	. . 3 |- ( ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) /\ x e. A ) -> R e. Mnd )
19	6	sselda 3224	. . 3 |- ( ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) /\ x e. A ) -> x e. B )
20		eqid 2229	. . . 4 |- ( +g ` R ) = ( +g ` R )
21		ress0g.0	. . . 4 |- .0. = ( 0g ` R )
22	3, 20, 21	mndlid 13468	. . 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 13469	. . 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 13416	1 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> .0. = ( 0g ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   _Vcvv 2799   C_ wss 3197   Fn wfn 5313   ` cfv 5318  (class class class)co 6001   Basecbs 13032   ↾_s cress 13033   +g cplusg 13110   0gc0g 13289   Mndcmnd 13449

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-pre-ltirr 8111  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-ltxr 8186  df-inn 9111  df-2 9169  df-ndx 13035  df-slot 13036  df-base 13038  df-sets 13039  df-iress 13040  df-plusg 13123  df-0g 13291  df-mgm 13389  df-sgrp 13435  df-mnd 13450

This theorem is referenced by:  submnd0  13477  zring0  14564