Theorem ress0g 13704

Description: 0g is unaffected by restriction. This is a bit more generic than submnd0 13705. (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 13365	. 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 13355	. . . . . 6 |- Base Fn _V
10	5	elexd 2829	. . . . . 6 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> R e. _V )
11		funfvex 5692	. . . . . . 7 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
12	11	funfni 5463	. . . . . 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 4255	. . 3 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> A e. _V )
16	2, 8, 15, 5	ressplusgd 13426	. 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 3242	. . 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 13696	. . 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 13697	. . 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 13646	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 3214   Fn wfn 5352   ` cfv 5357  (class class class)co 6058   Basecbs 13296   ↾_s cress 13297   +g cplusg 13374   0gc0g 13553   Mndcmnd 13677

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 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-ltadd 8259

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 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678

This theorem is referenced by:  submnd0  13705  zring0  14874