Theorem ress0g 13733

Description: 0g is unaffected by restriction. This is a bit more generic than submnd0 13734. (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 1028	. . 3 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> R e. Mnd )
6		simp3 1030	. . 3 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> A C_ B )
7	2, 4, 5, 6	ressbas2d 13399	. 2 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> A = ( Base ` S ) )
8		eqidd 2239	. . 3 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> ( +g ` R ) = ( +g ` R ) )
9		basfn 13389	. . . . . 6 |- Base Fn _V
10	5	elexd 2835	. . . . . 6 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> R e. _V )
11		funfvex 5707	. . . . . . 7 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
12	11	funfni 5478	. . . . . 6 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
13	9, 10, 12	sylancr 418	. . . . 5 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> ( Base ` R ) e. _V )
14	3, 13	eqeltrid 2325	. . . 4 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> B e. _V )
15	14, 6	ssexd 4268	. . 3 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> A e. _V )
16	2, 8, 15, 5	ressplusgd 13460	. 2 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> ( +g ` R ) = ( +g ` S ) )
17		simp2 1029	. 2 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> .0. e. A )
18		simpl1 1031	. . 3 |- ( ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) /\ x e. A ) -> R e. Mnd )
19	6	sselda 3248	. . 3 |- ( ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) /\ x e. A ) -> x e. B )
20		eqid 2238	. . . 4 |- ( +g ` R ) = ( +g ` R )
21		ress0g.0	. . . 4 |- .0. = ( 0g ` R )
22	3, 20, 21	mndlid 13725	. . 3 |- ( ( R e. Mnd /\ x e. B ) -> ( .0. ( +g ` R ) x ) = x )
23	18, 19, 22	syl2anc 415	. 2 |- ( ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) /\ x e. A ) -> ( .0. ( +g ` R ) x ) = x )
24	3, 20, 21	mndrid 13726	. . 3 |- ( ( R e. Mnd /\ x e. B ) -> ( x ( +g ` R ) .0. ) = x )
25	18, 19, 24	syl2anc 415	. 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 13680	1 |- ( ( R e. Mnd /\ .0. e. A /\ A C_ B ) -> .0. = ( 0g ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   _Vcvv 2821   C_ wss 3220   Fn wfn 5367   ` cfv 5372  (class class class)co 6075   Basecbs 13330   ↾_s cress 13331   +g cplusg 13408   0gc0g 13587   Mndcmnd 13706

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-iress 13338  df-plusg 13421  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707

This theorem is referenced by:  submnd0  13734  zring0  14907