Theorem grpsubval 12749

Description: Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 13-Dec-2014.)

Hypotheses

Ref	Expression
grpsubval.b	|- B = ( Base ` G )
grpsubval.p	|- .+ = ( +g ` G )
grpsubval.i	|- I = ( invg ` G )
grpsubval.m	|- .- = ( -g ` G )

Assertion

Ref	Expression
grpsubval	|- ( ( X e. B /\ Y e. B ) -> ( X .- Y ) = ( X .+ ( I ` Y ) ) )

Proof of Theorem grpsubval

Dummy variables x y z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpsubval.b	. . . . 5 |- B = ( Base ` G )
2	1	a1i 9	. . . 4 |- ( ( X e. B /\ Y e. B ) -> B = ( Base ` G ) )
3		simpl 108	. . . 4 |- ( ( X e. B /\ Y e. B ) -> X e. B )
4	2, 3	basmexd 12475	. . 3 |- ( ( X e. B /\ Y e. B ) -> G e. _V )
5		grpsubval.p	. . . 4 |- .+ = ( +g ` G )
6		grpsubval.i	. . . 4 |- I = ( invg ` G )
7		grpsubval.m	. . . 4 |- .- = ( -g ` G )
8	1, 5, 6, 7	grpsubfvalg 12748	. . 3 |- ( G e. _V -> .- = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )
9	4, 8	syl 14	. 2 |- ( ( X e. B /\ Y e. B ) -> .- = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )
10		oveq1 5860	. . . 4 |- ( x = X -> ( x .+ ( I ` y ) ) = ( X .+ ( I ` y ) ) )
11		fveq2 5496	. . . . 5 |- ( y = Y -> ( I ` y ) = ( I ` Y ) )
12	11	oveq2d 5869	. . . 4 |- ( y = Y -> ( X .+ ( I ` y ) ) = ( X .+ ( I ` Y ) ) )
13	10, 12	sylan9eq 2223	. . 3 |- ( ( x = X /\ y = Y ) -> ( x .+ ( I ` y ) ) = ( X .+ ( I ` Y ) ) )
14	13	adantl 275	. 2 |- ( ( ( X e. B /\ Y e. B ) /\ ( x = X /\ y = Y ) ) -> ( x .+ ( I ` y ) ) = ( X .+ ( I ` Y ) ) )
15		simpr 109	. 2 |- ( ( X e. B /\ Y e. B ) -> Y e. B )
16		plusgslid 12513	. . . . . 6 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
17	16	slotex 12443	. . . . 5 |- ( G e. _V -> ( +g ` G ) e. _V )
18	4, 17	syl 14	. . . 4 |- ( ( X e. B /\ Y e. B ) -> ( +g ` G ) e. _V )
19	5, 18	eqeltrid 2257	. . 3 |- ( ( X e. B /\ Y e. B ) -> .+ e. _V )
20		eqid 2170	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
21	1, 5, 20, 6	grpinvfvalg 12745	. . . . . 6 |- ( G e. _V -> I = ( z e. B |-> ( iota_ w e. B ( w .+ z ) = ( 0g ` G ) ) ) )
22	4, 21	syl 14	. . . . 5 |- ( ( X e. B /\ Y e. B ) -> I = ( z e. B |-> ( iota_ w e. B ( w .+ z ) = ( 0g ` G ) ) ) )
23		basfn 12473	. . . . . . . 8 |- Base Fn _V
24		funfvex 5513	. . . . . . . . 9 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
25	24	funfni 5298	. . . . . . . 8 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
26	23, 4, 25	sylancr 412	. . . . . . 7 |- ( ( X e. B /\ Y e. B ) -> ( Base ` G ) e. _V )
27	1, 26	eqeltrid 2257	. . . . . 6 |- ( ( X e. B /\ Y e. B ) -> B e. _V )
28	27	mptexd 5723	. . . . 5 |- ( ( X e. B /\ Y e. B ) -> ( z e. B |-> ( iota_ w e. B ( w .+ z ) = ( 0g ` G ) ) ) e. _V )
29	22, 28	eqeltrd 2247	. . . 4 |- ( ( X e. B /\ Y e. B ) -> I e. _V )
30		fvexg 5515	. . . 4 |- ( ( I e. _V /\ Y e. B ) -> ( I ` Y ) e. _V )
31	29, 30	sylancom 418	. . 3 |- ( ( X e. B /\ Y e. B ) -> ( I ` Y ) e. _V )
32		ovexg 5887	. . 3 |- ( ( X e. B /\ .+ e. _V /\ ( I ` Y ) e. _V ) -> ( X .+ ( I ` Y ) ) e. _V )
33	3, 19, 31, 32	syl3anc 1233	. 2 |- ( ( X e. B /\ Y e. B ) -> ( X .+ ( I ` Y ) ) e. _V )
34	9, 14, 3, 15, 33	ovmpod 5980	1 |- ( ( X e. B /\ Y e. B ) -> ( X .- Y ) = ( X .+ ( I ` Y ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   _Vcvv 2730   |-> cmpt 4050   Fn wfn 5193   ` cfv 5198   iota_crio 5808  (class class class)co 5853   e. cmpo 5855   Basecbs 12416   +g cplusg 12480   0gc0g 12596   invgcminusg 12709   -gcsg 12710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1re 7868  ax-addrcl 7871

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-inn 8879  df-2 8937  df-ndx 12419  df-slot 12420  df-base 12422  df-plusg 12493  df-minusg 12712  df-sbg 12713

This theorem is referenced by:  grpsubinv  12772