Theorem rabsubmgmd 41791

Description: Deduction for proving that a restricted class abstraction is a submagma. (Contributed by AV, 26-Feb-2020.)

Hypotheses

Ref	Expression
rabsubmgmd.b	|- B = ( Base ` M )
rabsubmgmd.p	|- .+ = ( +g ` M )
rabsubmgmd.m	|- ( ph -> M e. Mgm )
rabsubmgmd.cp	|- ( ( ph /\ ( ( x e. B /\ y e. B ) /\ ( th /\ ta ) ) ) -> et )
rabsubmgmd.th	|- ( z = x -> ( ps <-> th ) )
rabsubmgmd.ta	|- ( z = y -> ( ps <-> ta ) )
rabsubmgmd.et	|- ( z = ( x .+ y ) -> ( ps <-> et ) )

Assertion

Ref	Expression
rabsubmgmd	|- ( ph -> { z e. B | ps } e. (SubMgm ` M ) )

Distinct variable groups:   x, y, z, B   x, M, y   ph, x, y   ps, x, y   z, .+   et, z   ta, z   th, z

Allowed substitution hints:   ph( z)   ps( z)   th( x, y)   ta( x, y)   et( x, y)   .+ ( x, y)   M( z)

Proof of Theorem rabsubmgmd

Step	Hyp	Ref	Expression
1		ssrab2 3687	. . 3 |- { z e. B | ps } C_ B
2	1	a1i 11	. 2 |- ( ph -> { z e. B | ps } C_ B )
3		rabsubmgmd.th	. . . . . 6 |- ( z = x -> ( ps <-> th ) )
4	3	elrab 3363	. . . . 5 |- ( x e. { z e. B | ps } <-> ( x e. B /\ th ) )
5		rabsubmgmd.ta	. . . . . 6 |- ( z = y -> ( ps <-> ta ) )
6	5	elrab 3363	. . . . 5 |- ( y e. { z e. B | ps } <-> ( y e. B /\ ta ) )
7	4, 6	anbi12i 733	. . . 4 |- ( ( x e. { z e. B | ps } /\ y e. { z e. B | ps } ) <-> ( ( x e. B /\ th ) /\ ( y e. B /\ ta ) ) )
8		rabsubmgmd.m	. . . . . . 7 |- ( ph -> M e. Mgm )
9	8	adantr 481	. . . . . 6 |- ( ( ph /\ ( ( x e. B /\ th ) /\ ( y e. B /\ ta ) ) ) -> M e. Mgm )
10		simprll 802	. . . . . 6 |- ( ( ph /\ ( ( x e. B /\ th ) /\ ( y e. B /\ ta ) ) ) -> x e. B )
11		simprrl 804	. . . . . 6 |- ( ( ph /\ ( ( x e. B /\ th ) /\ ( y e. B /\ ta ) ) ) -> y e. B )
12		rabsubmgmd.b	. . . . . . 7 |- B = ( Base ` M )
13		rabsubmgmd.p	. . . . . . 7 |- .+ = ( +g ` M )
14	12, 13	mgmcl 17245	. . . . . 6 |- ( ( M e. Mgm /\ x e. B /\ y e. B ) -> ( x .+ y ) e. B )
15	9, 10, 11, 14	syl3anc 1326	. . . . 5 |- ( ( ph /\ ( ( x e. B /\ th ) /\ ( y e. B /\ ta ) ) ) -> ( x .+ y ) e. B )
16		simpl 473	. . . . . . . 8 |- ( ( x e. B /\ th ) -> x e. B )
17		simpl 473	. . . . . . . 8 |- ( ( y e. B /\ ta ) -> y e. B )
18	16, 17	anim12i 590	. . . . . . 7 |- ( ( ( x e. B /\ th ) /\ ( y e. B /\ ta ) ) -> ( x e. B /\ y e. B ) )
19		simpr 477	. . . . . . . 8 |- ( ( x e. B /\ th ) -> th )
20		simpr 477	. . . . . . . 8 |- ( ( y e. B /\ ta ) -> ta )
21	19, 20	anim12i 590	. . . . . . 7 |- ( ( ( x e. B /\ th ) /\ ( y e. B /\ ta ) ) -> ( th /\ ta ) )
22	18, 21	jca 554	. . . . . 6 |- ( ( ( x e. B /\ th ) /\ ( y e. B /\ ta ) ) -> ( ( x e. B /\ y e. B ) /\ ( th /\ ta ) ) )
23		rabsubmgmd.cp	. . . . . 6 |- ( ( ph /\ ( ( x e. B /\ y e. B ) /\ ( th /\ ta ) ) ) -> et )
24	22, 23	sylan2 491	. . . . 5 |- ( ( ph /\ ( ( x e. B /\ th ) /\ ( y e. B /\ ta ) ) ) -> et )
25		rabsubmgmd.et	. . . . . 6 |- ( z = ( x .+ y ) -> ( ps <-> et ) )
26	25	elrab 3363	. . . . 5 |- ( ( x .+ y ) e. { z e. B | ps } <-> ( ( x .+ y ) e. B /\ et ) )
27	15, 24, 26	sylanbrc 698	. . . 4 |- ( ( ph /\ ( ( x e. B /\ th ) /\ ( y e. B /\ ta ) ) ) -> ( x .+ y ) e. { z e. B | ps } )
28	7, 27	sylan2b 492	. . 3 |- ( ( ph /\ ( x e. { z e. B | ps } /\ y e. { z e. B | ps } ) ) -> ( x .+ y ) e. { z e. B | ps } )
29	28	ralrimivva 2971	. 2 |- ( ph -> A. x e. { z e. B | ps } A. y e. { z e. B | ps } ( x .+ y ) e. { z e. B | ps } )
30	12, 13	issubmgm 41789	. . 3 |- ( M e. Mgm -> ( { z e. B | ps } e. (SubMgm ` M ) <-> ( { z e. B | ps } C_ B /\ A. x e. { z e. B | ps } A. y e. { z e. B | ps } ( x .+ y ) e. { z e. B | ps } ) ) )
31	8, 30	syl 17	. 2 |- ( ph -> ( { z e. B | ps } e. (SubMgm ` M ) <-> ( { z e. B | ps } C_ B /\ A. x e. { z e. B | ps } A. y e. { z e. B | ps } ( x .+ y ) e. { z e. B | ps } ) ) )
32	2, 29, 31	mpbir2and 957	1 |- ( ph -> { z e. B | ps } e. (SubMgm ` M ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   C_ wss 3574   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941  Mgmcmgm 17240  SubMgmcsubmgm 41778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-mgm 17242  df-submgm 41780

This theorem is referenced by: (None)