Theorem issubmd 15579

Description: Deduction for proving a submonoid. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.)

Hypotheses

Ref	Expression
issubmd.b	|- B = ( Base ` M )
issubmd.p	|- .+ = ( +g ` M )
issubmd.z	|- .0. = ( 0g ` M )
issubmd.m	|- ( ph -> M e. Mnd )
issubmd.cz	|- ( ph -> ch )
issubmd.cp	|- ( ( ph /\ ( ( x e. B /\ y e. B ) /\ ( th /\ ta ) ) ) -> et )
issubmd.ch	|- ( z = .0. -> ( ps <-> ch ) )
issubmd.th	|- ( z = x -> ( ps <-> th ) )
issubmd.ta	|- ( z = y -> ( ps <-> ta ) )
issubmd.et	|- ( z = ( x .+ y ) -> ( ps <-> et ) )

Assertion

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

Distinct variable groups:   x, y, z, B   x, M, y   ph, x, y   ps, x, y   z, .+   z, .0.   ch, z   et, z   ta, z   th, z

Allowed substitution hints:   ph( z)   ps( z)   ch( x, y)   th( x, y)   ta( x, y)   et( x, y)   .+ ( x, y)   M( z)   .0. ( x, y)

Proof of Theorem issubmd

Step	Hyp	Ref	Expression
1		ssrab2 3535	. . 3 |- { z e. B | ps } C_ B
2	1	a1i 11	. 2 |- ( ph -> { z e. B | ps } C_ B )
3		issubmd.m	. . . 4 |- ( ph -> M e. Mnd )
4		issubmd.b	. . . . 5 |- B = ( Base ` M )
5		issubmd.z	. . . . 5 |- .0. = ( 0g ` M )
6	4, 5	mndidcl 15541	. . . 4 |- ( M e. Mnd -> .0. e. B )
7	3, 6	syl 16	. . 3 |- ( ph -> .0. e. B )
8		issubmd.cz	. . 3 |- ( ph -> ch )
9		issubmd.ch	. . . 4 |- ( z = .0. -> ( ps <-> ch ) )
10	9	elrab 3214	. . 3 |- ( .0. e. { z e. B | ps } <-> ( .0. e. B /\ ch ) )
11	7, 8, 10	sylanbrc 664	. 2 |- ( ph -> .0. e. { z e. B | ps } )
12		issubmd.th	. . . . . 6 |- ( z = x -> ( ps <-> th ) )
13	12	elrab 3214	. . . . 5 |- ( x e. { z e. B | ps } <-> ( x e. B /\ th ) )
14		issubmd.ta	. . . . . 6 |- ( z = y -> ( ps <-> ta ) )
15	14	elrab 3214	. . . . 5 |- ( y e. { z e. B | ps } <-> ( y e. B /\ ta ) )
16	13, 15	anbi12i 697	. . . 4 |- ( ( x e. { z e. B | ps } /\ y e. { z e. B | ps } ) <-> ( ( x e. B /\ th ) /\ ( y e. B /\ ta ) ) )
17	3	adantr 465	. . . . . 6 |- ( ( ph /\ ( ( x e. B /\ th ) /\ ( y e. B /\ ta ) ) ) -> M e. Mnd )
18		simprll 761	. . . . . 6 |- ( ( ph /\ ( ( x e. B /\ th ) /\ ( y e. B /\ ta ) ) ) -> x e. B )
19		simprrl 763	. . . . . 6 |- ( ( ph /\ ( ( x e. B /\ th ) /\ ( y e. B /\ ta ) ) ) -> y e. B )
20		issubmd.p	. . . . . . 7 |- .+ = ( +g ` M )
21	4, 20	mndcl 15522	. . . . . 6 |- ( ( M e. Mnd /\ x e. B /\ y e. B ) -> ( x .+ y ) e. B )
22	17, 18, 19, 21	syl3anc 1219	. . . . 5 |- ( ( ph /\ ( ( x e. B /\ th ) /\ ( y e. B /\ ta ) ) ) -> ( x .+ y ) e. B )
23		an4 820	. . . . . 6 |- ( ( ( x e. B /\ th ) /\ ( y e. B /\ ta ) ) <-> ( ( x e. B /\ y e. B ) /\ ( th /\ ta ) ) )
24		issubmd.cp	. . . . . 6 |- ( ( ph /\ ( ( x e. B /\ y e. B ) /\ ( th /\ ta ) ) ) -> et )
25	23, 24	sylan2b 475	. . . . 5 |- ( ( ph /\ ( ( x e. B /\ th ) /\ ( y e. B /\ ta ) ) ) -> et )
26		issubmd.et	. . . . . 6 |- ( z = ( x .+ y ) -> ( ps <-> et ) )
27	26	elrab 3214	. . . . 5 |- ( ( x .+ y ) e. { z e. B | ps } <-> ( ( x .+ y ) e. B /\ et ) )
28	22, 25, 27	sylanbrc 664	. . . 4 |- ( ( ph /\ ( ( x e. B /\ th ) /\ ( y e. B /\ ta ) ) ) -> ( x .+ y ) e. { z e. B | ps } )
29	16, 28	sylan2b 475	. . 3 |- ( ( ph /\ ( x e. { z e. B | ps } /\ y e. { z e. B | ps } ) ) -> ( x .+ y ) e. { z e. B | ps } )
30	29	ralrimivva 2904	. 2 |- ( ph -> A. x e. { z e. B | ps } A. y e. { z e. B | ps } ( x .+ y ) e. { z e. B | ps } )
31	4, 5, 20	issubm 15577	. . 3 |- ( M e. Mnd -> ( { z e. B | ps } e. (SubMnd ` M ) <-> ( { z e. B | ps } C_ B /\ .0. e. { z e. B | ps } /\ A. x e. { z e. B | ps } A. y e. { z e. B | ps } ( x .+ y ) e. { z e. B | ps } ) ) )
32	3, 31	syl 16	. 2 |- ( ph -> ( { z e. B | ps } e. (SubMnd ` M ) <-> ( { z e. B | ps } C_ B /\ .0. e. { z e. B | ps } /\ A. x e. { z e. B | ps } A. y e. { z e. B | ps } ( x .+ y ) e. { z e. B | ps } ) ) )
33	2, 11, 30, 32	mpbir3and 1171	1 |- ( ph -> { z e. B | ps } e. (SubMnd ` M ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   A.wral 2795   {crab 2799   C_ wss 3426   ` cfv 5516  (class class class)co 6190   Basecbs 14276   +g cplusg 14340   0gc0g 14480   Mndcmnd 15511  SubMndcsubmnd 15565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-iota 5479  df-fun 5518  df-fv 5524  df-riota 6151  df-ov 6193  df-0g 14482  df-mnd 15517  df-submnd 15567

This theorem is referenced by:  mrcmndind  15596