Theorem issubmd 16182

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 3571	. . 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 16140	. . . 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 3254	. . 3 |- ( .0. e. { z e. B | ps } <-> ( .0. e. B /\ ch ) )
11	7, 8, 10	sylanbrc 662	. 2 |- ( ph -> .0. e. { z e. B | ps } )
12		issubmd.th	. . . . . 6 |- ( z = x -> ( ps <-> th ) )
13	12	elrab 3254	. . . . 5 |- ( x e. { z e. B | ps } <-> ( x e. B /\ th ) )
14		issubmd.ta	. . . . . 6 |- ( z = y -> ( ps <-> ta ) )
15	14	elrab 3254	. . . . 5 |- ( y e. { z e. B | ps } <-> ( y e. B /\ ta ) )
16	13, 15	anbi12i 695	. . . 4 |- ( ( x e. { z e. B | ps } /\ y e. { z e. B | ps } ) <-> ( ( x e. B /\ th ) /\ ( y e. B /\ ta ) ) )
17	3	adantr 463	. . . . . 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 16131	. . . . . 6 |- ( ( M e. Mnd /\ x e. B /\ y e. B ) -> ( x .+ y ) e. B )
22	17, 18, 19, 21	syl3anc 1226	. . . . 5 |- ( ( ph /\ ( ( x e. B /\ th ) /\ ( y e. B /\ ta ) ) ) -> ( x .+ y ) e. B )
23		an4 822	. . . . . 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 473	. . . . 5 |- ( ( ph /\ ( ( x e. B /\ th ) /\ ( y e. B /\ ta ) ) ) -> et )
26		issubmd.et	. . . . . 6 |- ( z = ( x .+ y ) -> ( ps <-> et ) )
27	26	elrab 3254	. . . . 5 |- ( ( x .+ y ) e. { z e. B | ps } <-> ( ( x .+ y ) e. B /\ et ) )
28	22, 25, 27	sylanbrc 662	. . . 4 |- ( ( ph /\ ( ( x e. B /\ th ) /\ ( y e. B /\ ta ) ) ) -> ( x .+ y ) e. { z e. B | ps } )
29	16, 28	sylan2b 473	. . 3 |- ( ( ph /\ ( x e. { z e. B | ps } /\ y e. { z e. B | ps } ) ) -> ( x .+ y ) e. { z e. B | ps } )
30	29	ralrimivva 2875	. 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 16180	. . 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 1177	1 |- ( ph -> { z e. B | ps } e. (SubMnd ` M ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 971   = wceq 1398   e. wcel 1823   A.wral 2804   {crab 2808   C_ wss 3461   ` cfv 5570  (class class class)co 6270   Basecbs 14719   +g cplusg 14787   0gc0g 14932   Mndcmnd 16121  SubMndcsubmnd 16167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-riota 6232  df-ov 6273  df-0g 14934  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-submnd 16169

This theorem is referenced by:  mrcmndind  16199