Theorem issubmd 13046

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 3264	. . 3 |- { z e. B | ps } C_ B
2	1	a1i 9	. 2 |- ( ph -> { z e. B | ps } C_ B )
3		issubmd.ch	. . 3 |- ( z = .0. -> ( ps <-> ch ) )
4		issubmd.m	. . . 4 |- ( ph -> M e. Mnd )
5		issubmd.b	. . . . 5 |- B = ( Base ` M )
6		issubmd.z	. . . . 5 |- .0. = ( 0g ` M )
7	5, 6	mndidcl 13011	. . . 4 |- ( M e. Mnd -> .0. e. B )
8	4, 7	syl 14	. . 3 |- ( ph -> .0. e. B )
9		issubmd.cz	. . 3 |- ( ph -> ch )
10	3, 8, 9	elrabd 2918	. 2 |- ( ph -> .0. e. { z e. B | ps } )
11		issubmd.th	. . . . . 6 |- ( z = x -> ( ps <-> th ) )
12	11	elrab 2916	. . . . 5 |- ( x e. { z e. B | ps } <-> ( x e. B /\ th ) )
13		issubmd.ta	. . . . . 6 |- ( z = y -> ( ps <-> ta ) )
14	13	elrab 2916	. . . . 5 |- ( y e. { z e. B | ps } <-> ( y e. B /\ ta ) )
15	12, 14	anbi12i 460	. . . 4 |- ( ( x e. { z e. B | ps } /\ y e. { z e. B | ps } ) <-> ( ( x e. B /\ th ) /\ ( y e. B /\ ta ) ) )
16		issubmd.et	. . . . 5 |- ( z = ( x .+ y ) -> ( ps <-> et ) )
17	4	adantr 276	. . . . . 6 |- ( ( ph /\ ( ( x e. B /\ th ) /\ ( y e. B /\ ta ) ) ) -> M e. Mnd )
18		simprll 537	. . . . . 6 |- ( ( ph /\ ( ( x e. B /\ th ) /\ ( y e. B /\ ta ) ) ) -> x e. B )
19		simprrl 539	. . . . . 6 |- ( ( ph /\ ( ( x e. B /\ th ) /\ ( y e. B /\ ta ) ) ) -> y e. B )
20		issubmd.p	. . . . . . 7 |- .+ = ( +g ` M )
21	5, 20	mndcl 13004	. . . . . 6 |- ( ( M e. Mnd /\ x e. B /\ y e. B ) -> ( x .+ y ) e. B )
22	17, 18, 19, 21	syl3anc 1249	. . . . 5 |- ( ( ph /\ ( ( x e. B /\ th ) /\ ( y e. B /\ ta ) ) ) -> ( x .+ y ) e. B )
23		an4 586	. . . . . 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 287	. . . . 5 |- ( ( ph /\ ( ( x e. B /\ th ) /\ ( y e. B /\ ta ) ) ) -> et )
26	16, 22, 25	elrabd 2918	. . . 4 |- ( ( ph /\ ( ( x e. B /\ th ) /\ ( y e. B /\ ta ) ) ) -> ( x .+ y ) e. { z e. B | ps } )
27	15, 26	sylan2b 287	. . 3 |- ( ( ph /\ ( x e. { z e. B | ps } /\ y e. { z e. B | ps } ) ) -> ( x .+ y ) e. { z e. B | ps } )
28	27	ralrimivva 2576	. 2 |- ( ph -> A. x e. { z e. B | ps } A. y e. { z e. B | ps } ( x .+ y ) e. { z e. B | ps } )
29	5, 6, 20	issubm 13044	. . 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 } ) ) )
30	4, 29	syl 14	. 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 } ) ) )
31	2, 10, 28, 30	mpbir3and 1182	1 |- ( ph -> { z e. B | ps } e. (SubMnd ` M ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   A.wral 2472   {crab 2476   C_ wss 3153   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   0gc0g 12867   Mndcmnd 12997  SubMndcsubmnd 13030

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-riota 5873  df-ov 5921  df-inn 8983  df-2 9041  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-submnd 13032

This theorem is referenced by: (None)