Theorem issubmd 13106

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 3268	. . 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 13071	. . . 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 2922	. 2 |- ( ph -> .0. e. { z e. B | ps } )
11		issubmd.th	. . . . . 6 |- ( z = x -> ( ps <-> th ) )
12	11	elrab 2920	. . . . 5 |- ( x e. { z e. B | ps } <-> ( x e. B /\ th ) )
13		issubmd.ta	. . . . . 6 |- ( z = y -> ( ps <-> ta ) )
14	13	elrab 2920	. . . . 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 13064	. . . . . 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 2922	. . . 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 2579	. 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 13104	. . 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 2167   A.wral 2475   {crab 2479   C_ wss 3157   ` cfv 5258  (class class class)co 5922   Basecbs 12678   +g cplusg 12755   0gc0g 12927   Mndcmnd 13057  SubMndcsubmnd 13090

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-riota 5877  df-ov 5925  df-inn 8991  df-2 9049  df-ndx 12681  df-slot 12682  df-base 12684  df-plusg 12768  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-submnd 13092

This theorem is referenced by: (None)