Theorem imasmnd 13155

Description: The image structure of a monoid is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)

Hypotheses

Ref	Expression
imasmnd.u	|- ( ph -> U = ( F "s R ) )
imasmnd.v	|- ( ph -> V = ( Base ` R ) )
imasmnd.p	|- .+ = ( +g ` R )
imasmnd.f	|- ( ph -> F : V -onto-> B )
imasmnd.e	|- ( ( ph /\ ( a e. V /\ b e. V ) /\ ( p e. V /\ q e. V ) ) -> ( ( ( F ` a ) = ( F ` p ) /\ ( F ` b ) = ( F ` q ) ) -> ( F ` ( a .+ b ) ) = ( F ` ( p .+ q ) ) ) )
imasmnd.r	|- ( ph -> R e. Mnd )
imasmnd.z	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
imasmnd	|- ( ph -> ( U e. Mnd /\ ( F ` .0. ) = ( 0g ` U ) ) )

Distinct variable groups:   q, p, .+   a, b, p, q, ph   U, a, b, p, q   .0. , p, q   B, p, q   F, a, b, p, q   R, p, q   V, a, b, p, q

Allowed substitution hints:   B( a, b)   .+ ( a, b)   R( a, b)   .0. ( a, b)

Proof of Theorem imasmnd

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imasmnd.u	. 2 |- ( ph -> U = ( F "s R ) )
2		imasmnd.v	. 2 |- ( ph -> V = ( Base ` R ) )
3		imasmnd.p	. 2 |- .+ = ( +g ` R )
4		imasmnd.f	. 2 |- ( ph -> F : V -onto-> B )
5		imasmnd.e	. 2 |- ( ( ph /\ ( a e. V /\ b e. V ) /\ ( p e. V /\ q e. V ) ) -> ( ( ( F ` a ) = ( F ` p ) /\ ( F ` b ) = ( F ` q ) ) -> ( F ` ( a .+ b ) ) = ( F ` ( p .+ q ) ) ) )
6		imasmnd.r	. 2 |- ( ph -> R e. Mnd )
7	6	3ad2ant1 1020	. . . 4 |- ( ( ph /\ x e. V /\ y e. V ) -> R e. Mnd )
8		simp2 1000	. . . . 5 |- ( ( ph /\ x e. V /\ y e. V ) -> x e. V )
9	2	3ad2ant1 1020	. . . . 5 |- ( ( ph /\ x e. V /\ y e. V ) -> V = ( Base ` R ) )
10	8, 9	eleqtrd 2275	. . . 4 |- ( ( ph /\ x e. V /\ y e. V ) -> x e. ( Base ` R ) )
11		simp3 1001	. . . . 5 |- ( ( ph /\ x e. V /\ y e. V ) -> y e. V )
12	11, 9	eleqtrd 2275	. . . 4 |- ( ( ph /\ x e. V /\ y e. V ) -> y e. ( Base ` R ) )
13		eqid 2196	. . . . 5 |- ( Base ` R ) = ( Base ` R )
14	13, 3	mndcl 13125	. . . 4 |- ( ( R e. Mnd /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x .+ y ) e. ( Base ` R ) )
15	7, 10, 12, 14	syl3anc 1249	. . 3 |- ( ( ph /\ x e. V /\ y e. V ) -> ( x .+ y ) e. ( Base ` R ) )
16	15, 9	eleqtrrd 2276	. 2 |- ( ( ph /\ x e. V /\ y e. V ) -> ( x .+ y ) e. V )
17	6	adantr 276	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> R e. Mnd )
18	10	3adant3r3 1216	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> x e. ( Base ` R ) )
19	12	3adant3r3 1216	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> y e. ( Base ` R ) )
20		simpr3 1007	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> z e. V )
21	2	adantr 276	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> V = ( Base ` R ) )
22	20, 21	eleqtrd 2275	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> z e. ( Base ` R ) )
23	13, 3	mndass 13126	. . . 4 |- ( ( R e. Mnd /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )
24	17, 18, 19, 22, 23	syl13anc 1251	. . 3 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )
25	24	fveq2d 5565	. 2 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( F ` ( ( x .+ y ) .+ z ) ) = ( F ` ( x .+ ( y .+ z ) ) ) )
26		imasmnd.z	. . . . 5 |- .0. = ( 0g ` R )
27	13, 26	mndidcl 13132	. . . 4 |- ( R e. Mnd -> .0. e. ( Base ` R ) )
28	6, 27	syl 14	. . 3 |- ( ph -> .0. e. ( Base ` R ) )
29	28, 2	eleqtrrd 2276	. 2 |- ( ph -> .0. e. V )
30	2	eleq2d 2266	. . . . 5 |- ( ph -> ( x e. V <-> x e. ( Base ` R ) ) )
31	30	biimpa 296	. . . 4 |- ( ( ph /\ x e. V ) -> x e. ( Base ` R ) )
32	13, 3, 26	mndlid 13137	. . . 4 |- ( ( R e. Mnd /\ x e. ( Base ` R ) ) -> ( .0. .+ x ) = x )
33	6, 31, 32	syl2an2r 595	. . 3 |- ( ( ph /\ x e. V ) -> ( .0. .+ x ) = x )
34	33	fveq2d 5565	. 2 |- ( ( ph /\ x e. V ) -> ( F ` ( .0. .+ x ) ) = ( F ` x ) )
35	13, 3, 26	mndrid 13138	. . . 4 |- ( ( R e. Mnd /\ x e. ( Base ` R ) ) -> ( x .+ .0. ) = x )
36	6, 31, 35	syl2an2r 595	. . 3 |- ( ( ph /\ x e. V ) -> ( x .+ .0. ) = x )
37	36	fveq2d 5565	. 2 |- ( ( ph /\ x e. V ) -> ( F ` ( x .+ .0. ) ) = ( F ` x ) )
38	1, 2, 3, 4, 5, 6, 16, 25, 29, 34, 37	imasmnd2 13154	1 |- ( ph -> ( U e. Mnd /\ ( F ` .0. ) = ( 0g ` U ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2167   -onto->wfo 5257   ` cfv 5259  (class class class)co 5925   Basecbs 12703   +g cplusg 12780   0gc0g 12958   "_s cimas 13001   Mndcmnd 13118

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-in1 615  ax-in2 616  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-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-pre-ltirr 8008  ax-pre-lttrn 8010  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  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-ne 2368  df-nel 2463  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-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-tp 3631  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-ltxr 8083  df-inn 9008  df-2 9066  df-3 9067  df-ndx 12706  df-slot 12707  df-base 12709  df-plusg 12793  df-mulr 12794  df-0g 12960  df-iimas 13004  df-mgm 13058  df-sgrp 13104  df-mnd 13119

This theorem is referenced by:  imasmndf1  13156