Theorem imasmnd 13481

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 1042	. . . 4 |- ( ( ph /\ x e. V /\ y e. V ) -> R e. Mnd )
8		simp2 1022	. . . . 5 |- ( ( ph /\ x e. V /\ y e. V ) -> x e. V )
9	2	3ad2ant1 1042	. . . . 5 |- ( ( ph /\ x e. V /\ y e. V ) -> V = ( Base ` R ) )
10	8, 9	eleqtrd 2308	. . . 4 |- ( ( ph /\ x e. V /\ y e. V ) -> x e. ( Base ` R ) )
11		simp3 1023	. . . . 5 |- ( ( ph /\ x e. V /\ y e. V ) -> y e. V )
12	11, 9	eleqtrd 2308	. . . 4 |- ( ( ph /\ x e. V /\ y e. V ) -> y e. ( Base ` R ) )
13		eqid 2229	. . . . 5 |- ( Base ` R ) = ( Base ` R )
14	13, 3	mndcl 13451	. . . 4 |- ( ( R e. Mnd /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x .+ y ) e. ( Base ` R ) )
15	7, 10, 12, 14	syl3anc 1271	. . 3 |- ( ( ph /\ x e. V /\ y e. V ) -> ( x .+ y ) e. ( Base ` R ) )
16	15, 9	eleqtrrd 2309	. 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 1238	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> x e. ( Base ` R ) )
19	12	3adant3r3 1238	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> y e. ( Base ` R ) )
20		simpr3 1029	. . . . 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 2308	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> z e. ( Base ` R ) )
23	13, 3	mndass 13452	. . . 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 1273	. . 3 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )
25	24	fveq2d 5630	. 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 13458	. . . 4 |- ( R e. Mnd -> .0. e. ( Base ` R ) )
28	6, 27	syl 14	. . 3 |- ( ph -> .0. e. ( Base ` R ) )
29	28, 2	eleqtrrd 2309	. 2 |- ( ph -> .0. e. V )
30	2	eleq2d 2299	. . . . 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 13463	. . . 4 |- ( ( R e. Mnd /\ x e. ( Base ` R ) ) -> ( .0. .+ x ) = x )
33	6, 31, 32	syl2an2r 597	. . 3 |- ( ( ph /\ x e. V ) -> ( .0. .+ x ) = x )
34	33	fveq2d 5630	. 2 |- ( ( ph /\ x e. V ) -> ( F ` ( .0. .+ x ) ) = ( F ` x ) )
35	13, 3, 26	mndrid 13464	. . . 4 |- ( ( R e. Mnd /\ x e. ( Base ` R ) ) -> ( x .+ .0. ) = x )
36	6, 31, 35	syl2an2r 597	. . 3 |- ( ( ph /\ x e. V ) -> ( x .+ .0. ) = x )
37	36	fveq2d 5630	. 2 |- ( ( ph /\ x e. V ) -> ( F ` ( x .+ .0. ) ) = ( F ` x ) )
38	1, 2, 3, 4, 5, 6, 16, 25, 29, 34, 37	imasmnd2 13480	1 |- ( ph -> ( U e. Mnd /\ ( F ` .0. ) = ( 0g ` U ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   -onto->wfo 5315   ` cfv 5317  (class class class)co 6000   Basecbs 13027   +g cplusg 13105   0gc0g 13284   "_s cimas 13327   Mndcmnd 13444

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-pre-ltirr 8107  ax-pre-lttrn 8109  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-tp 3674  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-pnf 8179  df-mnf 8180  df-ltxr 8182  df-inn 9107  df-2 9165  df-3 9166  df-ndx 13030  df-slot 13031  df-base 13033  df-plusg 13118  df-mulr 13119  df-0g 13286  df-iimas 13330  df-mgm 13384  df-sgrp 13430  df-mnd 13445

This theorem is referenced by:  imasmndf1  13482