Theorem imasmnd 13599

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 1045	. . . 4 |- ( ( ph /\ x e. V /\ y e. V ) -> R e. Mnd )
8		simp2 1025	. . . . 5 |- ( ( ph /\ x e. V /\ y e. V ) -> x e. V )
9	2	3ad2ant1 1045	. . . . 5 |- ( ( ph /\ x e. V /\ y e. V ) -> V = ( Base ` R ) )
10	8, 9	eleqtrd 2310	. . . 4 |- ( ( ph /\ x e. V /\ y e. V ) -> x e. ( Base ` R ) )
11		simp3 1026	. . . . 5 |- ( ( ph /\ x e. V /\ y e. V ) -> y e. V )
12	11, 9	eleqtrd 2310	. . . 4 |- ( ( ph /\ x e. V /\ y e. V ) -> y e. ( Base ` R ) )
13		eqid 2231	. . . . 5 |- ( Base ` R ) = ( Base ` R )
14	13, 3	mndcl 13569	. . . 4 |- ( ( R e. Mnd /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x .+ y ) e. ( Base ` R ) )
15	7, 10, 12, 14	syl3anc 1274	. . 3 |- ( ( ph /\ x e. V /\ y e. V ) -> ( x .+ y ) e. ( Base ` R ) )
16	15, 9	eleqtrrd 2311	. 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 1241	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> x e. ( Base ` R ) )
19	12	3adant3r3 1241	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> y e. ( Base ` R ) )
20		simpr3 1032	. . . . 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 2310	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> z e. ( Base ` R ) )
23	13, 3	mndass 13570	. . . 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 1276	. . 3 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )
25	24	fveq2d 5652	. 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 13576	. . . 4 |- ( R e. Mnd -> .0. e. ( Base ` R ) )
28	6, 27	syl 14	. . 3 |- ( ph -> .0. e. ( Base ` R ) )
29	28, 2	eleqtrrd 2311	. 2 |- ( ph -> .0. e. V )
30	2	eleq2d 2301	. . . . 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 13581	. . . 4 |- ( ( R e. Mnd /\ x e. ( Base ` R ) ) -> ( .0. .+ x ) = x )
33	6, 31, 32	syl2an2r 599	. . 3 |- ( ( ph /\ x e. V ) -> ( .0. .+ x ) = x )
34	33	fveq2d 5652	. 2 |- ( ( ph /\ x e. V ) -> ( F ` ( .0. .+ x ) ) = ( F ` x ) )
35	13, 3, 26	mndrid 13582	. . . 4 |- ( ( R e. Mnd /\ x e. ( Base ` R ) ) -> ( x .+ .0. ) = x )
36	6, 31, 35	syl2an2r 599	. . 3 |- ( ( ph /\ x e. V ) -> ( x .+ .0. ) = x )
37	36	fveq2d 5652	. 2 |- ( ( ph /\ x e. V ) -> ( F ` ( x .+ .0. ) ) = ( F ` x ) )
38	1, 2, 3, 4, 5, 6, 16, 25, 29, 34, 37	imasmnd2 13598	1 |- ( ph -> ( U e. Mnd /\ ( F ` .0. ) = ( 0g ` U ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2202   -onto->wfo 5331   ` cfv 5333  (class class class)co 6028   Basecbs 13145   +g cplusg 13223   0gc0g 13402   "_s cimas 13445   Mndcmnd 13562

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-pre-ltirr 8187  ax-pre-lttrn 8189  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-tp 3681  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8258  df-mnf 8259  df-ltxr 8261  df-inn 9186  df-2 9244  df-3 9245  df-ndx 13148  df-slot 13149  df-base 13151  df-plusg 13236  df-mulr 13237  df-0g 13404  df-iimas 13448  df-mgm 13502  df-sgrp 13548  df-mnd 13563

This theorem is referenced by:  imasmndf1  13600