Theorem imasmnd 13737

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 1049	. . . 4 |- ( ( ph /\ x e. V /\ y e. V ) -> R e. Mnd )
8		simp2 1029	. . . . 5 |- ( ( ph /\ x e. V /\ y e. V ) -> x e. V )
9	2	3ad2ant1 1049	. . . . 5 |- ( ( ph /\ x e. V /\ y e. V ) -> V = ( Base ` R ) )
10	8, 9	eleqtrd 2317	. . . 4 |- ( ( ph /\ x e. V /\ y e. V ) -> x e. ( Base ` R ) )
11		simp3 1030	. . . . 5 |- ( ( ph /\ x e. V /\ y e. V ) -> y e. V )
12	11, 9	eleqtrd 2317	. . . 4 |- ( ( ph /\ x e. V /\ y e. V ) -> y e. ( Base ` R ) )
13		eqid 2238	. . . . 5 |- ( Base ` R ) = ( Base ` R )
14	13, 3	mndcl 13713	. . . 4 |- ( ( R e. Mnd /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x .+ y ) e. ( Base ` R ) )
15	7, 10, 12, 14	syl3anc 1278	. . 3 |- ( ( ph /\ x e. V /\ y e. V ) -> ( x .+ y ) e. ( Base ` R ) )
16	15, 9	eleqtrrd 2318	. 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 1245	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> x e. ( Base ` R ) )
19	12	3adant3r3 1245	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> y e. ( Base ` R ) )
20		simpr3 1036	. . . . 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 2317	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> z e. ( Base ` R ) )
23	13, 3	mndass 13714	. . . 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 1280	. . 3 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )
25	24	fveq2d 5694	. 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 13720	. . . 4 |- ( R e. Mnd -> .0. e. ( Base ` R ) )
28	6, 27	syl 14	. . 3 |- ( ph -> .0. e. ( Base ` R ) )
29	28, 2	eleqtrrd 2318	. 2 |- ( ph -> .0. e. V )
30	2	eleq2d 2308	. . . . 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 13725	. . . 4 |- ( ( R e. Mnd /\ x e. ( Base ` R ) ) -> ( .0. .+ x ) = x )
33	6, 31, 32	syl2an2r 603	. . 3 |- ( ( ph /\ x e. V ) -> ( .0. .+ x ) = x )
34	33	fveq2d 5694	. 2 |- ( ( ph /\ x e. V ) -> ( F ` ( .0. .+ x ) ) = ( F ` x ) )
35	13, 3, 26	mndrid 13726	. . . 4 |- ( ( R e. Mnd /\ x e. ( Base ` R ) ) -> ( x .+ .0. ) = x )
36	6, 31, 35	syl2an2r 603	. . 3 |- ( ( ph /\ x e. V ) -> ( x .+ .0. ) = x )
37	36	fveq2d 5694	. 2 |- ( ( ph /\ x e. V ) -> ( F ` ( x .+ .0. ) ) = ( F ` x ) )
38	1, 2, 3, 4, 5, 6, 16, 25, 29, 34, 37	imasmnd2 13736	1 |- ( ph -> ( U e. Mnd /\ ( F ` .0. ) = ( 0g ` U ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   -onto->wfo 5370   ` cfv 5372  (class class class)co 6075   Basecbs 13330   +g cplusg 13408   0gc0g 13587   "_s cimas 13599   Mndcmnd 13706

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-tp 3713  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-3 9343  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-mulr 13422  df-0g 13589  df-iimas 13601  df-mgm 13653  df-sgrp 13694  df-mnd 13707

This theorem is referenced by:  imasmndf1  13738