Theorem imasmnd 13360

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 1021	. . . 4 |- ( ( ph /\ x e. V /\ y e. V ) -> R e. Mnd )
8		simp2 1001	. . . . 5 |- ( ( ph /\ x e. V /\ y e. V ) -> x e. V )
9	2	3ad2ant1 1021	. . . . 5 |- ( ( ph /\ x e. V /\ y e. V ) -> V = ( Base ` R ) )
10	8, 9	eleqtrd 2285	. . . 4 |- ( ( ph /\ x e. V /\ y e. V ) -> x e. ( Base ` R ) )
11		simp3 1002	. . . . 5 |- ( ( ph /\ x e. V /\ y e. V ) -> y e. V )
12	11, 9	eleqtrd 2285	. . . 4 |- ( ( ph /\ x e. V /\ y e. V ) -> y e. ( Base ` R ) )
13		eqid 2206	. . . . 5 |- ( Base ` R ) = ( Base ` R )
14	13, 3	mndcl 13330	. . . 4 |- ( ( R e. Mnd /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x .+ y ) e. ( Base ` R ) )
15	7, 10, 12, 14	syl3anc 1250	. . 3 |- ( ( ph /\ x e. V /\ y e. V ) -> ( x .+ y ) e. ( Base ` R ) )
16	15, 9	eleqtrrd 2286	. 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 1217	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> x e. ( Base ` R ) )
19	12	3adant3r3 1217	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> y e. ( Base ` R ) )
20		simpr3 1008	. . . . 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 2285	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> z e. ( Base ` R ) )
23	13, 3	mndass 13331	. . . 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 1252	. . 3 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )
25	24	fveq2d 5593	. 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 13337	. . . 4 |- ( R e. Mnd -> .0. e. ( Base ` R ) )
28	6, 27	syl 14	. . 3 |- ( ph -> .0. e. ( Base ` R ) )
29	28, 2	eleqtrrd 2286	. 2 |- ( ph -> .0. e. V )
30	2	eleq2d 2276	. . . . 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 13342	. . . 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 5593	. 2 |- ( ( ph /\ x e. V ) -> ( F ` ( .0. .+ x ) ) = ( F ` x ) )
35	13, 3, 26	mndrid 13343	. . . 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 5593	. 2 |- ( ( ph /\ x e. V ) -> ( F ` ( x .+ .0. ) ) = ( F ` x ) )
38	1, 2, 3, 4, 5, 6, 16, 25, 29, 34, 37	imasmnd2 13359	1 |- ( ph -> ( U e. Mnd /\ ( F ` .0. ) = ( 0g ` U ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   e. wcel 2177   -onto->wfo 5278   ` cfv 5280  (class class class)co 5957   Basecbs 12907   +g cplusg 12984   0gc0g 13163   "_s cimas 13206   Mndcmnd 13323

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-addcom 8045  ax-addass 8047  ax-i2m1 8050  ax-0lt1 8051  ax-0id 8053  ax-rnegex 8054  ax-pre-ltirr 8057  ax-pre-lttrn 8059  ax-pre-ltadd 8061

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-tp 3646  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-pnf 8129  df-mnf 8130  df-ltxr 8132  df-inn 9057  df-2 9115  df-3 9116  df-ndx 12910  df-slot 12911  df-base 12913  df-plusg 12997  df-mulr 12998  df-0g 13165  df-iimas 13209  df-mgm 13263  df-sgrp 13309  df-mnd 13324

This theorem is referenced by:  imasmndf1  13361