Theorem fidomndrnglem 19306

Description: Lemma for fidomndrng 19307. (Contributed by Mario Carneiro, 15-Jun-2015.)

Hypotheses

Ref	Expression
fidomndrng.b	|- B = ( Base ` R )
fidomndrng.z	|- .0. = ( 0g ` R )
fidomndrng.o	|- .1. = ( 1r ` R )
fidomndrng.d	|- .|| = ( ||r ` R )
fidomndrng.t	|- .x. = ( .r ` R )
fidomndrng.r	|- ( ph -> R e. Domn )
fidomndrng.x	|- ( ph -> B e. Fin )
fidomndrng.a	|- ( ph -> A e. ( B \ { .0. } ) )
fidomndrng.f	|- F = ( x e. B |-> ( x .x. A ) )

Assertion

Ref	Expression
fidomndrnglem	|- ( ph -> A .|| .1. )

Distinct variable groups:   x, A   x, B   x, R   x, .x.

Allowed substitution hints:   ph( x)   .|| ( x)   .1. ( x)   F( x)   .0. ( x)

Proof of Theorem fidomndrnglem

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fidomndrng.a	. . . 4 |- ( ph -> A e. ( B \ { .0. } ) )
2	1	eldifad 3586	. . 3 |- ( ph -> A e. B )
3		eldifsni 4320	. . . . . . . . . . . 12 |- ( A e. ( B \ { .0. } ) -> A =/= .0. )
4	1, 3	syl 17	. . . . . . . . . . 11 |- ( ph -> A =/= .0. )
5	4	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ y e. B ) /\ ( F ` y ) = .0. ) -> A =/= .0. )
6		oveq1 6657	. . . . . . . . . . . . . . . . 17 |- ( x = y -> ( x .x. A ) = ( y .x. A ) )
7		fidomndrng.f	. . . . . . . . . . . . . . . . 17 |- F = ( x e. B |-> ( x .x. A ) )
8		ovex 6678	. . . . . . . . . . . . . . . . 17 |- ( y .x. A ) e. _V
9	6, 7, 8	fvmpt 6282	. . . . . . . . . . . . . . . 16 |- ( y e. B -> ( F ` y ) = ( y .x. A ) )
10	9	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( ph /\ y e. B ) -> ( F ` y ) = ( y .x. A ) )
11	10	eqeq1d 2624	. . . . . . . . . . . . . 14 |- ( ( ph /\ y e. B ) -> ( ( F ` y ) = .0. <-> ( y .x. A ) = .0. ) )
12		fidomndrng.r	. . . . . . . . . . . . . . . 16 |- ( ph -> R e. Domn )
13	12	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ y e. B ) -> R e. Domn )
14		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( ph /\ y e. B ) -> y e. B )
15	2	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ y e. B ) -> A e. B )
16		fidomndrng.b	. . . . . . . . . . . . . . . 16 |- B = ( Base ` R )
17		fidomndrng.t	. . . . . . . . . . . . . . . 16 |- .x. = ( .r ` R )
18		fidomndrng.z	. . . . . . . . . . . . . . . 16 |- .0. = ( 0g ` R )
19	16, 17, 18	domneq0 19297	. . . . . . . . . . . . . . 15 |- ( ( R e. Domn /\ y e. B /\ A e. B ) -> ( ( y .x. A ) = .0. <-> ( y = .0. \/ A = .0. ) ) )
20	13, 14, 15, 19	syl3anc 1326	. . . . . . . . . . . . . 14 |- ( ( ph /\ y e. B ) -> ( ( y .x. A ) = .0. <-> ( y = .0. \/ A = .0. ) ) )
21	11, 20	bitrd 268	. . . . . . . . . . . . 13 |- ( ( ph /\ y e. B ) -> ( ( F ` y ) = .0. <-> ( y = .0. \/ A = .0. ) ) )
22	21	biimpa 501	. . . . . . . . . . . 12 |- ( ( ( ph /\ y e. B ) /\ ( F ` y ) = .0. ) -> ( y = .0. \/ A = .0. ) )
23	22	ord 392	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. B ) /\ ( F ` y ) = .0. ) -> ( -. y = .0. -> A = .0. ) )
24	23	necon1ad 2811	. . . . . . . . . 10 |- ( ( ( ph /\ y e. B ) /\ ( F ` y ) = .0. ) -> ( A =/= .0. -> y = .0. ) )
25	5, 24	mpd 15	. . . . . . . . 9 |- ( ( ( ph /\ y e. B ) /\ ( F ` y ) = .0. ) -> y = .0. )
26	25	ex 450	. . . . . . . 8 |- ( ( ph /\ y e. B ) -> ( ( F ` y ) = .0. -> y = .0. ) )
27	26	ralrimiva 2966	. . . . . . 7 |- ( ph -> A. y e. B ( ( F ` y ) = .0. -> y = .0. ) )
28		domnring 19296	. . . . . . . . . . 11 |- ( R e. Domn -> R e. Ring )
29	12, 28	syl 17	. . . . . . . . . 10 |- ( ph -> R e. Ring )
30	16, 17	ringrghm 18605	. . . . . . . . . 10 |- ( ( R e. Ring /\ A e. B ) -> ( x e. B |-> ( x .x. A ) ) e. ( R GrpHom R ) )
31	29, 2, 30	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( x e. B |-> ( x .x. A ) ) e. ( R GrpHom R ) )
32	7, 31	syl5eqel 2705	. . . . . . . 8 |- ( ph -> F e. ( R GrpHom R ) )
33	16, 16, 18, 18	ghmf1 17689	. . . . . . . 8 |- ( F e. ( R GrpHom R ) -> ( F : B -1-1-> B <-> A. y e. B ( ( F ` y ) = .0. -> y = .0. ) ) )
34	32, 33	syl 17	. . . . . . 7 |- ( ph -> ( F : B -1-1-> B <-> A. y e. B ( ( F ` y ) = .0. -> y = .0. ) ) )
35	27, 34	mpbird 247	. . . . . 6 |- ( ph -> F : B -1-1-> B )
36		fidomndrng.x	. . . . . . . 8 |- ( ph -> B e. Fin )
37		enrefg 7987	. . . . . . . 8 |- ( B e. Fin -> B ~~ B )
38	36, 37	syl 17	. . . . . . 7 |- ( ph -> B ~~ B )
39		f1finf1o 8187	. . . . . . 7 |- ( ( B ~~ B /\ B e. Fin ) -> ( F : B -1-1-> B <-> F : B -1-1-onto-> B ) )
40	38, 36, 39	syl2anc 693	. . . . . 6 |- ( ph -> ( F : B -1-1-> B <-> F : B -1-1-onto-> B ) )
41	35, 40	mpbid 222	. . . . 5 |- ( ph -> F : B -1-1-onto-> B )
42		f1ocnv 6149	. . . . 5 |- ( F : B -1-1-onto-> B -> `' F : B -1-1-onto-> B )
43		f1of 6137	. . . . 5 |- ( `' F : B -1-1-onto-> B -> `' F : B --> B )
44	41, 42, 43	3syl 18	. . . 4 |- ( ph -> `' F : B --> B )
45		fidomndrng.o	. . . . . 6 |- .1. = ( 1r ` R )
46	16, 45	ringidcl 18568	. . . . 5 |- ( R e. Ring -> .1. e. B )
47	29, 46	syl 17	. . . 4 |- ( ph -> .1. e. B )
48	44, 47	ffvelrnd 6360	. . 3 |- ( ph -> ( `' F ` .1. ) e. B )
49		fidomndrng.d	. . . 4 |- .|| = ( ||r ` R )
50	16, 49, 17	dvdsrmul 18648	. . 3 |- ( ( A e. B /\ ( `' F ` .1. ) e. B ) -> A .|| ( ( `' F ` .1. ) .x. A ) )
51	2, 48, 50	syl2anc 693	. 2 |- ( ph -> A .|| ( ( `' F ` .1. ) .x. A ) )
52		oveq1 6657	. . . . 5 |- ( y = ( `' F ` .1. ) -> ( y .x. A ) = ( ( `' F ` .1. ) .x. A ) )
53	6	cbvmptv 4750	. . . . . 6 |- ( x e. B |-> ( x .x. A ) ) = ( y e. B |-> ( y .x. A ) )
54	7, 53	eqtri 2644	. . . . 5 |- F = ( y e. B |-> ( y .x. A ) )
55		ovex 6678	. . . . 5 |- ( ( `' F ` .1. ) .x. A ) e. _V
56	52, 54, 55	fvmpt 6282	. . . 4 |- ( ( `' F ` .1. ) e. B -> ( F ` ( `' F ` .1. ) ) = ( ( `' F ` .1. ) .x. A ) )
57	48, 56	syl 17	. . 3 |- ( ph -> ( F ` ( `' F ` .1. ) ) = ( ( `' F ` .1. ) .x. A ) )
58		f1ocnvfv2 6533	. . . 4 |- ( ( F : B -1-1-onto-> B /\ .1. e. B ) -> ( F ` ( `' F ` .1. ) ) = .1. )
59	41, 47, 58	syl2anc 693	. . 3 |- ( ph -> ( F ` ( `' F ` .1. ) ) = .1. )
60	57, 59	eqtr3d 2658	. 2 |- ( ph -> ( ( `' F ` .1. ) .x. A ) = .1. )
61	51, 60	breqtrd 4679	1 |- ( ph -> A .|| .1. )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   \ cdif 3571   {csn 4177   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   -->wf 5884   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   ~~ cen 7952   Fincfn 7955   Basecbs 15857   .rcmulr 15942   0gc0g 16100   GrpHom cghm 17657   1rcur 18501   Ringcrg 18547   ||_rcdsr 18638  Domncdomn 19280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-ghm 17658  df-mgp 18490  df-ur 18502  df-ring 18549  df-dvdsr 18641  df-nzr 19258  df-domn 19284

This theorem is referenced by:  fidomndrng  19307