Theorem mvrf1fi 15215

Description: The power series variable function is injective if the base ring is nonzero. (Contributed by Mario Carneiro, 29-Dec-2014.)

Hypotheses

Ref	Expression
mvrf.s	|- S = ( I mPwSer R )
mvrf.v	|- V = ( I mVar R )
mvrf.b	|- B = ( Base ` S )
mvrffi.i	|- ( ph -> I e. Fin )
mvrf.r	|- ( ph -> R e. Ring )
mvrf1.z	|- .0. = ( 0g ` R )
mvrf1.o	|- .1. = ( 1r ` R )
mvrf1.n	|- ( ph -> .1. =/= .0. )

Assertion

Ref	Expression
mvrf1fi	|- ( ph -> V : I -1-1-> B )

Proof of Theorem mvrf1fi

Dummy variables h x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mvrf.s	. . 3 |- S = ( I mPwSer R )
2		mvrf.v	. . 3 |- V = ( I mVar R )
3		mvrf.b	. . 3 |- B = ( Base ` S )
4		mvrffi.i	. . 3 |- ( ph -> I e. Fin )
5		mvrf.r	. . 3 |- ( ph -> R e. Ring )
6	1, 2, 3, 4, 5	mvrffi 15214	. 2 |- ( ph -> V : I --> B )
7		mvrf1.n	. . . . . 6 |- ( ph -> .1. =/= .0. )
8	7	adantr 276	. . . . 5 |- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) ) -> .1. =/= .0. )
9	4	adantr 276	. . . . . . 7 |- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) ) -> I e. Fin )
10		simprll 543	. . . . . . 7 |- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) ) -> x e. I )
11		simprlr 544	. . . . . . 7 |- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) ) -> y e. I )
12		fidceq 7176	. . . . . . 7 |- ( ( I e. Fin /\ x e. I /\ y e. I ) -> DECID x = y )
13	9, 10, 11, 12	syl3anc 1278	. . . . . 6 |- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) ) -> DECID x = y )
14		simp2r 1055	. . . . . . . . . . 11 |- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> ( V ` x ) = ( V ` y ) )
15	14	fveq1d 5700	. . . . . . . . . 10 |- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> ( ( V ` x ) ` ( z e. I |-> if ( z = x , 1 , 0 ) ) ) = ( ( V ` y ) ` ( z e. I |-> if ( z = x , 1 , 0 ) ) ) )
16		eqid 2238	. . . . . . . . . . 11 |- { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin }
17		mvrf1.z	. . . . . . . . . . 11 |- .0. = ( 0g ` R )
18		mvrf1.o	. . . . . . . . . . 11 |- .1. = ( 1r ` R )
19	4	3ad2ant1 1049	. . . . . . . . . . 11 |- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> I e. Fin )
20	5	3ad2ant1 1049	. . . . . . . . . . 11 |- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> R e. Ring )
21		simp2ll 1095	. . . . . . . . . . 11 |- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> x e. I )
22	2, 16, 17, 18, 19, 20, 21	mvridfi 15213	. . . . . . . . . 10 |- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> ( ( V ` x ) ` ( z e. I |-> if ( z = x , 1 , 0 ) ) ) = .1. )
23		simp2lr 1096	. . . . . . . . . . 11 |- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> y e. I )
24		1nn0 9591	. . . . . . . . . . . . 13 |- 1 e. NN0
25	24	a1i 9	. . . . . . . . . . . 12 |- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> 1 e. NN0 )
26	16	snifpsrbagfi 15170	. . . . . . . . . . . 12 |- ( ( I e. Fin /\ 1 e. NN0 /\ x e. I ) -> ( z e. I |-> if ( z = x , 1 , 0 ) ) e. { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } )
27	19, 25, 21, 26	syl3anc 1278	. . . . . . . . . . 11 |- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> ( z e. I |-> if ( z = x , 1 , 0 ) ) e. { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } )
28	2, 16, 17, 18, 19, 20, 23, 27	mvrval2 15212	. . . . . . . . . 10 |- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> ( ( V ` y ) ` ( z e. I |-> if ( z = x , 1 , 0 ) ) ) = if ( ( z e. I |-> if ( z = x , 1 , 0 ) ) = ( z e. I |-> if ( z = y , 1 , 0 ) ) , .1. , .0. ) )
29	15, 22, 28	3eqtr3d 2279	. . . . . . . . 9 |- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> .1. = if ( ( z e. I |-> if ( z = x , 1 , 0 ) ) = ( z e. I |-> if ( z = y , 1 , 0 ) ) , .1. , .0. ) )
30		simp3 1030	. . . . . . . . . . 11 |- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> -. x = y )
31		mpteqb 5799	. . . . . . . . . . . . . . 15 |- ( A. z e. I if ( z = x , 1 , 0 ) e. _V -> ( ( z e. I |-> if ( z = x , 1 , 0 ) ) = ( z e. I |-> if ( z = y , 1 , 0 ) ) <-> A. z e. I if ( z = x , 1 , 0 ) = if ( z = y , 1 , 0 ) ) )
32		1ex 8328	. . . . . . . . . . . . . . . . 17 |- 1 e. _V
33		c0ex 8327	. . . . . . . . . . . . . . . . 17 |- 0 e. _V
34	32, 33	ifex 4634	. . . . . . . . . . . . . . . 16 |- if ( z = x , 1 , 0 ) e. _V
35	34	a1i 9	. . . . . . . . . . . . . . 15 |- ( z e. I -> if ( z = x , 1 , 0 ) e. _V )
36	31, 35	mprg 2607	. . . . . . . . . . . . . 14 |- ( ( z e. I |-> if ( z = x , 1 , 0 ) ) = ( z e. I |-> if ( z = y , 1 , 0 ) ) <-> A. z e. I if ( z = x , 1 , 0 ) = if ( z = y , 1 , 0 ) )
37		iftrue 3645	. . . . . . . . . . . . . . . 16 |- ( z = x -> if ( z = x , 1 , 0 ) = 1 )
38		eqeq1 2245	. . . . . . . . . . . . . . . . 17 |- ( z = x -> ( z = y <-> x = y ) )
39	38	ifbid 3662	. . . . . . . . . . . . . . . 16 |- ( z = x -> if ( z = y , 1 , 0 ) = if ( x = y , 1 , 0 ) )
40	37, 39	eqeq12d 2253	. . . . . . . . . . . . . . 15 |- ( z = x -> ( if ( z = x , 1 , 0 ) = if ( z = y , 1 , 0 ) <-> 1 = if ( x = y , 1 , 0 ) ) )
41	40	rspcv 2925	. . . . . . . . . . . . . 14 |- ( x e. I -> ( A. z e. I if ( z = x , 1 , 0 ) = if ( z = y , 1 , 0 ) -> 1 = if ( x = y , 1 , 0 ) ) )
42	36, 41	biimtrid 152	. . . . . . . . . . . . 13 |- ( x e. I -> ( ( z e. I |-> if ( z = x , 1 , 0 ) ) = ( z e. I |-> if ( z = y , 1 , 0 ) ) -> 1 = if ( x = y , 1 , 0 ) ) )
43	21, 42	syl 14	. . . . . . . . . . . 12 |- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> ( ( z e. I |-> if ( z = x , 1 , 0 ) ) = ( z e. I |-> if ( z = y , 1 , 0 ) ) -> 1 = if ( x = y , 1 , 0 ) ) )
44		1ne0 9382	. . . . . . . . . . . . . 14 |- 1 =/= 0
45		eqeq1 2245	. . . . . . . . . . . . . . 15 |- ( 1 = if ( x = y , 1 , 0 ) -> ( 1 = 0 <-> if ( x = y , 1 , 0 ) = 0 ) )
46	45	necon3abid 2459	. . . . . . . . . . . . . 14 |- ( 1 = if ( x = y , 1 , 0 ) -> ( 1 =/= 0 <-> -. if ( x = y , 1 , 0 ) = 0 ) )
47	44, 46	mpbii 148	. . . . . . . . . . . . 13 |- ( 1 = if ( x = y , 1 , 0 ) -> -. if ( x = y , 1 , 0 ) = 0 )
48		iffalse 3648	. . . . . . . . . . . . 13 |- ( -. x = y -> if ( x = y , 1 , 0 ) = 0 )
49	47, 48	nsyl 637	. . . . . . . . . . . 12 |- ( 1 = if ( x = y , 1 , 0 ) -> -. -. x = y )
50	43, 49	syl6 33	. . . . . . . . . . 11 |- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> ( ( z e. I |-> if ( z = x , 1 , 0 ) ) = ( z e. I |-> if ( z = y , 1 , 0 ) ) -> -. -. x = y ) )
51	30, 50	mt2d 634	. . . . . . . . . 10 |- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> -. ( z e. I |-> if ( z = x , 1 , 0 ) ) = ( z e. I |-> if ( z = y , 1 , 0 ) ) )
52	51	iffalsed 3650	. . . . . . . . 9 |- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> if ( ( z e. I |-> if ( z = x , 1 , 0 ) ) = ( z e. I |-> if ( z = y , 1 , 0 ) ) , .1. , .0. ) = .0. )
53	29, 52	eqtrd 2271	. . . . . . . 8 |- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) /\ -. x = y ) -> .1. = .0. )
54	53	3expia 1236	. . . . . . 7 |- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) ) -> ( -. x = y -> .1. = .0. ) )
55	54	necon3ad 2462	. . . . . 6 |- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) ) -> ( .1. =/= .0. -> -. -. x = y ) )
56		notnotrdc 855	. . . . . 6 |- (DECID x = y -> ( -. -. x = y -> x = y ) )
57	13, 55, 56	sylsyld 58	. . . . 5 |- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) ) -> ( .1. =/= .0. -> x = y ) )
58	8, 57	mpd 13	. . . 4 |- ( ( ph /\ ( ( x e. I /\ y e. I ) /\ ( V ` x ) = ( V ` y ) ) ) -> x = y )
59	58	expr 375	. . 3 |- ( ( ph /\ ( x e. I /\ y e. I ) ) -> ( ( V ` x ) = ( V ` y ) -> x = y ) )
60	59	ralrimivva 2632	. 2 |- ( ph -> A. x e. I A. y e. I ( ( V ` x ) = ( V ` y ) -> x = y ) )
61		dff13 5977	. 2 |- ( V : I -1-1-> B <-> ( V : I --> B /\ A. x e. I A. y e. I ( ( V ` x ) = ( V ` y ) -> x = y ) ) )
62	6, 60, 61	sylanbrc 421	1 |- ( ph -> V : I -1-1-> B )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 846   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   A.wral 2528   {crab 2532   _Vcvv 2821   ifcif 3638   |-> cmpt 4193   `'ccnv 4775   "cima 4779   -->wf 5376   -1-1->wf1 5377   ` cfv 5380  (class class class)co 6088   ^m cmap 6926   Fincfn 7027   0cc0 8186   1c1 8187   NNcn 9314   NN0cn0 9575   Basecbs 13411   0gc0g 13670   1rcur 14370   Ringcrg 14408   mPwSer cmps 15155   mVar cmvr 15156

This proof depends on 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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-addcom 8286  ax-addass 8288  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-0id 8294  ax-rnegex 8295  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-apti 8301  ax-pre-ltadd 8302

This proof depends on definitions:  df-bi 117  df-dc 847  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-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-of 6305  df-1st 6378  df-2nd 6379  df-1o 6691  df-er 6811  df-map 6928  df-ixp 6986  df-en 7028  df-fin 7030  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-inn 9315  df-2 9373  df-3 9374  df-4 9375  df-5 9376  df-6 9377  df-7 9378  df-8 9379  df-9 9380  df-n0 9576  df-z 9657  df-uz 9939  df-fz 10430  df-struct 13413  df-ndx 13414  df-slot 13415  df-base 13417  df-sets 13418  df-plusg 13504  df-mulr 13505  df-sca 13507  df-vsca 13508  df-tset 13510  df-rest 13655  df-topn 13656  df-0g 13672  df-topgen 13674  df-pt 13675  df-mgm 13736  df-sgrp 13777  df-mnd 13790  df-grp 13868  df-mgp 14326  df-ur 14371  df-ring 14410  df-psr 15158  df-mvr 15159

This theorem is used by: (None)