Theorem gexval 17993

Description: Value of the exponent of a group. (Contributed by Mario Carneiro, 23-Apr-2016.) (Revised by AV, 26-Sep-2020.)

Hypotheses

Ref	Expression
gexval.1	|- X = ( Base ` G )
gexval.2	|- .x. = (.g ` G )
gexval.3	|- .0. = ( 0g ` G )
gexval.4	|- E = (gEx ` G )
gexval.i	|- I = { y e. NN | A. x e. X ( y .x. x ) = .0. }

Assertion

Ref	Expression
gexval	|- ( G e. V -> E = if ( I = (/) , 0 , inf ( I , RR , < ) ) )

Distinct variable groups:   x, y, .0.   x, G, y   x, V, y   x, .x. , y   x, X

Allowed substitution hints:   E( x, y)   I( x, y)   X( y)

Proof of Theorem gexval

Dummy variables g i are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gexval.4	. 2 |- E = (gEx ` G )
2		df-gex 17949	. . . 4 |- gEx = ( g e. _V |-> [_ { y e. NN | A. x e. ( Base ` g ) ( y (.g ` g ) x ) = ( 0g ` g ) } / i ]_ if ( i = (/) , 0 , inf ( i , RR , < ) ) )
3	2	a1i 11	. . 3 |- ( G e. V -> gEx = ( g e. _V |-> [_ { y e. NN | A. x e. ( Base ` g ) ( y (.g ` g ) x ) = ( 0g ` g ) } / i ]_ if ( i = (/) , 0 , inf ( i , RR , < ) ) ) )
4		nnex 11026	. . . . . 6 |- NN e. _V
5	4	rabex 4813	. . . . 5 |- { y e. NN | A. x e. ( Base ` g ) ( y (.g ` g ) x ) = ( 0g ` g ) } e. _V
6	5	a1i 11	. . . 4 |- ( ( G e. V /\ g = G ) -> { y e. NN | A. x e. ( Base ` g ) ( y (.g ` g ) x ) = ( 0g ` g ) } e. _V )
7		simpr 477	. . . . . . . . . . . . 13 |- ( ( G e. V /\ g = G ) -> g = G )
8	7	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( G e. V /\ g = G ) -> ( Base ` g ) = ( Base ` G ) )
9		gexval.1	. . . . . . . . . . . 12 |- X = ( Base ` G )
10	8, 9	syl6eqr 2674	. . . . . . . . . . 11 |- ( ( G e. V /\ g = G ) -> ( Base ` g ) = X )
11	7	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( ( G e. V /\ g = G ) -> (.g ` g ) = (.g ` G ) )
12		gexval.2	. . . . . . . . . . . . . 14 |- .x. = (.g ` G )
13	11, 12	syl6eqr 2674	. . . . . . . . . . . . 13 |- ( ( G e. V /\ g = G ) -> (.g ` g ) = .x. )
14	13	oveqd 6667	. . . . . . . . . . . 12 |- ( ( G e. V /\ g = G ) -> ( y (.g ` g ) x ) = ( y .x. x ) )
15	7	fveq2d 6195	. . . . . . . . . . . . 13 |- ( ( G e. V /\ g = G ) -> ( 0g ` g ) = ( 0g ` G ) )
16		gexval.3	. . . . . . . . . . . . 13 |- .0. = ( 0g ` G )
17	15, 16	syl6eqr 2674	. . . . . . . . . . . 12 |- ( ( G e. V /\ g = G ) -> ( 0g ` g ) = .0. )
18	14, 17	eqeq12d 2637	. . . . . . . . . . 11 |- ( ( G e. V /\ g = G ) -> ( ( y (.g ` g ) x ) = ( 0g ` g ) <-> ( y .x. x ) = .0. ) )
19	10, 18	raleqbidv 3152	. . . . . . . . . 10 |- ( ( G e. V /\ g = G ) -> ( A. x e. ( Base ` g ) ( y (.g ` g ) x ) = ( 0g ` g ) <-> A. x e. X ( y .x. x ) = .0. ) )
20	19	rabbidv 3189	. . . . . . . . 9 |- ( ( G e. V /\ g = G ) -> { y e. NN | A. x e. ( Base ` g ) ( y (.g ` g ) x ) = ( 0g ` g ) } = { y e. NN | A. x e. X ( y .x. x ) = .0. } )
21		gexval.i	. . . . . . . . 9 |- I = { y e. NN | A. x e. X ( y .x. x ) = .0. }
22	20, 21	syl6eqr 2674	. . . . . . . 8 |- ( ( G e. V /\ g = G ) -> { y e. NN | A. x e. ( Base ` g ) ( y (.g ` g ) x ) = ( 0g ` g ) } = I )
23	22	eqeq2d 2632	. . . . . . 7 |- ( ( G e. V /\ g = G ) -> ( i = { y e. NN | A. x e. ( Base ` g ) ( y (.g ` g ) x ) = ( 0g ` g ) } <-> i = I ) )
24	23	biimpa 501	. . . . . 6 |- ( ( ( G e. V /\ g = G ) /\ i = { y e. NN | A. x e. ( Base ` g ) ( y (.g ` g ) x ) = ( 0g ` g ) } ) -> i = I )
25	24	eqeq1d 2624	. . . . 5 |- ( ( ( G e. V /\ g = G ) /\ i = { y e. NN | A. x e. ( Base ` g ) ( y (.g ` g ) x ) = ( 0g ` g ) } ) -> ( i = (/) <-> I = (/) ) )
26	24	infeq1d 8383	. . . . 5 |- ( ( ( G e. V /\ g = G ) /\ i = { y e. NN | A. x e. ( Base ` g ) ( y (.g ` g ) x ) = ( 0g ` g ) } ) -> inf ( i , RR , < ) = inf ( I , RR , < ) )
27	25, 26	ifbieq2d 4111	. . . 4 |- ( ( ( G e. V /\ g = G ) /\ i = { y e. NN | A. x e. ( Base ` g ) ( y (.g ` g ) x ) = ( 0g ` g ) } ) -> if ( i = (/) , 0 , inf ( i , RR , < ) ) = if ( I = (/) , 0 , inf ( I , RR , < ) ) )
28	6, 27	csbied 3560	. . 3 |- ( ( G e. V /\ g = G ) -> [_ { y e. NN | A. x e. ( Base ` g ) ( y (.g ` g ) x ) = ( 0g ` g ) } / i ]_ if ( i = (/) , 0 , inf ( i , RR , < ) ) = if ( I = (/) , 0 , inf ( I , RR , < ) ) )
29		elex 3212	. . 3 |- ( G e. V -> G e. _V )
30		c0ex 10034	. . . . 5 |- 0 e. _V
31		ltso 10118	. . . . . 6 |- < Or RR
32	31	infex 8399	. . . . 5 |- inf ( I , RR , < ) e. _V
33	30, 32	ifex 4156	. . . 4 |- if ( I = (/) , 0 , inf ( I , RR , < ) ) e. _V
34	33	a1i 11	. . 3 |- ( G e. V -> if ( I = (/) , 0 , inf ( I , RR , < ) ) e. _V )
35	3, 28, 29, 34	fvmptd 6288	. 2 |- ( G e. V -> (gEx ` G ) = if ( I = (/) , 0 , inf ( I , RR , < ) ) )
36	1, 35	syl5eq 2668	1 |- ( G e. V -> E = if ( I = (/) , 0 , inf ( I , RR , < ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   [_csb 3533   (/)c0 3915   ifcif 4086   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650  infcinf 8347   RRcr 9935   0cc0 9936   < clt 10074   NNcn 11020   Basecbs 15857   0gc0g 16100  ._gcmg 17540  gExcgex 17945

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-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-i2m1 10004  ax-1ne0 10005  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011

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-ov 6653  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-ltxr 10079  df-nn 11021  df-gex 17949

This theorem is referenced by:  gexlem1  17994  gexlem2  17997