Theorem efgmnvl 18127

Description: The inversion function on the generators is an involution. (Contributed by Mario Carneiro, 1-Oct-2015.)

Hypothesis

Ref	Expression
efgmval.m	|- M = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. )

Assertion

Ref	Expression
efgmnvl	|- ( A e. ( I X. 2o ) -> ( M ` ( M ` A ) ) = A )

Distinct variable group:   y, z, I

Allowed substitution hints:   A( y, z)   M( y, z)

Proof of Theorem efgmnvl

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxp2 5132	. 2 |- ( A e. ( I X. 2o ) <-> E. a e. I E. b e. 2o A = <. a , b >. )
2		efgmval.m	. . . . . . . 8 |- M = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. )
3	2	efgmval 18125	. . . . . . 7 |- ( ( a e. I /\ b e. 2o ) -> ( a M b ) = <. a , ( 1o \ b ) >. )
4	3	fveq2d 6195	. . . . . 6 |- ( ( a e. I /\ b e. 2o ) -> ( M ` ( a M b ) ) = ( M ` <. a , ( 1o \ b ) >. ) )
5		df-ov 6653	. . . . . 6 |- ( a M ( 1o \ b ) ) = ( M ` <. a , ( 1o \ b ) >. )
6	4, 5	syl6eqr 2674	. . . . 5 |- ( ( a e. I /\ b e. 2o ) -> ( M ` ( a M b ) ) = ( a M ( 1o \ b ) ) )
7		2oconcl 7583	. . . . . 6 |- ( b e. 2o -> ( 1o \ b ) e. 2o )
8	2	efgmval 18125	. . . . . 6 |- ( ( a e. I /\ ( 1o \ b ) e. 2o ) -> ( a M ( 1o \ b ) ) = <. a , ( 1o \ ( 1o \ b ) ) >. )
9	7, 8	sylan2 491	. . . . 5 |- ( ( a e. I /\ b e. 2o ) -> ( a M ( 1o \ b ) ) = <. a , ( 1o \ ( 1o \ b ) ) >. )
10		1on 7567	. . . . . . . . . . 11 |- 1o e. On
11	10	onordi 5832	. . . . . . . . . 10 |- Ord 1o
12		ordtr 5737	. . . . . . . . . 10 |- ( Ord 1o -> Tr 1o )
13		trsucss 5811	. . . . . . . . . 10 |- ( Tr 1o -> ( b e. suc 1o -> b C_ 1o ) )
14	11, 12, 13	mp2b 10	. . . . . . . . 9 |- ( b e. suc 1o -> b C_ 1o )
15		df-2o 7561	. . . . . . . . 9 |- 2o = suc 1o
16	14, 15	eleq2s 2719	. . . . . . . 8 |- ( b e. 2o -> b C_ 1o )
17	16	adantl 482	. . . . . . 7 |- ( ( a e. I /\ b e. 2o ) -> b C_ 1o )
18		dfss4 3858	. . . . . . 7 |- ( b C_ 1o <-> ( 1o \ ( 1o \ b ) ) = b )
19	17, 18	sylib 208	. . . . . 6 |- ( ( a e. I /\ b e. 2o ) -> ( 1o \ ( 1o \ b ) ) = b )
20	19	opeq2d 4409	. . . . 5 |- ( ( a e. I /\ b e. 2o ) -> <. a , ( 1o \ ( 1o \ b ) ) >. = <. a , b >. )
21	6, 9, 20	3eqtrd 2660	. . . 4 |- ( ( a e. I /\ b e. 2o ) -> ( M ` ( a M b ) ) = <. a , b >. )
22		fveq2 6191	. . . . . . 7 |- ( A = <. a , b >. -> ( M ` A ) = ( M ` <. a , b >. ) )
23		df-ov 6653	. . . . . . 7 |- ( a M b ) = ( M ` <. a , b >. )
24	22, 23	syl6eqr 2674	. . . . . 6 |- ( A = <. a , b >. -> ( M ` A ) = ( a M b ) )
25	24	fveq2d 6195	. . . . 5 |- ( A = <. a , b >. -> ( M ` ( M ` A ) ) = ( M ` ( a M b ) ) )
26		id 22	. . . . 5 |- ( A = <. a , b >. -> A = <. a , b >. )
27	25, 26	eqeq12d 2637	. . . 4 |- ( A = <. a , b >. -> ( ( M ` ( M ` A ) ) = A <-> ( M ` ( a M b ) ) = <. a , b >. ) )
28	21, 27	syl5ibrcom 237	. . 3 |- ( ( a e. I /\ b e. 2o ) -> ( A = <. a , b >. -> ( M ` ( M ` A ) ) = A ) )
29	28	rexlimivv 3036	. 2 |- ( E. a e. I E. b e. 2o A = <. a , b >. -> ( M ` ( M ` A ) ) = A )
30	1, 29	sylbi 207	1 |- ( A e. ( I X. 2o ) -> ( M ` ( M ` A ) ) = A )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   \ cdif 3571   C_ wss 3574   <.cop 4183   Tr wtr 4752   X. cxp 5112   Ord word 5722   suc csuc 5725   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1oc1o 7553   2oc2o 7554

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-pr 4906  ax-un 6949

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-br 4654  df-opab 4713  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-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1o 7560  df-2o 7561

This theorem is referenced by:  efginvrel1  18141  efgredlemc  18158