Theorem symgmatr01lem 20459

Description: Lemma for symgmatr01 20460. (Contributed by AV, 3-Jan-2019.)

Hypothesis

Ref	Expression
symgmatr01.p	|- P = ( Base ` ( SymGrp ` N ) )

Assertion

Ref	Expression
symgmatr01lem	|- ( ( K e. N /\ L e. N ) -> ( Q e. ( P \ { q e. P | ( q ` K ) = L } ) -> E. k e. N if ( k = K , if ( ( Q ` k ) = L , A , B ) , ( k M ( Q ` k ) ) ) = B ) )

Distinct variable groups:   A, k   B, k   k, q, L   k, K, q   k, M   k, N   P, k, q   Q, k, q

Allowed substitution hints:   A( q)   B( q)   M( q)   N( q)

Proof of Theorem symgmatr01lem

Step	Hyp	Ref	Expression
1		simpll 790	. . 3 |- ( ( ( K e. N /\ L e. N ) /\ Q e. ( P \ { q e. P | ( q ` K ) = L } ) ) -> K e. N )
2		eqeq1 2626	. . . . . 6 |- ( k = K -> ( k = K <-> K = K ) )
3		fveq2 6191	. . . . . . . 8 |- ( k = K -> ( Q ` k ) = ( Q ` K ) )
4	3	eqeq1d 2624	. . . . . . 7 |- ( k = K -> ( ( Q ` k ) = L <-> ( Q ` K ) = L ) )
5	4	ifbid 4108	. . . . . 6 |- ( k = K -> if ( ( Q ` k ) = L , A , B ) = if ( ( Q ` K ) = L , A , B ) )
6		id 22	. . . . . . 7 |- ( k = K -> k = K )
7	6, 3	oveq12d 6668	. . . . . 6 |- ( k = K -> ( k M ( Q ` k ) ) = ( K M ( Q ` K ) ) )
8	2, 5, 7	ifbieq12d 4113	. . . . 5 |- ( k = K -> if ( k = K , if ( ( Q ` k ) = L , A , B ) , ( k M ( Q ` k ) ) ) = if ( K = K , if ( ( Q ` K ) = L , A , B ) , ( K M ( Q ` K ) ) ) )
9	8	eqeq1d 2624	. . . 4 |- ( k = K -> ( if ( k = K , if ( ( Q ` k ) = L , A , B ) , ( k M ( Q ` k ) ) ) = B <-> if ( K = K , if ( ( Q ` K ) = L , A , B ) , ( K M ( Q ` K ) ) ) = B ) )
10	9	adantl 482	. . 3 |- ( ( ( ( K e. N /\ L e. N ) /\ Q e. ( P \ { q e. P | ( q ` K ) = L } ) ) /\ k = K ) -> ( if ( k = K , if ( ( Q ` k ) = L , A , B ) , ( k M ( Q ` k ) ) ) = B <-> if ( K = K , if ( ( Q ` K ) = L , A , B ) , ( K M ( Q ` K ) ) ) = B ) )
11		eqidd 2623	. . . . 5 |- ( ( ( K e. N /\ L e. N ) /\ Q e. ( P \ { q e. P | ( q ` K ) = L } ) ) -> K = K )
12	11	iftrued 4094	. . . 4 |- ( ( ( K e. N /\ L e. N ) /\ Q e. ( P \ { q e. P | ( q ` K ) = L } ) ) -> if ( K = K , if ( ( Q ` K ) = L , A , B ) , ( K M ( Q ` K ) ) ) = if ( ( Q ` K ) = L , A , B ) )
13		eldif 3584	. . . . . . 7 |- ( Q e. ( P \ { q e. P | ( q ` K ) = L } ) <-> ( Q e. P /\ -. Q e. { q e. P | ( q ` K ) = L } ) )
14		ianor 509	. . . . . . . . . 10 |- ( -. ( Q e. P /\ ( Q ` K ) = L ) <-> ( -. Q e. P \/ -. ( Q ` K ) = L ) )
15		fveq1 6190	. . . . . . . . . . . 12 |- ( q = Q -> ( q ` K ) = ( Q ` K ) )
16	15	eqeq1d 2624	. . . . . . . . . . 11 |- ( q = Q -> ( ( q ` K ) = L <-> ( Q ` K ) = L ) )
17	16	elrab 3363	. . . . . . . . . 10 |- ( Q e. { q e. P | ( q ` K ) = L } <-> ( Q e. P /\ ( Q ` K ) = L ) )
18	14, 17	xchnxbir 323	. . . . . . . . 9 |- ( -. Q e. { q e. P | ( q ` K ) = L } <-> ( -. Q e. P \/ -. ( Q ` K ) = L ) )
19		pm2.21 120	. . . . . . . . . 10 |- ( -. Q e. P -> ( Q e. P -> -. ( Q ` K ) = L ) )
20		ax-1 6	. . . . . . . . . 10 |- ( -. ( Q ` K ) = L -> ( Q e. P -> -. ( Q ` K ) = L ) )
21	19, 20	jaoi 394	. . . . . . . . 9 |- ( ( -. Q e. P \/ -. ( Q ` K ) = L ) -> ( Q e. P -> -. ( Q ` K ) = L ) )
22	18, 21	sylbi 207	. . . . . . . 8 |- ( -. Q e. { q e. P | ( q ` K ) = L } -> ( Q e. P -> -. ( Q ` K ) = L ) )
23	22	impcom 446	. . . . . . 7 |- ( ( Q e. P /\ -. Q e. { q e. P | ( q ` K ) = L } ) -> -. ( Q ` K ) = L )
24	13, 23	sylbi 207	. . . . . 6 |- ( Q e. ( P \ { q e. P | ( q ` K ) = L } ) -> -. ( Q ` K ) = L )
25	24	adantl 482	. . . . 5 |- ( ( ( K e. N /\ L e. N ) /\ Q e. ( P \ { q e. P | ( q ` K ) = L } ) ) -> -. ( Q ` K ) = L )
26	25	iffalsed 4097	. . . 4 |- ( ( ( K e. N /\ L e. N ) /\ Q e. ( P \ { q e. P | ( q ` K ) = L } ) ) -> if ( ( Q ` K ) = L , A , B ) = B )
27	12, 26	eqtrd 2656	. . 3 |- ( ( ( K e. N /\ L e. N ) /\ Q e. ( P \ { q e. P | ( q ` K ) = L } ) ) -> if ( K = K , if ( ( Q ` K ) = L , A , B ) , ( K M ( Q ` K ) ) ) = B )
28	1, 10, 27	rspcedvd 3317	. 2 |- ( ( ( K e. N /\ L e. N ) /\ Q e. ( P \ { q e. P | ( q ` K ) = L } ) ) -> E. k e. N if ( k = K , if ( ( Q ` k ) = L , A , B ) , ( k M ( Q ` k ) ) ) = B )
29	28	ex 450	1 |- ( ( K e. N /\ L e. N ) -> ( Q e. ( P \ { q e. P | ( q ` K ) = L } ) -> E. k e. N if ( k = K , if ( ( Q ` k ) = L , A , B ) , ( k M ( Q ` k ) ) ) = B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   \ cdif 3571   ifcif 4086   ` cfv 5888  (class class class)co 6650   Basecbs 15857   SymGrpcsymg 17797

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  symgmatr01  20460