Theorem 1idprl 7703

Description: Lemma for 1idpr 7705. (Contributed by Jim Kingdon, 13-Dec-2019.)

Assertion

Ref	Expression
1idprl	|- ( A e. P. -> ( 1st ` ( A .P. 1P ) ) = ( 1st ` A ) )

Proof of Theorem 1idprl

Dummy variables x y z w v u f g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 3213	. . . . . 6 |- ( 1st ` 1P ) C_ ( 1st ` 1P )
2		rexss 3260	. . . . . 6 |- ( ( 1st ` 1P ) C_ ( 1st ` 1P ) -> ( E. g e. ( 1st ` 1P ) x = ( f .Q g ) <-> E. g e. ( 1st ` 1P ) ( g e. ( 1st ` 1P ) /\ x = ( f .Q g ) ) ) )
3	1, 2	ax-mp 5	. . . . 5 |- ( E. g e. ( 1st ` 1P ) x = ( f .Q g ) <-> E. g e. ( 1st ` 1P ) ( g e. ( 1st ` 1P ) /\ x = ( f .Q g ) ) )
4		1pr 7667	. . . . . . . . . . 11 |- 1P e. P.
5		prop 7588	. . . . . . . . . . . 12 |- ( 1P e. P. -> <. ( 1st ` 1P ) , ( 2nd ` 1P ) >. e. P. )
6		elprnql 7594	. . . . . . . . . . . 12 |- ( ( <. ( 1st ` 1P ) , ( 2nd ` 1P ) >. e. P. /\ g e. ( 1st ` 1P ) ) -> g e. Q. )
7	5, 6	sylan 283	. . . . . . . . . . 11 |- ( ( 1P e. P. /\ g e. ( 1st ` 1P ) ) -> g e. Q. )
8	4, 7	mpan 424	. . . . . . . . . 10 |- ( g e. ( 1st ` 1P ) -> g e. Q. )
9		prop 7588	. . . . . . . . . . . 12 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
10		elprnql 7594	. . . . . . . . . . . 12 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ f e. ( 1st ` A ) ) -> f e. Q. )
11	9, 10	sylan 283	. . . . . . . . . . 11 |- ( ( A e. P. /\ f e. ( 1st ` A ) ) -> f e. Q. )
12		breq1 4047	. . . . . . . . . . . . 13 |- ( x = ( f .Q g ) -> ( x <Q f <-> ( f .Q g ) <Q f ) )
13	12	3ad2ant3 1023	. . . . . . . . . . . 12 |- ( ( f e. Q. /\ g e. Q. /\ x = ( f .Q g ) ) -> ( x <Q f <-> ( f .Q g ) <Q f ) )
14		1prl 7668	. . . . . . . . . . . . . . 15 |- ( 1st ` 1P ) = { g | g <Q 1Q }
15	14	abeq2i 2316	. . . . . . . . . . . . . 14 |- ( g e. ( 1st ` 1P ) <-> g <Q 1Q )
16		1nq 7479	. . . . . . . . . . . . . . . . 17 |- 1Q e. Q.
17		ltmnqg 7514	. . . . . . . . . . . . . . . . 17 |- ( ( g e. Q. /\ 1Q e. Q. /\ f e. Q. ) -> ( g <Q 1Q <-> ( f .Q g ) <Q ( f .Q 1Q ) ) )
18	16, 17	mp3an2 1338	. . . . . . . . . . . . . . . 16 |- ( ( g e. Q. /\ f e. Q. ) -> ( g <Q 1Q <-> ( f .Q g ) <Q ( f .Q 1Q ) ) )
19	18	ancoms 268	. . . . . . . . . . . . . . 15 |- ( ( f e. Q. /\ g e. Q. ) -> ( g <Q 1Q <-> ( f .Q g ) <Q ( f .Q 1Q ) ) )
20		mulidnq 7502	. . . . . . . . . . . . . . . . 17 |- ( f e. Q. -> ( f .Q 1Q ) = f )
21	20	breq2d 4056	. . . . . . . . . . . . . . . 16 |- ( f e. Q. -> ( ( f .Q g ) <Q ( f .Q 1Q ) <-> ( f .Q g ) <Q f ) )
22	21	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( f e. Q. /\ g e. Q. ) -> ( ( f .Q g ) <Q ( f .Q 1Q ) <-> ( f .Q g ) <Q f ) )
23	19, 22	bitrd 188	. . . . . . . . . . . . . 14 |- ( ( f e. Q. /\ g e. Q. ) -> ( g <Q 1Q <-> ( f .Q g ) <Q f ) )
24	15, 23	bitr2id 193	. . . . . . . . . . . . 13 |- ( ( f e. Q. /\ g e. Q. ) -> ( ( f .Q g ) <Q f <-> g e. ( 1st ` 1P ) ) )
25	24	3adant3 1020	. . . . . . . . . . . 12 |- ( ( f e. Q. /\ g e. Q. /\ x = ( f .Q g ) ) -> ( ( f .Q g ) <Q f <-> g e. ( 1st ` 1P ) ) )
26	13, 25	bitrd 188	. . . . . . . . . . 11 |- ( ( f e. Q. /\ g e. Q. /\ x = ( f .Q g ) ) -> ( x <Q f <-> g e. ( 1st ` 1P ) ) )
27	11, 26	syl3an1 1283	. . . . . . . . . 10 |- ( ( ( A e. P. /\ f e. ( 1st ` A ) ) /\ g e. Q. /\ x = ( f .Q g ) ) -> ( x <Q f <-> g e. ( 1st ` 1P ) ) )
28	8, 27	syl3an2 1284	. . . . . . . . 9 |- ( ( ( A e. P. /\ f e. ( 1st ` A ) ) /\ g e. ( 1st ` 1P ) /\ x = ( f .Q g ) ) -> ( x <Q f <-> g e. ( 1st ` 1P ) ) )
29	28	3expia 1208	. . . . . . . 8 |- ( ( ( A e. P. /\ f e. ( 1st ` A ) ) /\ g e. ( 1st ` 1P ) ) -> ( x = ( f .Q g ) -> ( x <Q f <-> g e. ( 1st ` 1P ) ) ) )
30	29	pm5.32rd 451	. . . . . . 7 |- ( ( ( A e. P. /\ f e. ( 1st ` A ) ) /\ g e. ( 1st ` 1P ) ) -> ( ( x <Q f /\ x = ( f .Q g ) ) <-> ( g e. ( 1st ` 1P ) /\ x = ( f .Q g ) ) ) )
31	30	rexbidva 2503	. . . . . 6 |- ( ( A e. P. /\ f e. ( 1st ` A ) ) -> ( E. g e. ( 1st ` 1P ) ( x <Q f /\ x = ( f .Q g ) ) <-> E. g e. ( 1st ` 1P ) ( g e. ( 1st ` 1P ) /\ x = ( f .Q g ) ) ) )
32		r19.42v 2663	. . . . . 6 |- ( E. g e. ( 1st ` 1P ) ( x <Q f /\ x = ( f .Q g ) ) <-> ( x <Q f /\ E. g e. ( 1st ` 1P ) x = ( f .Q g ) ) )
33	31, 32	bitr3di 195	. . . . 5 |- ( ( A e. P. /\ f e. ( 1st ` A ) ) -> ( E. g e. ( 1st ` 1P ) ( g e. ( 1st ` 1P ) /\ x = ( f .Q g ) ) <-> ( x <Q f /\ E. g e. ( 1st ` 1P ) x = ( f .Q g ) ) ) )
34	3, 33	bitrid 192	. . . 4 |- ( ( A e. P. /\ f e. ( 1st ` A ) ) -> ( E. g e. ( 1st ` 1P ) x = ( f .Q g ) <-> ( x <Q f /\ E. g e. ( 1st ` 1P ) x = ( f .Q g ) ) ) )
35	34	rexbidva 2503	. . 3 |- ( A e. P. -> ( E. f e. ( 1st ` A ) E. g e. ( 1st ` 1P ) x = ( f .Q g ) <-> E. f e. ( 1st ` A ) ( x <Q f /\ E. g e. ( 1st ` 1P ) x = ( f .Q g ) ) ) )
36		df-imp 7582	. . . . 5 |- .P. = ( y e. P. , z e. P. |-> <. { w e. Q. | E. u e. Q. E. v e. Q. ( u e. ( 1st ` y ) /\ v e. ( 1st ` z ) /\ w = ( u .Q v ) ) } , { w e. Q. | E. u e. Q. E. v e. Q. ( u e. ( 2nd ` y ) /\ v e. ( 2nd ` z ) /\ w = ( u .Q v ) ) } >. )
37		mulclnq 7489	. . . . 5 |- ( ( u e. Q. /\ v e. Q. ) -> ( u .Q v ) e. Q. )
38	36, 37	genpelvl 7625	. . . 4 |- ( ( A e. P. /\ 1P e. P. ) -> ( x e. ( 1st ` ( A .P. 1P ) ) <-> E. f e. ( 1st ` A ) E. g e. ( 1st ` 1P ) x = ( f .Q g ) ) )
39	4, 38	mpan2 425	. . 3 |- ( A e. P. -> ( x e. ( 1st ` ( A .P. 1P ) ) <-> E. f e. ( 1st ` A ) E. g e. ( 1st ` 1P ) x = ( f .Q g ) ) )
40		prnmaxl 7601	. . . . . . 7 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ x e. ( 1st ` A ) ) -> E. f e. ( 1st ` A ) x <Q f )
41	9, 40	sylan 283	. . . . . 6 |- ( ( A e. P. /\ x e. ( 1st ` A ) ) -> E. f e. ( 1st ` A ) x <Q f )
42		ltrelnq 7478	. . . . . . . . . . . . 13 |- <Q C_ ( Q. X. Q. )
43	42	brel 4727	. . . . . . . . . . . 12 |- ( x <Q f -> ( x e. Q. /\ f e. Q. ) )
44		ltmnqg 7514	. . . . . . . . . . . . . . . 16 |- ( ( y e. Q. /\ z e. Q. /\ w e. Q. ) -> ( y <Q z <-> ( w .Q y ) <Q ( w .Q z ) ) )
45	44	adantl 277	. . . . . . . . . . . . . . 15 |- ( ( ( x e. Q. /\ f e. Q. ) /\ ( y e. Q. /\ z e. Q. /\ w e. Q. ) ) -> ( y <Q z <-> ( w .Q y ) <Q ( w .Q z ) ) )
46		simpl 109	. . . . . . . . . . . . . . 15 |- ( ( x e. Q. /\ f e. Q. ) -> x e. Q. )
47		simpr 110	. . . . . . . . . . . . . . 15 |- ( ( x e. Q. /\ f e. Q. ) -> f e. Q. )
48		recclnq 7505	. . . . . . . . . . . . . . . 16 |- ( f e. Q. -> ( *Q ` f ) e. Q. )
49	48	adantl 277	. . . . . . . . . . . . . . 15 |- ( ( x e. Q. /\ f e. Q. ) -> ( *Q ` f ) e. Q. )
50		mulcomnqg 7496	. . . . . . . . . . . . . . . 16 |- ( ( y e. Q. /\ z e. Q. ) -> ( y .Q z ) = ( z .Q y ) )
51	50	adantl 277	. . . . . . . . . . . . . . 15 |- ( ( ( x e. Q. /\ f e. Q. ) /\ ( y e. Q. /\ z e. Q. ) ) -> ( y .Q z ) = ( z .Q y ) )
52	45, 46, 47, 49, 51	caovord2d 6116	. . . . . . . . . . . . . 14 |- ( ( x e. Q. /\ f e. Q. ) -> ( x <Q f <-> ( x .Q ( *Q ` f ) ) <Q ( f .Q ( *Q ` f ) ) ) )
53		recidnq 7506	. . . . . . . . . . . . . . . 16 |- ( f e. Q. -> ( f .Q ( *Q ` f ) ) = 1Q )
54	53	breq2d 4056	. . . . . . . . . . . . . . 15 |- ( f e. Q. -> ( ( x .Q ( *Q ` f ) ) <Q ( f .Q ( *Q ` f ) ) <-> ( x .Q ( *Q ` f ) ) <Q 1Q ) )
55	54	adantl 277	. . . . . . . . . . . . . 14 |- ( ( x e. Q. /\ f e. Q. ) -> ( ( x .Q ( *Q ` f ) ) <Q ( f .Q ( *Q ` f ) ) <-> ( x .Q ( *Q ` f ) ) <Q 1Q ) )
56	52, 55	bitrd 188	. . . . . . . . . . . . 13 |- ( ( x e. Q. /\ f e. Q. ) -> ( x <Q f <-> ( x .Q ( *Q ` f ) ) <Q 1Q ) )
57	56	biimpd 144	. . . . . . . . . . . 12 |- ( ( x e. Q. /\ f e. Q. ) -> ( x <Q f -> ( x .Q ( *Q ` f ) ) <Q 1Q ) )
58	43, 57	mpcom 36	. . . . . . . . . . 11 |- ( x <Q f -> ( x .Q ( *Q ` f ) ) <Q 1Q )
59		mulclnq 7489	. . . . . . . . . . . . . 14 |- ( ( x e. Q. /\ ( *Q ` f ) e. Q. ) -> ( x .Q ( *Q ` f ) ) e. Q. )
60	48, 59	sylan2 286	. . . . . . . . . . . . 13 |- ( ( x e. Q. /\ f e. Q. ) -> ( x .Q ( *Q ` f ) ) e. Q. )
61	43, 60	syl 14	. . . . . . . . . . . 12 |- ( x <Q f -> ( x .Q ( *Q ` f ) ) e. Q. )
62		breq1 4047	. . . . . . . . . . . . 13 |- ( g = ( x .Q ( *Q ` f ) ) -> ( g <Q 1Q <-> ( x .Q ( *Q ` f ) ) <Q 1Q ) )
63	62, 14	elab2g 2920	. . . . . . . . . . . 12 |- ( ( x .Q ( *Q ` f ) ) e. Q. -> ( ( x .Q ( *Q ` f ) ) e. ( 1st ` 1P ) <-> ( x .Q ( *Q ` f ) ) <Q 1Q ) )
64	61, 63	syl 14	. . . . . . . . . . 11 |- ( x <Q f -> ( ( x .Q ( *Q ` f ) ) e. ( 1st ` 1P ) <-> ( x .Q ( *Q ` f ) ) <Q 1Q ) )
65	58, 64	mpbird 167	. . . . . . . . . 10 |- ( x <Q f -> ( x .Q ( *Q ` f ) ) e. ( 1st ` 1P ) )
66		mulassnqg 7497	. . . . . . . . . . . . . 14 |- ( ( y e. Q. /\ z e. Q. /\ w e. Q. ) -> ( ( y .Q z ) .Q w ) = ( y .Q ( z .Q w ) ) )
67	66	adantl 277	. . . . . . . . . . . . 13 |- ( ( ( x e. Q. /\ f e. Q. ) /\ ( y e. Q. /\ z e. Q. /\ w e. Q. ) ) -> ( ( y .Q z ) .Q w ) = ( y .Q ( z .Q w ) ) )
68	47, 46, 49, 51, 67	caov12d 6128	. . . . . . . . . . . 12 |- ( ( x e. Q. /\ f e. Q. ) -> ( f .Q ( x .Q ( *Q ` f ) ) ) = ( x .Q ( f .Q ( *Q ` f ) ) ) )
69	53	oveq2d 5960	. . . . . . . . . . . . 13 |- ( f e. Q. -> ( x .Q ( f .Q ( *Q ` f ) ) ) = ( x .Q 1Q ) )
70	69	adantl 277	. . . . . . . . . . . 12 |- ( ( x e. Q. /\ f e. Q. ) -> ( x .Q ( f .Q ( *Q ` f ) ) ) = ( x .Q 1Q ) )
71		mulidnq 7502	. . . . . . . . . . . . 13 |- ( x e. Q. -> ( x .Q 1Q ) = x )
72	71	adantr 276	. . . . . . . . . . . 12 |- ( ( x e. Q. /\ f e. Q. ) -> ( x .Q 1Q ) = x )
73	68, 70, 72	3eqtrrd 2243	. . . . . . . . . . 11 |- ( ( x e. Q. /\ f e. Q. ) -> x = ( f .Q ( x .Q ( *Q ` f ) ) ) )
74	43, 73	syl 14	. . . . . . . . . 10 |- ( x <Q f -> x = ( f .Q ( x .Q ( *Q ` f ) ) ) )
75		oveq2 5952	. . . . . . . . . . . 12 |- ( g = ( x .Q ( *Q ` f ) ) -> ( f .Q g ) = ( f .Q ( x .Q ( *Q ` f ) ) ) )
76	75	eqeq2d 2217	. . . . . . . . . . 11 |- ( g = ( x .Q ( *Q ` f ) ) -> ( x = ( f .Q g ) <-> x = ( f .Q ( x .Q ( *Q ` f ) ) ) ) )
77	76	rspcev 2877	. . . . . . . . . 10 |- ( ( ( x .Q ( *Q ` f ) ) e. ( 1st ` 1P ) /\ x = ( f .Q ( x .Q ( *Q ` f ) ) ) ) -> E. g e. ( 1st ` 1P ) x = ( f .Q g ) )
78	65, 74, 77	syl2anc 411	. . . . . . . . 9 |- ( x <Q f -> E. g e. ( 1st ` 1P ) x = ( f .Q g ) )
79	78	a1i 9	. . . . . . . 8 |- ( f e. ( 1st ` A ) -> ( x <Q f -> E. g e. ( 1st ` 1P ) x = ( f .Q g ) ) )
80	79	ancld 325	. . . . . . 7 |- ( f e. ( 1st ` A ) -> ( x <Q f -> ( x <Q f /\ E. g e. ( 1st ` 1P ) x = ( f .Q g ) ) ) )
81	80	reximia 2601	. . . . . 6 |- ( E. f e. ( 1st ` A ) x <Q f -> E. f e. ( 1st ` A ) ( x <Q f /\ E. g e. ( 1st ` 1P ) x = ( f .Q g ) ) )
82	41, 81	syl 14	. . . . 5 |- ( ( A e. P. /\ x e. ( 1st ` A ) ) -> E. f e. ( 1st ` A ) ( x <Q f /\ E. g e. ( 1st ` 1P ) x = ( f .Q g ) ) )
83	82	ex 115	. . . 4 |- ( A e. P. -> ( x e. ( 1st ` A ) -> E. f e. ( 1st ` A ) ( x <Q f /\ E. g e. ( 1st ` 1P ) x = ( f .Q g ) ) ) )
84		prcdnql 7597	. . . . . . 7 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ f e. ( 1st ` A ) ) -> ( x <Q f -> x e. ( 1st ` A ) ) )
85	9, 84	sylan 283	. . . . . 6 |- ( ( A e. P. /\ f e. ( 1st ` A ) ) -> ( x <Q f -> x e. ( 1st ` A ) ) )
86	85	adantrd 279	. . . . 5 |- ( ( A e. P. /\ f e. ( 1st ` A ) ) -> ( ( x <Q f /\ E. g e. ( 1st ` 1P ) x = ( f .Q g ) ) -> x e. ( 1st ` A ) ) )
87	86	rexlimdva 2623	. . . 4 |- ( A e. P. -> ( E. f e. ( 1st ` A ) ( x <Q f /\ E. g e. ( 1st ` 1P ) x = ( f .Q g ) ) -> x e. ( 1st ` A ) ) )
88	83, 87	impbid 129	. . 3 |- ( A e. P. -> ( x e. ( 1st ` A ) <-> E. f e. ( 1st ` A ) ( x <Q f /\ E. g e. ( 1st ` 1P ) x = ( f .Q g ) ) ) )
89	35, 39, 88	3bitr4d 220	. 2 |- ( A e. P. -> ( x e. ( 1st ` ( A .P. 1P ) ) <-> x e. ( 1st ` A ) ) )
90	89	eqrdv 2203	1 |- ( A e. P. -> ( 1st ` ( A .P. 1P ) ) = ( 1st ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2176   E.wrex 2485   C_ wss 3166   <.cop 3636   class class class wbr 4044   ` cfv 5271  (class class class)co 5944   1stc1st 6224   2ndc2nd 6225   Q.cnq 7393   1Qc1q 7394   .Q cmq 7396   *Qcrq 7397   <Q cltq 7398   P.cnp 7404   1Pc1p 7405   .P. cmp 7407

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-eprel 4336  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-irdg 6456  df-1o 6502  df-oadd 6506  df-omul 6507  df-er 6620  df-ec 6622  df-qs 6626  df-ni 7417  df-pli 7418  df-mi 7419  df-lti 7420  df-plpq 7457  df-mpq 7458  df-enq 7460  df-nqqs 7461  df-plqqs 7462  df-mqqs 7463  df-1nqqs 7464  df-rq 7465  df-ltnqqs 7466  df-inp 7579  df-i1p 7580  df-imp 7582

This theorem is referenced by:  1idpr  7705