Theorem 1idprl 7800

Description: Lemma for 1idpr 7802. (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 3245	. . . . . 6 |- ( 1st ` 1P ) C_ ( 1st ` 1P )
2		rexss 3292	. . . . . 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 7764	. . . . . . . . . . 11 |- 1P e. P.
5		prop 7685	. . . . . . . . . . . 12 |- ( 1P e. P. -> <. ( 1st ` 1P ) , ( 2nd ` 1P ) >. e. P. )
6		elprnql 7691	. . . . . . . . . . . 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 7685	. . . . . . . . . . . 12 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
10		elprnql 7691	. . . . . . . . . . . 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 4089	. . . . . . . . . . . . 13 |- ( x = ( f .Q g ) -> ( x <Q f <-> ( f .Q g ) <Q f ) )
13	12	3ad2ant3 1044	. . . . . . . . . . . 12 |- ( ( f e. Q. /\ g e. Q. /\ x = ( f .Q g ) ) -> ( x <Q f <-> ( f .Q g ) <Q f ) )
14		1prl 7765	. . . . . . . . . . . . . . 15 |- ( 1st ` 1P ) = { g | g <Q 1Q }
15	14	abeq2i 2340	. . . . . . . . . . . . . 14 |- ( g e. ( 1st ` 1P ) <-> g <Q 1Q )
16		1nq 7576	. . . . . . . . . . . . . . . . 17 |- 1Q e. Q.
17		ltmnqg 7611	. . . . . . . . . . . . . . . . 17 |- ( ( g e. Q. /\ 1Q e. Q. /\ f e. Q. ) -> ( g <Q 1Q <-> ( f .Q g ) <Q ( f .Q 1Q ) ) )
18	16, 17	mp3an2 1359	. . . . . . . . . . . . . . . 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 7599	. . . . . . . . . . . . . . . . 17 |- ( f e. Q. -> ( f .Q 1Q ) = f )
21	20	breq2d 4098	. . . . . . . . . . . . . . . 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 1041	. . . . . . . . . . . 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 1304	. . . . . . . . . 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 1305	. . . . . . . . 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 1229	. . . . . . . 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 2527	. . . . . 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 2688	. . . . . 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 2527	. . 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 7679	. . . . 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 7586	. . . . 5 |- ( ( u e. Q. /\ v e. Q. ) -> ( u .Q v ) e. Q. )
38	36, 37	genpelvl 7722	. . . 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 7698	. . . . . . 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 7575	. . . . . . . . . . . . 13 |- <Q C_ ( Q. X. Q. )
43	42	brel 4776	. . . . . . . . . . . 12 |- ( x <Q f -> ( x e. Q. /\ f e. Q. ) )
44		ltmnqg 7611	. . . . . . . . . . . . . . . 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 7602	. . . . . . . . . . . . . . . 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 7593	. . . . . . . . . . . . . . . 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 6187	. . . . . . . . . . . . . 14 |- ( ( x e. Q. /\ f e. Q. ) -> ( x <Q f <-> ( x .Q ( *Q ` f ) ) <Q ( f .Q ( *Q ` f ) ) ) )
53		recidnq 7603	. . . . . . . . . . . . . . . 16 |- ( f e. Q. -> ( f .Q ( *Q ` f ) ) = 1Q )
54	53	breq2d 4098	. . . . . . . . . . . . . . 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 7586	. . . . . . . . . . . . . 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 4089	. . . . . . . . . . . . 13 |- ( g = ( x .Q ( *Q ` f ) ) -> ( g <Q 1Q <-> ( x .Q ( *Q ` f ) ) <Q 1Q ) )
63	62, 14	elab2g 2951	. . . . . . . . . . . 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 7594	. . . . . . . . . . . . . 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 6199	. . . . . . . . . . . 12 |- ( ( x e. Q. /\ f e. Q. ) -> ( f .Q ( x .Q ( *Q ` f ) ) ) = ( x .Q ( f .Q ( *Q ` f ) ) ) )
69	53	oveq2d 6029	. . . . . . . . . . . . 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 7599	. . . . . . . . . . . . 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 2267	. . . . . . . . . . 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 6021	. . . . . . . . . . . 12 |- ( g = ( x .Q ( *Q ` f ) ) -> ( f .Q g ) = ( f .Q ( x .Q ( *Q ` f ) ) ) )
76	75	eqeq2d 2241	. . . . . . . . . . 11 |- ( g = ( x .Q ( *Q ` f ) ) -> ( x = ( f .Q g ) <-> x = ( f .Q ( x .Q ( *Q ` f ) ) ) ) )
77	76	rspcev 2908	. . . . . . . . . 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 2625	. . . . . 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 7694	. . . . . . 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 2648	. . . 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 2227	1 |- ( A e. P. -> ( 1st ` ( A .P. 1P ) ) = ( 1st ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   E.wrex 2509   C_ wss 3198   <.cop 3670   class class class wbr 4086   ` cfv 5324  (class class class)co 6013   1stc1st 6296   2ndc2nd 6297   Q.cnq 7490   1Qc1q 7491   .Q cmq 7493   *Qcrq 7494   <Q cltq 7495   P.cnp 7501   1Pc1p 7502   .P. cmp 7504

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-1o 6577  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7514  df-pli 7515  df-mi 7516  df-lti 7517  df-plpq 7554  df-mpq 7555  df-enq 7557  df-nqqs 7558  df-plqqs 7559  df-mqqs 7560  df-1nqqs 7561  df-rq 7562  df-ltnqqs 7563  df-inp 7676  df-i1p 7677  df-imp 7679

This theorem is referenced by:  1idpr  7802