Theorem 1idpru 6567

Description: Lemma for 1idpr 6568. (Contributed by Jim Kingdon, 13-Dec-2019.)

Assertion

Ref	Expression
1idpru	|- (A e. P. -> ( 2nd ` (A .P. 1P)) = ( 2nd ` A))

Proof of Theorem 1idpru

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

Step	Hyp	Ref	Expression
1		ssid 2958	. . . . . 6 |- ( 2nd ` 1P) C_ ( 2nd ` 1P)
2		rexss 3001	. . . . . 6 |- (( 2nd ` 1P) C_ ( 2nd ` 1P) -> (E. h e. ( 2nd ` 1P)x = (f .Q h) <-> E. h e. ( 2nd ` 1P)( h e. ( 2nd ` 1P) /\ x = (f .Q h))))
3	1, 2	ax-mp 7	. . . . 5 |- (E. h e. ( 2nd ` 1P)x = (f .Q h) <-> E. h e. ( 2nd ` 1P)( h e. ( 2nd ` 1P) /\ x = (f .Q h)))
4		r19.42v 2461	. . . . . 6 |- (E. h e. ( 2nd ` 1P)(f <Q x /\ x = (f .Q h)) <-> (f <Q x /\ E. h e. ( 2nd ` 1P)x = (f .Q h)))
5		1pr 6535	. . . . . . . . . . 11 |- 1P e. P.
6		prop 6458	. . . . . . . . . . . 12 |- ( 1P e. P. -> <.( 1st ` 1P) , ( 2nd ` 1P) >. e. P.)
7		elprnqu 6465	. . . . . . . . . . . 12 |- (( <.( 1st ` 1P) , ( 2nd ` 1P) >. e. P. /\ h e. ( 2nd ` 1P)) -> h e. Q.)
8	6, 7	sylan 267	. . . . . . . . . . 11 |- (( 1P e. P. /\ h e. ( 2nd ` 1P)) -> h e. Q.)
9	5, 8	mpan 400	. . . . . . . . . 10 |- ( h e. ( 2nd ` 1P) -> h e. Q.)
10		prop 6458	. . . . . . . . . . . 12 |- (A e. P. -> <.( 1st ` A) , ( 2nd ` A) >. e. P.)
11		elprnqu 6465	. . . . . . . . . . . 12 |- (( <.( 1st ` A) , ( 2nd ` A) >. e. P. /\ f e. ( 2nd ` A)) -> f e. Q.)
12	10, 11	sylan 267	. . . . . . . . . . 11 |- ((A e. P. /\ f e. ( 2nd ` A)) -> f e. Q.)
13		breq2 3759	. . . . . . . . . . . . 13 |- (x = (f .Q h) -> (f <Q x <-> f <Q (f .Q h)))
14	13	3ad2ant3 926	. . . . . . . . . . . 12 |- ((f e. Q. /\ h e. Q. /\ x = (f .Q h)) -> (f <Q x <-> f <Q (f .Q h)))
15		1pru 6537	. . . . . . . . . . . . . . 15 |- ( 2nd ` 1P) = { h | 1Q <Q h }
16	15	abeq2i 2145	. . . . . . . . . . . . . 14 |- ( h e. ( 2nd ` 1P) <-> 1Q <Q h)
17		1nq 6350	. . . . . . . . . . . . . . . . 17 |- 1Q e. Q.
18		ltmnqg 6385	. . . . . . . . . . . . . . . . 17 |- (( 1Q e. Q. /\ h e. Q. /\ f e. Q.) -> ( 1Q <Q h <-> (f .Q 1Q) <Q (f .Q h)))
19	17, 18	mp3an1 1218	. . . . . . . . . . . . . . . 16 |- (( h e. Q. /\ f e. Q.) -> ( 1Q <Q h <-> (f .Q 1Q) <Q (f .Q h)))
20	19	ancoms 255	. . . . . . . . . . . . . . 15 |- ((f e. Q. /\ h e. Q.) -> ( 1Q <Q h <-> (f .Q 1Q) <Q (f .Q h)))
21		mulidnq 6373	. . . . . . . . . . . . . . . . 17 |- (f e. Q. -> (f .Q 1Q) = f)
22	21	breq1d 3765	. . . . . . . . . . . . . . . 16 |- (f e. Q. -> ((f .Q 1Q) <Q (f .Q h) <-> f <Q (f .Q h)))
23	22	adantr 261	. . . . . . . . . . . . . . 15 |- ((f e. Q. /\ h e. Q.) -> ((f .Q 1Q) <Q (f .Q h) <-> f <Q (f .Q h)))
24	20, 23	bitrd 177	. . . . . . . . . . . . . 14 |- ((f e. Q. /\ h e. Q.) -> ( 1Q <Q h <-> f <Q (f .Q h)))
25	16, 24	syl5rbb 182	. . . . . . . . . . . . 13 |- ((f e. Q. /\ h e. Q.) -> (f <Q (f .Q h) <-> h e. ( 2nd ` 1P)))
26	25	3adant3 923	. . . . . . . . . . . 12 |- ((f e. Q. /\ h e. Q. /\ x = (f .Q h)) -> (f <Q (f .Q h) <-> h e. ( 2nd ` 1P)))
27	14, 26	bitrd 177	. . . . . . . . . . 11 |- ((f e. Q. /\ h e. Q. /\ x = (f .Q h)) -> (f <Q x <-> h e. ( 2nd ` 1P)))
28	12, 27	syl3an1 1167	. . . . . . . . . 10 |- (((A e. P. /\ f e. ( 2nd ` A)) /\ h e. Q. /\ x = (f .Q h)) -> (f <Q x <-> h e. ( 2nd ` 1P)))
29	9, 28	syl3an2 1168	. . . . . . . . 9 |- (((A e. P. /\ f e. ( 2nd ` A)) /\ h e. ( 2nd ` 1P) /\ x = (f .Q h)) -> (f <Q x <-> h e. ( 2nd ` 1P)))
30	29	3expia 1105	. . . . . . . 8 |- (((A e. P. /\ f e. ( 2nd ` A)) /\ h e. ( 2nd ` 1P)) -> (x = (f .Q h) -> (f <Q x <-> h e. ( 2nd ` 1P))))
31	30	pm5.32rd 424	. . . . . . 7 |- (((A e. P. /\ f e. ( 2nd ` A)) /\ h e. ( 2nd ` 1P)) -> ((f <Q x /\ x = (f .Q h)) <-> ( h e. ( 2nd ` 1P) /\ x = (f .Q h))))
32	31	rexbidva 2317	. . . . . 6 |- ((A e. P. /\ f e. ( 2nd ` A)) -> (E. h e. ( 2nd ` 1P)(f <Q x /\ x = (f .Q h)) <-> E. h e. ( 2nd ` 1P)( h e. ( 2nd ` 1P) /\ x = (f .Q h))))
33	4, 32	syl5rbbr 184	. . . . 5 |- ((A e. P. /\ f e. ( 2nd ` A)) -> (E. h e. ( 2nd ` 1P)( h e. ( 2nd ` 1P) /\ x = (f .Q h)) <-> (f <Q x /\ E. h e. ( 2nd ` 1P)x = (f .Q h))))
34	3, 33	syl5bb 181	. . . 4 |- ((A e. P. /\ f e. ( 2nd ` A)) -> (E. h e. ( 2nd ` 1P)x = (f .Q h) <-> (f <Q x /\ E. h e. ( 2nd ` 1P)x = (f .Q h))))
35	34	rexbidva 2317	. . 3 |- (A e. P. -> (E.f e. ( 2nd ` A)E. h e. ( 2nd ` 1P)x = (f .Q h) <-> E.f e. ( 2nd ` A)(f <Q x /\ E. h e. ( 2nd ` 1P)x = (f .Q h))))
36		df-imp 6452	. . . . 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 6360	. . . . 5 |- ((u e. Q. /\ v e. Q.) -> (u .Q v) e. Q.)
38	36, 37	genpelvu 6496	. . . 4 |- ((A e. P. /\ 1P e. P.) -> (x e. ( 2nd ` (A .P. 1P)) <-> E.f e. ( 2nd ` A)E. h e. ( 2nd ` 1P)x = (f .Q h)))
39	5, 38	mpan2 401	. . 3 |- (A e. P. -> (x e. ( 2nd ` (A .P. 1P)) <-> E.f e. ( 2nd ` A)E. h e. ( 2nd ` 1P)x = (f .Q h)))
40		prnminu 6472	. . . . . . 7 |- (( <.( 1st ` A) , ( 2nd ` A) >. e. P. /\ x e. ( 2nd ` A)) -> E.f e. ( 2nd ` A)f <Q x)
41	10, 40	sylan 267	. . . . . 6 |- ((A e. P. /\ x e. ( 2nd ` A)) -> E.f e. ( 2nd ` A)f <Q x)
42		ltrelnq 6349	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q.)
43	42	brel 4335	. . . . . . . . . . . . 13 |- (f <Q x -> (f e. Q. /\ x e. Q.))
44	43	ancomd 254	. . . . . . . . . . . 12 |- (f <Q x -> (x e. Q. /\ f e. Q.))
45		ltmnqg 6385	. . . . . . . . . . . . . . . 16 |- ((y e. Q. /\ z e. Q. /\ w e. Q.) -> (y <Q z <-> (w .Q y) <Q (w .Q z)))
46	45	adantl 262	. . . . . . . . . . . . . . 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)))
47		simpr 103	. . . . . . . . . . . . . . 15 |- ((x e. Q. /\ f e. Q.) -> f e. Q.)
48		simpl 102	. . . . . . . . . . . . . . 15 |- ((x e. Q. /\ f e. Q.) -> x e. Q.)
49		recclnq 6376	. . . . . . . . . . . . . . . 16 |- (f e. Q. -> ( *Q ` f) e. Q.)
50	49	adantl 262	. . . . . . . . . . . . . . 15 |- ((x e. Q. /\ f e. Q.) -> ( *Q ` f) e. Q.)
51		mulcomnqg 6367	. . . . . . . . . . . . . . . 16 |- ((y e. Q. /\ z e. Q.) -> (y .Q z) = (z .Q y))
52	51	adantl 262	. . . . . . . . . . . . . . 15 |- (((x e. Q. /\ f e. Q.) /\ (y e. Q. /\ z e. Q.)) -> (y .Q z) = (z .Q y))
53	46, 47, 48, 50, 52	caovord2d 5612	. . . . . . . . . . . . . 14 |- ((x e. Q. /\ f e. Q.) -> (f <Q x <-> (f .Q ( *Q ` f)) <Q (x .Q ( *Q ` f))))
54		recidnq 6377	. . . . . . . . . . . . . . . 16 |- (f e. Q. -> (f .Q ( *Q ` f)) = 1Q)
55	54	breq1d 3765	. . . . . . . . . . . . . . 15 |- (f e. Q. -> ((f .Q ( *Q ` f)) <Q (x .Q ( *Q ` f)) <-> 1Q <Q (x .Q ( *Q ` f))))
56	55	adantl 262	. . . . . . . . . . . . . 14 |- ((x e. Q. /\ f e. Q.) -> ((f .Q ( *Q ` f)) <Q (x .Q ( *Q ` f)) <-> 1Q <Q (x .Q ( *Q ` f))))
57	53, 56	bitrd 177	. . . . . . . . . . . . 13 |- ((x e. Q. /\ f e. Q.) -> (f <Q x <-> 1Q <Q (x .Q ( *Q ` f))))
58	57	biimpd 132	. . . . . . . . . . . 12 |- ((x e. Q. /\ f e. Q.) -> (f <Q x -> 1Q <Q (x .Q ( *Q ` f))))
59	44, 58	mpcom 32	. . . . . . . . . . 11 |- (f <Q x -> 1Q <Q (x .Q ( *Q ` f)))
60		mulclnq 6360	. . . . . . . . . . . . 13 |- ((x e. Q. /\ ( *Q ` f) e. Q.) -> (x .Q ( *Q ` f)) e. Q.)
61	49, 60	sylan2 270	. . . . . . . . . . . 12 |- ((x e. Q. /\ f e. Q.) -> (x .Q ( *Q ` f)) e. Q.)
62		breq2 3759	. . . . . . . . . . . . 13 |- ( h = (x .Q ( *Q ` f)) -> ( 1Q <Q h <-> 1Q <Q (x .Q ( *Q ` f))))
63	62, 15	elab2g 2683	. . . . . . . . . . . 12 |- ((x .Q ( *Q ` f)) e. Q. -> ((x .Q ( *Q ` f)) e. ( 2nd ` 1P) <-> 1Q <Q (x .Q ( *Q ` f))))
64	44, 61, 63	3syl 17	. . . . . . . . . . 11 |- (f <Q x -> ((x .Q ( *Q ` f)) e. ( 2nd ` 1P) <-> 1Q <Q (x .Q ( *Q ` f))))
65	59, 64	mpbird 156	. . . . . . . . . 10 |- (f <Q x -> (x .Q ( *Q ` f)) e. ( 2nd ` 1P))
66		mulassnqg 6368	. . . . . . . . . . . . . 14 |- ((y e. Q. /\ z e. Q. /\ w e. Q.) -> ((y .Q z) .Q w) = (y .Q (z .Q w)))
67	66	adantl 262	. . . . . . . . . . . . 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, 48, 50, 52, 67	caov12d 5624	. . . . . . . . . . . 12 |- ((x e. Q. /\ f e. Q.) -> (f .Q (x .Q ( *Q ` f))) = (x .Q (f .Q ( *Q ` f))))
69	54	oveq2d 5471	. . . . . . . . . . . . 13 |- (f e. Q. -> (x .Q (f .Q ( *Q ` f))) = (x .Q 1Q))
70	69	adantl 262	. . . . . . . . . . . 12 |- ((x e. Q. /\ f e. Q.) -> (x .Q (f .Q ( *Q ` f))) = (x .Q 1Q))
71		mulidnq 6373	. . . . . . . . . . . . 13 |- (x e. Q. -> (x .Q 1Q) = x)
72	71	adantr 261	. . . . . . . . . . . 12 |- ((x e. Q. /\ f e. Q.) -> (x .Q 1Q) = x)
73	68, 70, 72	3eqtrrd 2074	. . . . . . . . . . 11 |- ((x e. Q. /\ f e. Q.) -> x = (f .Q (x .Q ( *Q ` f))))
74	44, 73	syl 14	. . . . . . . . . 10 |- (f <Q x -> x = (f .Q (x .Q ( *Q ` f))))
75		oveq2 5463	. . . . . . . . . . . 12 |- ( h = (x .Q ( *Q ` f)) -> (f .Q h) = (f .Q (x .Q ( *Q ` f))))
76	75	eqeq2d 2048	. . . . . . . . . . 11 |- ( h = (x .Q ( *Q ` f)) -> (x = (f .Q h) <-> x = (f .Q (x .Q ( *Q ` f)))))
77	76	rspcev 2650	. . . . . . . . . 10 |- (((x .Q ( *Q ` f)) e. ( 2nd ` 1P) /\ x = (f .Q (x .Q ( *Q ` f)))) -> E. h e. ( 2nd ` 1P)x = (f .Q h))
78	65, 74, 77	syl2anc 391	. . . . . . . . 9 |- (f <Q x -> E. h e. ( 2nd ` 1P)x = (f .Q h))
79	78	a1i 9	. . . . . . . 8 |- (f e. ( 2nd ` A) -> (f <Q x -> E. h e. ( 2nd ` 1P)x = (f .Q h)))
80	79	ancld 308	. . . . . . 7 |- (f e. ( 2nd ` A) -> (f <Q x -> (f <Q x /\ E. h e. ( 2nd ` 1P)x = (f .Q h))))
81	80	reximia 2408	. . . . . 6 |- (E.f e. ( 2nd ` A)f <Q x -> E.f e. ( 2nd ` A)(f <Q x /\ E. h e. ( 2nd ` 1P)x = (f .Q h)))
82	41, 81	syl 14	. . . . 5 |- ((A e. P. /\ x e. ( 2nd ` A)) -> E.f e. ( 2nd ` A)(f <Q x /\ E. h e. ( 2nd ` 1P)x = (f .Q h)))
83	82	ex 108	. . . 4 |- (A e. P. -> (x e. ( 2nd ` A) -> E.f e. ( 2nd ` A)(f <Q x /\ E. h e. ( 2nd ` 1P)x = (f .Q h))))
84		prcunqu 6468	. . . . . . 7 |- (( <.( 1st ` A) , ( 2nd ` A) >. e. P. /\ f e. ( 2nd ` A)) -> (f <Q x -> x e. ( 2nd ` A)))
85	10, 84	sylan 267	. . . . . 6 |- ((A e. P. /\ f e. ( 2nd ` A)) -> (f <Q x -> x e. ( 2nd ` A)))
86	85	adantrd 264	. . . . 5 |- ((A e. P. /\ f e. ( 2nd ` A)) -> ((f <Q x /\ E. h e. ( 2nd ` 1P)x = (f .Q h)) -> x e. ( 2nd ` A)))
87	86	rexlimdva 2427	. . . 4 |- (A e. P. -> (E.f e. ( 2nd ` A)(f <Q x /\ E. h e. ( 2nd ` 1P)x = (f .Q h)) -> x e. ( 2nd ` A)))
88	83, 87	impbid 120	. . 3 |- (A e. P. -> (x e. ( 2nd ` A) <-> E.f e. ( 2nd ` A)(f <Q x /\ E. h e. ( 2nd ` 1P)x = (f .Q h))))
89	35, 39, 88	3bitr4d 209	. 2 |- (A e. P. -> (x e. ( 2nd ` (A .P. 1P)) <-> x e. ( 2nd ` A)))
90	89	eqrdv 2035	1 |- (A e. P. -> ( 2nd ` (A .P. 1P)) = ( 2nd ` A))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   /\ w3a 884   = wceq 1242   e. wcel 1390  E.wrex 2301   C_ wss 2911   <.cop 3370   class class class wbr 3755   ` cfv 4845  (class class class)co 5455   1stc1st 5707   2ndc2nd 5708   Q.cnq 6264   1Qc1q 6265   .Q cmq 6267   *Qcrq 6268   <Q cltq 6269   P.cnp 6275   1Pc1p 6276   .P. cmp 6278

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-inp 6449  df-i1p 6450  df-imp 6452

This theorem is referenced by:  1idpr  6568