Theorem mulnqpru 7370

Description: Lemma to prove upward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.)

Assertion

Ref	Expression
mulnqpru	|- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( ( G .Q H ) <Q X -> X e. ( 2nd ` ( A .P. B ) ) ) )

Proof of Theorem mulnqpru

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

Step	Hyp	Ref	Expression
1		ltmnqg 7202	. . . . . . 7 |- ( ( y e. Q. /\ z e. Q. /\ w e. Q. ) -> ( y <Q z <-> ( w .Q y ) <Q ( w .Q z ) ) )
2	1	adantl 275	. . . . . 6 |- ( ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) /\ ( y e. Q. /\ z e. Q. /\ w e. Q. ) ) -> ( y <Q z <-> ( w .Q y ) <Q ( w .Q z ) ) )
3		prop 7276	. . . . . . . . 9 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
4		elprnqu 7283	. . . . . . . . 9 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ G e. ( 2nd ` A ) ) -> G e. Q. )
5	3, 4	sylan 281	. . . . . . . 8 |- ( ( A e. P. /\ G e. ( 2nd ` A ) ) -> G e. Q. )
6	5	ad2antrr 479	. . . . . . 7 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> G e. Q. )
7		prop 7276	. . . . . . . . 9 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
8		elprnqu 7283	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ H e. ( 2nd ` B ) ) -> H e. Q. )
9	7, 8	sylan 281	. . . . . . . 8 |- ( ( B e. P. /\ H e. ( 2nd ` B ) ) -> H e. Q. )
10	9	ad2antlr 480	. . . . . . 7 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> H e. Q. )
11		mulclnq 7177	. . . . . . 7 |- ( ( G e. Q. /\ H e. Q. ) -> ( G .Q H ) e. Q. )
12	6, 10, 11	syl2anc 408	. . . . . 6 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( G .Q H ) e. Q. )
13		simpr 109	. . . . . 6 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> X e. Q. )
14		recclnq 7193	. . . . . . 7 |- ( H e. Q. -> ( *Q ` H ) e. Q. )
15	10, 14	syl 14	. . . . . 6 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( *Q ` H ) e. Q. )
16		mulcomnqg 7184	. . . . . . 7 |- ( ( y e. Q. /\ z e. Q. ) -> ( y .Q z ) = ( z .Q y ) )
17	16	adantl 275	. . . . . 6 |- ( ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) /\ ( y e. Q. /\ z e. Q. ) ) -> ( y .Q z ) = ( z .Q y ) )
18	2, 12, 13, 15, 17	caovord2d 5933	. . . . 5 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( ( G .Q H ) <Q X <-> ( ( G .Q H ) .Q ( *Q ` H ) ) <Q ( X .Q ( *Q ` H ) ) ) )
19		mulassnqg 7185	. . . . . . . 8 |- ( ( G e. Q. /\ H e. Q. /\ ( *Q ` H ) e. Q. ) -> ( ( G .Q H ) .Q ( *Q ` H ) ) = ( G .Q ( H .Q ( *Q ` H ) ) ) )
20	6, 10, 15, 19	syl3anc 1216	. . . . . . 7 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( ( G .Q H ) .Q ( *Q ` H ) ) = ( G .Q ( H .Q ( *Q ` H ) ) ) )
21		recidnq 7194	. . . . . . . . 9 |- ( H e. Q. -> ( H .Q ( *Q ` H ) ) = 1Q )
22	21	oveq2d 5783	. . . . . . . 8 |- ( H e. Q. -> ( G .Q ( H .Q ( *Q ` H ) ) ) = ( G .Q 1Q ) )
23	10, 22	syl 14	. . . . . . 7 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( G .Q ( H .Q ( *Q ` H ) ) ) = ( G .Q 1Q ) )
24		mulidnq 7190	. . . . . . . 8 |- ( G e. Q. -> ( G .Q 1Q ) = G )
25	6, 24	syl 14	. . . . . . 7 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( G .Q 1Q ) = G )
26	20, 23, 25	3eqtrd 2174	. . . . . 6 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( ( G .Q H ) .Q ( *Q ` H ) ) = G )
27	26	breq1d 3934	. . . . 5 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( ( ( G .Q H ) .Q ( *Q ` H ) ) <Q ( X .Q ( *Q ` H ) ) <-> G <Q ( X .Q ( *Q ` H ) ) ) )
28	18, 27	bitrd 187	. . . 4 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( ( G .Q H ) <Q X <-> G <Q ( X .Q ( *Q ` H ) ) ) )
29		prcunqu 7286	. . . . . 6 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ G e. ( 2nd ` A ) ) -> ( G <Q ( X .Q ( *Q ` H ) ) -> ( X .Q ( *Q ` H ) ) e. ( 2nd ` A ) ) )
30	3, 29	sylan 281	. . . . 5 |- ( ( A e. P. /\ G e. ( 2nd ` A ) ) -> ( G <Q ( X .Q ( *Q ` H ) ) -> ( X .Q ( *Q ` H ) ) e. ( 2nd ` A ) ) )
31	30	ad2antrr 479	. . . 4 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( G <Q ( X .Q ( *Q ` H ) ) -> ( X .Q ( *Q ` H ) ) e. ( 2nd ` A ) ) )
32	28, 31	sylbid 149	. . 3 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( ( G .Q H ) <Q X -> ( X .Q ( *Q ` H ) ) e. ( 2nd ` A ) ) )
33		df-imp 7270	. . . . . . . . 9 |- .P. = ( w e. P. , v e. P. |-> <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` w ) /\ z e. ( 1st ` v ) /\ x = ( y .Q z ) ) } , { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` w ) /\ z e. ( 2nd ` v ) /\ x = ( y .Q z ) ) } >. )
34		mulclnq 7177	. . . . . . . . 9 |- ( ( y e. Q. /\ z e. Q. ) -> ( y .Q z ) e. Q. )
35	33, 34	genppreclu 7316	. . . . . . . 8 |- ( ( A e. P. /\ B e. P. ) -> ( ( ( X .Q ( *Q ` H ) ) e. ( 2nd ` A ) /\ H e. ( 2nd ` B ) ) -> ( ( X .Q ( *Q ` H ) ) .Q H ) e. ( 2nd ` ( A .P. B ) ) ) )
36	35	exp4b 364	. . . . . . 7 |- ( A e. P. -> ( B e. P. -> ( ( X .Q ( *Q ` H ) ) e. ( 2nd ` A ) -> ( H e. ( 2nd ` B ) -> ( ( X .Q ( *Q ` H ) ) .Q H ) e. ( 2nd ` ( A .P. B ) ) ) ) ) )
37	36	com34 83	. . . . . 6 |- ( A e. P. -> ( B e. P. -> ( H e. ( 2nd ` B ) -> ( ( X .Q ( *Q ` H ) ) e. ( 2nd ` A ) -> ( ( X .Q ( *Q ` H ) ) .Q H ) e. ( 2nd ` ( A .P. B ) ) ) ) ) )
38	37	imp32 255	. . . . 5 |- ( ( A e. P. /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) -> ( ( X .Q ( *Q ` H ) ) e. ( 2nd ` A ) -> ( ( X .Q ( *Q ` H ) ) .Q H ) e. ( 2nd ` ( A .P. B ) ) ) )
39	38	adantlr 468	. . . 4 |- ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) -> ( ( X .Q ( *Q ` H ) ) e. ( 2nd ` A ) -> ( ( X .Q ( *Q ` H ) ) .Q H ) e. ( 2nd ` ( A .P. B ) ) ) )
40	39	adantr 274	. . 3 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( ( X .Q ( *Q ` H ) ) e. ( 2nd ` A ) -> ( ( X .Q ( *Q ` H ) ) .Q H ) e. ( 2nd ` ( A .P. B ) ) ) )
41	32, 40	syld 45	. 2 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( ( G .Q H ) <Q X -> ( ( X .Q ( *Q ` H ) ) .Q H ) e. ( 2nd ` ( A .P. B ) ) ) )
42		mulassnqg 7185	. . . . 5 |- ( ( X e. Q. /\ ( *Q ` H ) e. Q. /\ H e. Q. ) -> ( ( X .Q ( *Q ` H ) ) .Q H ) = ( X .Q ( ( *Q ` H ) .Q H ) ) )
43	13, 15, 10, 42	syl3anc 1216	. . . 4 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( ( X .Q ( *Q ` H ) ) .Q H ) = ( X .Q ( ( *Q ` H ) .Q H ) ) )
44		mulcomnqg 7184	. . . . . . 7 |- ( ( ( *Q ` H ) e. Q. /\ H e. Q. ) -> ( ( *Q ` H ) .Q H ) = ( H .Q ( *Q ` H ) ) )
45	15, 10, 44	syl2anc 408	. . . . . 6 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( ( *Q ` H ) .Q H ) = ( H .Q ( *Q ` H ) ) )
46	10, 21	syl 14	. . . . . 6 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( H .Q ( *Q ` H ) ) = 1Q )
47	45, 46	eqtrd 2170	. . . . 5 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( ( *Q ` H ) .Q H ) = 1Q )
48	47	oveq2d 5783	. . . 4 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( X .Q ( ( *Q ` H ) .Q H ) ) = ( X .Q 1Q ) )
49		mulidnq 7190	. . . . 5 |- ( X e. Q. -> ( X .Q 1Q ) = X )
50	49	adantl 275	. . . 4 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( X .Q 1Q ) = X )
51	43, 48, 50	3eqtrd 2174	. . 3 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( ( X .Q ( *Q ` H ) ) .Q H ) = X )
52	51	eleq1d 2206	. 2 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( ( ( X .Q ( *Q ` H ) ) .Q H ) e. ( 2nd ` ( A .P. B ) ) <-> X e. ( 2nd ` ( A .P. B ) ) ) )
53	41, 52	sylibd 148	1 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( ( G .Q H ) <Q X -> X e. ( 2nd ` ( A .P. B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   <.cop 3525   class class class wbr 3924   ` cfv 5118  (class class class)co 5767   1stc1st 6029   2ndc2nd 6030   Q.cnq 7081   1Qc1q 7082   .Q cmq 7084   *Qcrq 7085   <Q cltq 7086   P.cnp 7092   .P. cmp 7095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-mi 7107  df-lti 7108  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-inp 7267  df-imp 7270

This theorem is referenced by:  mullocprlem  7371  mulclpr  7373