Theorem mulnqprl 7491

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

Assertion

Ref	Expression
mulnqprl	|- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( X <Q ( G .Q H ) -> X e. ( 1st ` ( A .P. B ) ) ) )

Proof of Theorem mulnqprl

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

Step	Hyp	Ref	Expression
1		ltmnqg 7324	. . . . . . 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. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) /\ ( y e. Q. /\ z e. Q. /\ w e. Q. ) ) -> ( y <Q z <-> ( w .Q y ) <Q ( w .Q z ) ) )
3		simpr 109	. . . . . 6 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> X e. Q. )
4		prop 7398	. . . . . . . . 9 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
5		elprnql 7404	. . . . . . . . 9 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ G e. ( 1st ` A ) ) -> G e. Q. )
6	4, 5	sylan 281	. . . . . . . 8 |- ( ( A e. P. /\ G e. ( 1st ` A ) ) -> G e. Q. )
7	6	ad2antrr 480	. . . . . . 7 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> G e. Q. )
8		prop 7398	. . . . . . . . 9 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
9		elprnql 7404	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ H e. ( 1st ` B ) ) -> H e. Q. )
10	8, 9	sylan 281	. . . . . . . 8 |- ( ( B e. P. /\ H e. ( 1st ` B ) ) -> H e. Q. )
11	10	ad2antlr 481	. . . . . . 7 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> H e. Q. )
12		mulclnq 7299	. . . . . . 7 |- ( ( G e. Q. /\ H e. Q. ) -> ( G .Q H ) e. Q. )
13	7, 11, 12	syl2anc 409	. . . . . 6 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( G .Q H ) e. Q. )
14		recclnq 7315	. . . . . . 7 |- ( H e. Q. -> ( *Q ` H ) e. Q. )
15	11, 14	syl 14	. . . . . 6 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( *Q ` H ) e. Q. )
16		mulcomnqg 7306	. . . . . . 7 |- ( ( y e. Q. /\ z e. Q. ) -> ( y .Q z ) = ( z .Q y ) )
17	16	adantl 275	. . . . . 6 |- ( ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) /\ ( y e. Q. /\ z e. Q. ) ) -> ( y .Q z ) = ( z .Q y ) )
18	2, 3, 13, 15, 17	caovord2d 5993	. . . . 5 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( X <Q ( G .Q H ) <-> ( X .Q ( *Q ` H ) ) <Q ( ( G .Q H ) .Q ( *Q ` H ) ) ) )
19		mulassnqg 7307	. . . . . . . 8 |- ( ( G e. Q. /\ H e. Q. /\ ( *Q ` H ) e. Q. ) -> ( ( G .Q H ) .Q ( *Q ` H ) ) = ( G .Q ( H .Q ( *Q ` H ) ) ) )
20	7, 11, 15, 19	syl3anc 1220	. . . . . . 7 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( ( G .Q H ) .Q ( *Q ` H ) ) = ( G .Q ( H .Q ( *Q ` H ) ) ) )
21		recidnq 7316	. . . . . . . . 9 |- ( H e. Q. -> ( H .Q ( *Q ` H ) ) = 1Q )
22	21	oveq2d 5843	. . . . . . . 8 |- ( H e. Q. -> ( G .Q ( H .Q ( *Q ` H ) ) ) = ( G .Q 1Q ) )
23	11, 22	syl 14	. . . . . . 7 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( G .Q ( H .Q ( *Q ` H ) ) ) = ( G .Q 1Q ) )
24		mulidnq 7312	. . . . . . . 8 |- ( G e. Q. -> ( G .Q 1Q ) = G )
25	7, 24	syl 14	. . . . . . 7 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( G .Q 1Q ) = G )
26	20, 23, 25	3eqtrd 2194	. . . . . 6 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( ( G .Q H ) .Q ( *Q ` H ) ) = G )
27	26	breq2d 3979	. . . . 5 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( ( X .Q ( *Q ` H ) ) <Q ( ( G .Q H ) .Q ( *Q ` H ) ) <-> ( X .Q ( *Q ` H ) ) <Q G ) )
28	18, 27	bitrd 187	. . . 4 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( X <Q ( G .Q H ) <-> ( X .Q ( *Q ` H ) ) <Q G ) )
29		prcdnql 7407	. . . . . 6 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ G e. ( 1st ` A ) ) -> ( ( X .Q ( *Q ` H ) ) <Q G -> ( X .Q ( *Q ` H ) ) e. ( 1st ` A ) ) )
30	4, 29	sylan 281	. . . . 5 |- ( ( A e. P. /\ G e. ( 1st ` A ) ) -> ( ( X .Q ( *Q ` H ) ) <Q G -> ( X .Q ( *Q ` H ) ) e. ( 1st ` A ) ) )
31	30	ad2antrr 480	. . . 4 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( ( X .Q ( *Q ` H ) ) <Q G -> ( X .Q ( *Q ` H ) ) e. ( 1st ` A ) ) )
32	28, 31	sylbid 149	. . 3 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( X <Q ( G .Q H ) -> ( X .Q ( *Q ` H ) ) e. ( 1st ` A ) ) )
33		df-imp 7392	. . . . . . . . 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 7299	. . . . . . . . 9 |- ( ( y e. Q. /\ z e. Q. ) -> ( y .Q z ) e. Q. )
35	33, 34	genpprecll 7437	. . . . . . . 8 |- ( ( A e. P. /\ B e. P. ) -> ( ( ( X .Q ( *Q ` H ) ) e. ( 1st ` A ) /\ H e. ( 1st ` B ) ) -> ( ( X .Q ( *Q ` H ) ) .Q H ) e. ( 1st ` ( A .P. B ) ) ) )
36	35	exp4b 365	. . . . . . 7 |- ( A e. P. -> ( B e. P. -> ( ( X .Q ( *Q ` H ) ) e. ( 1st ` A ) -> ( H e. ( 1st ` B ) -> ( ( X .Q ( *Q ` H ) ) .Q H ) e. ( 1st ` ( A .P. B ) ) ) ) ) )
37	36	com34 83	. . . . . 6 |- ( A e. P. -> ( B e. P. -> ( H e. ( 1st ` B ) -> ( ( X .Q ( *Q ` H ) ) e. ( 1st ` A ) -> ( ( X .Q ( *Q ` H ) ) .Q H ) e. ( 1st ` ( A .P. B ) ) ) ) ) )
38	37	imp32 255	. . . . 5 |- ( ( A e. P. /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) -> ( ( X .Q ( *Q ` H ) ) e. ( 1st ` A ) -> ( ( X .Q ( *Q ` H ) ) .Q H ) e. ( 1st ` ( A .P. B ) ) ) )
39	38	adantlr 469	. . . 4 |- ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) -> ( ( X .Q ( *Q ` H ) ) e. ( 1st ` A ) -> ( ( X .Q ( *Q ` H ) ) .Q H ) e. ( 1st ` ( A .P. B ) ) ) )
40	39	adantr 274	. . 3 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( ( X .Q ( *Q ` H ) ) e. ( 1st ` A ) -> ( ( X .Q ( *Q ` H ) ) .Q H ) e. ( 1st ` ( A .P. B ) ) ) )
41	32, 40	syld 45	. 2 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( X <Q ( G .Q H ) -> ( ( X .Q ( *Q ` H ) ) .Q H ) e. ( 1st ` ( A .P. B ) ) ) )
42		mulassnqg 7307	. . . . 5 |- ( ( X e. Q. /\ ( *Q ` H ) e. Q. /\ H e. Q. ) -> ( ( X .Q ( *Q ` H ) ) .Q H ) = ( X .Q ( ( *Q ` H ) .Q H ) ) )
43	3, 15, 11, 42	syl3anc 1220	. . . 4 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( ( X .Q ( *Q ` H ) ) .Q H ) = ( X .Q ( ( *Q ` H ) .Q H ) ) )
44		mulcomnqg 7306	. . . . . . 7 |- ( ( ( *Q ` H ) e. Q. /\ H e. Q. ) -> ( ( *Q ` H ) .Q H ) = ( H .Q ( *Q ` H ) ) )
45	15, 11, 44	syl2anc 409	. . . . . 6 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( ( *Q ` H ) .Q H ) = ( H .Q ( *Q ` H ) ) )
46	11, 21	syl 14	. . . . . 6 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( H .Q ( *Q ` H ) ) = 1Q )
47	45, 46	eqtrd 2190	. . . . 5 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( ( *Q ` H ) .Q H ) = 1Q )
48	47	oveq2d 5843	. . . 4 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( X .Q ( ( *Q ` H ) .Q H ) ) = ( X .Q 1Q ) )
49		mulidnq 7312	. . . . 5 |- ( X e. Q. -> ( X .Q 1Q ) = X )
50	49	adantl 275	. . . 4 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( X .Q 1Q ) = X )
51	43, 48, 50	3eqtrd 2194	. . 3 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( ( X .Q ( *Q ` H ) ) .Q H ) = X )
52	51	eleq1d 2226	. 2 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( ( ( X .Q ( *Q ` H ) ) .Q H ) e. ( 1st ` ( A .P. B ) ) <-> X e. ( 1st ` ( A .P. B ) ) ) )
53	41, 52	sylibd 148	1 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( X <Q ( G .Q H ) -> X e. ( 1st ` ( A .P. B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 963   = wceq 1335   e. wcel 2128   <.cop 3564   class class class wbr 3967   ` cfv 5173  (class class class)co 5827   1stc1st 6089   2ndc2nd 6090   Q.cnq 7203   1Qc1q 7204   .Q cmq 7206   *Qcrq 7207   <Q cltq 7208   P.cnp 7214   .P. cmp 7217

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4082  ax-sep 4085  ax-nul 4093  ax-pow 4138  ax-pr 4172  ax-un 4396  ax-setind 4499  ax-iinf 4550

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4029  df-mpt 4030  df-tr 4066  df-eprel 4252  df-id 4256  df-iord 4329  df-on 4331  df-suc 4334  df-iom 4553  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-rn 4600  df-res 4601  df-ima 4602  df-iota 5138  df-fun 5175  df-fn 5176  df-f 5177  df-f1 5178  df-fo 5179  df-f1o 5180  df-fv 5181  df-ov 5830  df-oprab 5831  df-mpo 5832  df-1st 6091  df-2nd 6092  df-recs 6255  df-irdg 6320  df-1o 6366  df-oadd 6370  df-omul 6371  df-er 6483  df-ec 6485  df-qs 6489  df-ni 7227  df-mi 7229  df-lti 7230  df-mpq 7268  df-enq 7270  df-nqqs 7271  df-mqqs 7273  df-1nqqs 7274  df-rq 7275  df-ltnqqs 7276  df-inp 7389  df-imp 7392

This theorem is referenced by:  mullocprlem  7493  mulclpr  7495