Theorem mulnqprl 7530

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 7363	. . . . . . 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 7437	. . . . . . . . 9 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
5		elprnql 7443	. . . . . . . . 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 485	. . . . . . 7 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> G e. Q. )
8		prop 7437	. . . . . . . . 9 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
9		elprnql 7443	. . . . . . . . 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 486	. . . . . . 7 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> H e. Q. )
12		mulclnq 7338	. . . . . . 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 7354	. . . . . . 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 7345	. . . . . . 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 6022	. . . . 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 7346	. . . . . . . 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 1233	. . . . . . 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 7355	. . . . . . . . 9 |- ( H e. Q. -> ( H .Q ( *Q ` H ) ) = 1Q )
22	21	oveq2d 5869	. . . . . . . 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 7351	. . . . . . . 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 2207	. . . . . 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 4001	. . . . 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 7446	. . . . . 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 485	. . . 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 7431	. . . . . . . . 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 7338	. . . . . . . . 9 |- ( ( y e. Q. /\ z e. Q. ) -> ( y .Q z ) e. Q. )
35	33, 34	genpprecll 7476	. . . . . . . 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 474	. . . 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 7346	. . . . 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 1233	. . . 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 7345	. . . . . . 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 2203	. . . . 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 5869	. . . 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 7351	. . . . 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 2207	. . 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 2239	. 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 973   = wceq 1348   e. wcel 2141   <.cop 3586   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   1stc1st 6117   2ndc2nd 6118   Q.cnq 7242   1Qc1q 7243   .Q cmq 7245   *Qcrq 7246   <Q cltq 7247   P.cnp 7253   .P. cmp 7256

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-mi 7268  df-lti 7269  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-inp 7428  df-imp 7431

This theorem is referenced by:  mullocprlem  7532  mulclpr  7534