Theorem mulnqprl 7558

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 7391	. . . . . . 7 |- ( ( y e. Q. /\ z e. Q. /\ w e. Q. ) -> ( y <Q z <-> ( w .Q y ) <Q ( w .Q z ) ) )
2	1	adantl 277	. . . . . 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 110	. . . . . 6 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> X e. Q. )
4		prop 7465	. . . . . . . . 9 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
5		elprnql 7471	. . . . . . . . 9 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ G e. ( 1st ` A ) ) -> G e. Q. )
6	4, 5	sylan 283	. . . . . . . 8 |- ( ( A e. P. /\ G e. ( 1st ` A ) ) -> G e. Q. )
7	6	ad2antrr 488	. . . . . . 7 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> G e. Q. )
8		prop 7465	. . . . . . . . 9 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
9		elprnql 7471	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ H e. ( 1st ` B ) ) -> H e. Q. )
10	8, 9	sylan 283	. . . . . . . 8 |- ( ( B e. P. /\ H e. ( 1st ` B ) ) -> H e. Q. )
11	10	ad2antlr 489	. . . . . . 7 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> H e. Q. )
12		mulclnq 7366	. . . . . . 7 |- ( ( G e. Q. /\ H e. Q. ) -> ( G .Q H ) e. Q. )
13	7, 11, 12	syl2anc 411	. . . . . 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 7382	. . . . . . 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 7373	. . . . . . 7 |- ( ( y e. Q. /\ z e. Q. ) -> ( y .Q z ) = ( z .Q y ) )
17	16	adantl 277	. . . . . 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 6038	. . . . 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 7374	. . . . . . . 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 1238	. . . . . . 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 7383	. . . . . . . . 9 |- ( H e. Q. -> ( H .Q ( *Q ` H ) ) = 1Q )
22	21	oveq2d 5885	. . . . . . . 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 7379	. . . . . . . 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 2214	. . . . . 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 4012	. . . . 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 188	. . . 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 7474	. . . . . 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 283	. . . . 5 |- ( ( A e. P. /\ G e. ( 1st ` A ) ) -> ( ( X .Q ( *Q ` H ) ) <Q G -> ( X .Q ( *Q ` H ) ) e. ( 1st ` A ) ) )
31	30	ad2antrr 488	. . . 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 150	. . 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 7459	. . . . . . . . 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 7366	. . . . . . . . 9 |- ( ( y e. Q. /\ z e. Q. ) -> ( y .Q z ) e. Q. )
35	33, 34	genpprecll 7504	. . . . . . . 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 367	. . . . . . 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 257	. . . . 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 477	. . . 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 276	. . 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 7374	. . . . 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 1238	. . . 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 7373	. . . . . . 7 |- ( ( ( *Q ` H ) e. Q. /\ H e. Q. ) -> ( ( *Q ` H ) .Q H ) = ( H .Q ( *Q ` H ) ) )
45	15, 11, 44	syl2anc 411	. . . . . 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 2210	. . . . 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 5885	. . . 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 7379	. . . . 5 |- ( X e. Q. -> ( X .Q 1Q ) = X )
50	49	adantl 277	. . . 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 2214	. . 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 2246	. 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 149	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 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   <.cop 3594   class class class wbr 4000   ` cfv 5212  (class class class)co 5869   1stc1st 6133   2ndc2nd 6134   Q.cnq 7270   1Qc1q 7271   .Q cmq 7273   *Qcrq 7274   <Q cltq 7275   P.cnp 7281   .P. cmp 7284

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-mi 7296  df-lti 7297  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-inp 7456  df-imp 7459

This theorem is referenced by:  mullocprlem  7560  mulclpr  7562