Theorem mulnqpru 7629

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 7461	. . . . . . 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. ( 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 7535	. . . . . . . . 9 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
4		elprnqu 7542	. . . . . . . . 9 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ G e. ( 2nd ` A ) ) -> G e. Q. )
5	3, 4	sylan 283	. . . . . . . 8 |- ( ( A e. P. /\ G e. ( 2nd ` A ) ) -> G e. Q. )
6	5	ad2antrr 488	. . . . . . 7 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> G e. Q. )
7		prop 7535	. . . . . . . . 9 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
8		elprnqu 7542	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ H e. ( 2nd ` B ) ) -> H e. Q. )
9	7, 8	sylan 283	. . . . . . . 8 |- ( ( B e. P. /\ H e. ( 2nd ` B ) ) -> H e. Q. )
10	9	ad2antlr 489	. . . . . . 7 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> H e. Q. )
11		mulclnq 7436	. . . . . . 7 |- ( ( G e. Q. /\ H e. Q. ) -> ( G .Q H ) e. Q. )
12	6, 10, 11	syl2anc 411	. . . . . 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 110	. . . . . 6 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> X e. Q. )
14		recclnq 7452	. . . . . . 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 7443	. . . . . . 7 |- ( ( y e. Q. /\ z e. Q. ) -> ( y .Q z ) = ( z .Q y ) )
17	16	adantl 277	. . . . . 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 6088	. . . . 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 7444	. . . . . . . 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 1249	. . . . . . 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 7453	. . . . . . . . 9 |- ( H e. Q. -> ( H .Q ( *Q ` H ) ) = 1Q )
22	21	oveq2d 5934	. . . . . . . 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 7449	. . . . . . . 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 2230	. . . . . 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 4039	. . . . 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 188	. . . 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 7545	. . . . . 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 283	. . . . 5 |- ( ( A e. P. /\ G e. ( 2nd ` A ) ) -> ( G <Q ( X .Q ( *Q ` H ) ) -> ( X .Q ( *Q ` H ) ) e. ( 2nd ` A ) ) )
31	30	ad2antrr 488	. . . 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 150	. . 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 7529	. . . . . . . . 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 7436	. . . . . . . . 9 |- ( ( y e. Q. /\ z e. Q. ) -> ( y .Q z ) e. Q. )
35	33, 34	genppreclu 7575	. . . . . . . 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 367	. . . . . . 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 257	. . . . 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 477	. . . 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 276	. . 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 7444	. . . . 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 1249	. . . 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 7443	. . . . . . 7 |- ( ( ( *Q ` H ) e. Q. /\ H e. Q. ) -> ( ( *Q ` H ) .Q H ) = ( H .Q ( *Q ` H ) ) )
45	15, 10, 44	syl2anc 411	. . . . . 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 2226	. . . . 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 5934	. . . 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 7449	. . . . 5 |- ( X e. Q. -> ( X .Q 1Q ) = X )
50	49	adantl 277	. . . 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 2230	. . 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 2262	. 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 149	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 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   <.cop 3621   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   1stc1st 6191   2ndc2nd 6192   Q.cnq 7340   1Qc1q 7341   .Q cmq 7343   *Qcrq 7344   <Q cltq 7345   P.cnp 7351   .P. cmp 7354

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-mi 7366  df-lti 7367  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413  df-inp 7526  df-imp 7529

This theorem is referenced by:  mullocprlem  7630  mulclpr  7632