Theorem addnqpru 7740

Description: Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.)

Assertion

Ref	Expression
addnqpru	|- ( ( ( ( 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 addnqpru

Dummy variables r q s x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prop 7685	. . . . . 6 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		addnqprulem 7738	. . . . . 6 |- ( ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ G e. ( 2nd ` A ) ) /\ X e. Q. ) -> ( ( G +Q H ) <Q X -> ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q G ) e. ( 2nd ` A ) ) )
3	1, 2	sylanl1 402	. . . . 5 |- ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ X e. Q. ) -> ( ( G +Q H ) <Q X -> ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q G ) e. ( 2nd ` A ) ) )
4	3	adantlr 477	. . . 4 |- ( ( ( ( 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 ` ( G +Q H ) ) ) .Q G ) e. ( 2nd ` A ) ) )
5		prop 7685	. . . . . 6 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
6		addnqprulem 7738	. . . . . 6 |- ( ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ H e. ( 2nd ` B ) ) /\ X e. Q. ) -> ( ( G +Q H ) <Q X -> ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q H ) e. ( 2nd ` B ) ) )
7	5, 6	sylanl1 402	. . . . 5 |- ( ( ( B e. P. /\ H e. ( 2nd ` B ) ) /\ X e. Q. ) -> ( ( G +Q H ) <Q X -> ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q H ) e. ( 2nd ` B ) ) )
8	7	adantll 476	. . . 4 |- ( ( ( ( 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 ` ( G +Q H ) ) ) .Q H ) e. ( 2nd ` B ) ) )
9	4, 8	jcad 307	. . 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 ` ( G +Q H ) ) ) .Q G ) e. ( 2nd ` A ) /\ ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q H ) e. ( 2nd ` B ) ) ) )
10		simpl 109	. . . 4 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) )
11		simpl 109	. . . . 5 |- ( ( A e. P. /\ G e. ( 2nd ` A ) ) -> A e. P. )
12		simpl 109	. . . . 5 |- ( ( B e. P. /\ H e. ( 2nd ` B ) ) -> B e. P. )
13	11, 12	anim12i 338	. . . 4 |- ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) -> ( A e. P. /\ B e. P. ) )
14		df-iplp 7678	. . . . 5 |- +P. = ( x e. P. , y e. P. |-> <. { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 1st ` x ) /\ s e. ( 1st ` y ) /\ q = ( r +Q s ) ) } , { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 2nd ` x ) /\ s e. ( 2nd ` y ) /\ q = ( r +Q s ) ) } >. )
15		addclnq 7585	. . . . 5 |- ( ( r e. Q. /\ s e. Q. ) -> ( r +Q s ) e. Q. )
16	14, 15	genppreclu 7725	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( ( ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q G ) e. ( 2nd ` A ) /\ ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q H ) e. ( 2nd ` B ) ) -> ( ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q G ) +Q ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q H ) ) e. ( 2nd ` ( A +P. B ) ) ) )
17	10, 13, 16	3syl 17	. . 3 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( ( ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q G ) e. ( 2nd ` A ) /\ ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q H ) e. ( 2nd ` B ) ) -> ( ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q G ) +Q ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q H ) ) e. ( 2nd ` ( A +P. B ) ) ) )
18	9, 17	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 ` ( G +Q H ) ) ) .Q G ) +Q ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q H ) ) e. ( 2nd ` ( A +P. B ) ) ) )
19		simpr 110	. . . . 5 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> X e. Q. )
20		elprnqu 7692	. . . . . . . . 9 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ G e. ( 2nd ` A ) ) -> G e. Q. )
21	1, 20	sylan 283	. . . . . . . 8 |- ( ( A e. P. /\ G e. ( 2nd ` A ) ) -> G e. Q. )
22	21	ad2antrr 488	. . . . . . 7 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> G e. Q. )
23		elprnqu 7692	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ H e. ( 2nd ` B ) ) -> H e. Q. )
24	5, 23	sylan 283	. . . . . . . 8 |- ( ( B e. P. /\ H e. ( 2nd ` B ) ) -> H e. Q. )
25	24	ad2antlr 489	. . . . . . 7 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> H e. Q. )
26		addclnq 7585	. . . . . . 7 |- ( ( G e. Q. /\ H e. Q. ) -> ( G +Q H ) e. Q. )
27	22, 25, 26	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. )
28		recclnq 7602	. . . . . 6 |- ( ( G +Q H ) e. Q. -> ( *Q ` ( G +Q H ) ) e. Q. )
29	27, 28	syl 14	. . . . 5 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( *Q ` ( G +Q H ) ) e. Q. )
30		mulassnqg 7594	. . . . 5 |- ( ( X e. Q. /\ ( *Q ` ( G +Q H ) ) e. Q. /\ ( G +Q H ) e. Q. ) -> ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q ( G +Q H ) ) = ( X .Q ( ( *Q ` ( G +Q H ) ) .Q ( G +Q H ) ) ) )
31	19, 29, 27, 30	syl3anc 1271	. . . 4 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q ( G +Q H ) ) = ( X .Q ( ( *Q ` ( G +Q H ) ) .Q ( G +Q H ) ) ) )
32		mulclnq 7586	. . . . . 6 |- ( ( X e. Q. /\ ( *Q ` ( G +Q H ) ) e. Q. ) -> ( X .Q ( *Q ` ( G +Q H ) ) ) e. Q. )
33	19, 29, 32	syl2anc 411	. . . . 5 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( X .Q ( *Q ` ( G +Q H ) ) ) e. Q. )
34		distrnqg 7597	. . . . 5 |- ( ( ( X .Q ( *Q ` ( G +Q H ) ) ) e. Q. /\ G e. Q. /\ H e. Q. ) -> ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q ( G +Q H ) ) = ( ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q G ) +Q ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q H ) ) )
35	33, 22, 25, 34	syl3anc 1271	. . . 4 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q ( G +Q H ) ) = ( ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q G ) +Q ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q H ) ) )
36		mulcomnqg 7593	. . . . . . . 8 |- ( ( ( *Q ` ( G +Q H ) ) e. Q. /\ ( G +Q H ) e. Q. ) -> ( ( *Q ` ( G +Q H ) ) .Q ( G +Q H ) ) = ( ( G +Q H ) .Q ( *Q ` ( G +Q H ) ) ) )
37	29, 27, 36	syl2anc 411	. . . . . . 7 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( ( *Q ` ( G +Q H ) ) .Q ( G +Q H ) ) = ( ( G +Q H ) .Q ( *Q ` ( G +Q H ) ) ) )
38		recidnq 7603	. . . . . . . 8 |- ( ( G +Q H ) e. Q. -> ( ( G +Q H ) .Q ( *Q ` ( G +Q H ) ) ) = 1Q )
39	27, 38	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 ` ( G +Q H ) ) ) = 1Q )
40	37, 39	eqtrd 2262	. . . . . 6 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( ( *Q ` ( G +Q H ) ) .Q ( G +Q H ) ) = 1Q )
41	40	oveq2d 6029	. . . . 5 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( X .Q ( ( *Q ` ( G +Q H ) ) .Q ( G +Q H ) ) ) = ( X .Q 1Q ) )
42		mulidnq 7599	. . . . . 6 |- ( X e. Q. -> ( X .Q 1Q ) = X )
43	42	adantl 277	. . . . 5 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( X .Q 1Q ) = X )
44	41, 43	eqtrd 2262	. . . 4 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( X .Q ( ( *Q ` ( G +Q H ) ) .Q ( G +Q H ) ) ) = X )
45	31, 35, 44	3eqtr3d 2270	. . 3 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q G ) +Q ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q H ) ) = X )
46	45	eleq1d 2298	. 2 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> ( ( ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q G ) +Q ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q H ) ) e. ( 2nd ` ( A +P. B ) ) <-> X e. ( 2nd ` ( A +P. B ) ) ) )
47	18, 46	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   = wceq 1395   e. wcel 2200   <.cop 3670   class class class wbr 4086   ` cfv 5324  (class class class)co 6013   1stc1st 6296   2ndc2nd 6297   Q.cnq 7490   1Qc1q 7491   +Q cplq 7492   .Q cmq 7493   *Qcrq 7494   <Q cltq 7495   P.cnp 7501   +P. cpp 7503

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-1o 6577  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7514  df-pli 7515  df-mi 7516  df-lti 7517  df-plpq 7554  df-mpq 7555  df-enq 7557  df-nqqs 7558  df-plqqs 7559  df-mqqs 7560  df-1nqqs 7561  df-rq 7562  df-ltnqqs 7563  df-inp 7676  df-iplp 7678

This theorem is referenced by:  addlocprlemeq  7743  addlocprlemgt  7744  addclpr  7747