Theorem addnqpru 7897

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 7842	. . . . . 6 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		addnqprulem 7895	. . . . . 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 406	. . . . 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 481	. . . 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 7842	. . . . . 6 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
6		addnqprulem 7895	. . . . . 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 406	. . . . 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 480	. . . 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 7835	. . . . 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 7742	. . . . 5 |- ( ( r e. Q. /\ s e. Q. ) -> ( r +Q s ) e. Q. )
16	14, 15	genppreclu 7882	. . . 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 7849	. . . . . . . . 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 492	. . . . . . 7 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> G e. Q. )
23		elprnqu 7849	. . . . . . . . 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 493	. . . . . . 7 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> H e. Q. )
26		addclnq 7742	. . . . . . 7 |- ( ( G e. Q. /\ H e. Q. ) -> ( G +Q H ) e. Q. )
27	22, 25, 26	syl2anc 415	. . . . . 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 7759	. . . . . 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 7751	. . . . 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 1278	. . . 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 7743	. . . . . 6 |- ( ( X e. Q. /\ ( *Q ` ( G +Q H ) ) e. Q. ) -> ( X .Q ( *Q ` ( G +Q H ) ) ) e. Q. )
33	19, 29, 32	syl2anc 415	. . . . 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 7754	. . . . 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 1278	. . . 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 7750	. . . . . . . 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 415	. . . . . . 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 7760	. . . . . . . 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 2271	. . . . . 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 6101	. . . . 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 7756	. . . . . 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 2271	. . . 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 2279	. . 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 2307	. 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 ) ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   <.cop 3712   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   1stc1st 6372   2ndc2nd 6373   Q.cnq 7647   1Qc1q 7648   +Q cplq 7649   .Q cmq 7650   *Qcrq 7651   <Q cltq 7652   P.cnp 7658   +P. cpp 7660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-eprel 4434  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-1o 6687  df-oadd 6691  df-omul 6692  df-er 6807  df-ec 6809  df-qs 6813  df-ni 7671  df-pli 7672  df-mi 7673  df-lti 7674  df-plpq 7711  df-mpq 7712  df-enq 7714  df-nqqs 7715  df-plqqs 7716  df-mqqs 7717  df-1nqqs 7718  df-rq 7719  df-ltnqqs 7720  df-inp 7833  df-iplp 7835

This theorem is used by:  addlocprlemeq  7900  addlocprlemgt  7901  addclpr  7904