Theorem addnqprl 7596

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

Assertion

Ref	Expression
addnqprl	|- ( ( ( ( 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 addnqprl

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

Step	Hyp	Ref	Expression
1		prop 7542	. . . . . 6 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		addnqprllem 7594	. . . . . 6 |- ( ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ G e. ( 1st ` A ) ) /\ X e. Q. ) -> ( X <Q ( G +Q H ) -> ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q G ) e. ( 1st ` A ) ) )
3	1, 2	sylanl1 402	. . . . 5 |- ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ X e. Q. ) -> ( X <Q ( G +Q H ) -> ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q G ) e. ( 1st ` A ) ) )
4	3	adantlr 477	. . . 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 ` ( G +Q H ) ) ) .Q G ) e. ( 1st ` A ) ) )
5		prop 7542	. . . . . 6 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
6		addnqprllem 7594	. . . . . 6 |- ( ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ H e. ( 1st ` B ) ) /\ X e. Q. ) -> ( X <Q ( G +Q H ) -> ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q H ) e. ( 1st ` B ) ) )
7	5, 6	sylanl1 402	. . . . 5 |- ( ( ( B e. P. /\ H e. ( 1st ` B ) ) /\ X e. Q. ) -> ( X <Q ( G +Q H ) -> ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q H ) e. ( 1st ` B ) ) )
8	7	adantll 476	. . . 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 ` ( G +Q H ) ) ) .Q H ) e. ( 1st ` B ) ) )
9	4, 8	jcad 307	. . 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 ` ( G +Q H ) ) ) .Q G ) e. ( 1st ` A ) /\ ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q H ) e. ( 1st ` B ) ) ) )
10		simpl 109	. . . 4 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) )
11		simpl 109	. . . . 5 |- ( ( A e. P. /\ G e. ( 1st ` A ) ) -> A e. P. )
12		simpl 109	. . . . 5 |- ( ( B e. P. /\ H e. ( 1st ` B ) ) -> B e. P. )
13	11, 12	anim12i 338	. . . 4 |- ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) -> ( A e. P. /\ B e. P. ) )
14		df-iplp 7535	. . . . 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 7442	. . . . 5 |- ( ( r e. Q. /\ s e. Q. ) -> ( r +Q s ) e. Q. )
16	14, 15	genpprecll 7581	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( ( ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q G ) e. ( 1st ` A ) /\ ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q H ) e. ( 1st ` B ) ) -> ( ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q G ) +Q ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q H ) ) e. ( 1st ` ( A +P. B ) ) ) )
17	10, 13, 16	3syl 17	. . 3 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( ( ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q G ) e. ( 1st ` A ) /\ ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q H ) e. ( 1st ` B ) ) -> ( ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q G ) +Q ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q H ) ) e. ( 1st ` ( A +P. B ) ) ) )
18	9, 17	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 ` ( G +Q H ) ) ) .Q G ) +Q ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q H ) ) e. ( 1st ` ( A +P. B ) ) ) )
19		simpr 110	. . . . 5 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> X e. Q. )
20		elprnql 7548	. . . . . . . . 9 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ G e. ( 1st ` A ) ) -> G e. Q. )
21	1, 20	sylan 283	. . . . . . . 8 |- ( ( A e. P. /\ G e. ( 1st ` A ) ) -> G e. Q. )
22	21	ad2antrr 488	. . . . . . 7 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> G e. Q. )
23		elprnql 7548	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ H e. ( 1st ` B ) ) -> H e. Q. )
24	5, 23	sylan 283	. . . . . . . 8 |- ( ( B e. P. /\ H e. ( 1st ` B ) ) -> H e. Q. )
25	24	ad2antlr 489	. . . . . . 7 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> H e. Q. )
26		addclnq 7442	. . . . . . 7 |- ( ( G e. Q. /\ H e. Q. ) -> ( G +Q H ) e. Q. )
27	22, 25, 26	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. )
28		recclnq 7459	. . . . . 6 |- ( ( G +Q H ) e. Q. -> ( *Q ` ( G +Q H ) ) e. Q. )
29	27, 28	syl 14	. . . . 5 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( *Q ` ( G +Q H ) ) e. Q. )
30		mulassnqg 7451	. . . . 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 1249	. . . 4 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` 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 7443	. . . . . 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. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( X .Q ( *Q ` ( G +Q H ) ) ) e. Q. )
34		distrnqg 7454	. . . . 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 1249	. . . 4 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` 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 7450	. . . . . . . 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. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( ( *Q ` ( G +Q H ) ) .Q ( G +Q H ) ) = ( ( G +Q H ) .Q ( *Q ` ( G +Q H ) ) ) )
38		recidnq 7460	. . . . . . . 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. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( ( G +Q H ) .Q ( *Q ` ( G +Q H ) ) ) = 1Q )
40	37, 39	eqtrd 2229	. . . . . 6 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( ( *Q ` ( G +Q H ) ) .Q ( G +Q H ) ) = 1Q )
41	40	oveq2d 5938	. . . . 5 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( X .Q ( ( *Q ` ( G +Q H ) ) .Q ( G +Q H ) ) ) = ( X .Q 1Q ) )
42		mulidnq 7456	. . . . . 6 |- ( X e. Q. -> ( X .Q 1Q ) = X )
43	42	adantl 277	. . . . 5 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( X .Q 1Q ) = X )
44	41, 43	eqtrd 2229	. . . 4 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( X .Q ( ( *Q ` ( G +Q H ) ) .Q ( G +Q H ) ) ) = X )
45	31, 35, 44	3eqtr3d 2237	. . 3 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` 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 2265	. 2 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( ( ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q G ) +Q ( ( X .Q ( *Q ` ( G +Q H ) ) ) .Q H ) ) e. ( 1st ` ( A +P. B ) ) <-> X e. ( 1st ` ( A +P. B ) ) ) )
47	18, 46	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   = wceq 1364   e. wcel 2167   <.cop 3625   class class class wbr 4033   ` cfv 5258  (class class class)co 5922   1stc1st 6196   2ndc2nd 6197   Q.cnq 7347   1Qc1q 7348   +Q cplq 7349   .Q cmq 7350   *Qcrq 7351   <Q cltq 7352   P.cnp 7358   +P. cpp 7360

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-eprel 4324  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-1o 6474  df-oadd 6478  df-omul 6479  df-er 6592  df-ec 6594  df-qs 6598  df-ni 7371  df-pli 7372  df-mi 7373  df-lti 7374  df-plpq 7411  df-mpq 7412  df-enq 7414  df-nqqs 7415  df-plqqs 7416  df-mqqs 7417  df-1nqqs 7418  df-rq 7419  df-ltnqqs 7420  df-inp 7533  df-iplp 7535

This theorem is referenced by:  addlocprlemlt  7598  addclpr  7604