Theorem addnqpru 7338

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 7283	. . . . . 6 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		addnqprulem 7336	. . . . . 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 399	. . . . 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 468	. . . 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 7283	. . . . . 6 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
6		addnqprulem 7336	. . . . . 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 399	. . . . 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 467	. . . 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 305	. . 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 108	. . . 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 108	. . . . 5 |- ( ( A e. P. /\ G e. ( 2nd ` A ) ) -> A e. P. )
12		simpl 108	. . . . 5 |- ( ( B e. P. /\ H e. ( 2nd ` B ) ) -> B e. P. )
13	11, 12	anim12i 336	. . . 4 |- ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) -> ( A e. P. /\ B e. P. ) )
14		df-iplp 7276	. . . . 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 7183	. . . . 5 |- ( ( r e. Q. /\ s e. Q. ) -> ( r +Q s ) e. Q. )
16	14, 15	genppreclu 7323	. . . 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 109	. . . . 5 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> X e. Q. )
20		elprnqu 7290	. . . . . . . . 9 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ G e. ( 2nd ` A ) ) -> G e. Q. )
21	1, 20	sylan 281	. . . . . . . 8 |- ( ( A e. P. /\ G e. ( 2nd ` A ) ) -> G e. Q. )
22	21	ad2antrr 479	. . . . . . 7 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> G e. Q. )
23		elprnqu 7290	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ H e. ( 2nd ` B ) ) -> H e. Q. )
24	5, 23	sylan 281	. . . . . . . 8 |- ( ( B e. P. /\ H e. ( 2nd ` B ) ) -> H e. Q. )
25	24	ad2antlr 480	. . . . . . 7 |- ( ( ( ( A e. P. /\ G e. ( 2nd ` A ) ) /\ ( B e. P. /\ H e. ( 2nd ` B ) ) ) /\ X e. Q. ) -> H e. Q. )
26		addclnq 7183	. . . . . . 7 |- ( ( G e. Q. /\ H e. Q. ) -> ( G +Q H ) e. Q. )
27	22, 25, 26	syl2anc 408	. . . . . 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 7200	. . . . . 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 7192	. . . . 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 1216	. . . 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 7184	. . . . . 6 |- ( ( X e. Q. /\ ( *Q ` ( G +Q H ) ) e. Q. ) -> ( X .Q ( *Q ` ( G +Q H ) ) ) e. Q. )
33	19, 29, 32	syl2anc 408	. . . . 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 7195	. . . . 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 1216	. . . 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 7191	. . . . . . . 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 408	. . . . . . 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 7201	. . . . . . . 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 2172	. . . . . 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 5790	. . . . 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 7197	. . . . . 6 |- ( X e. Q. -> ( X .Q 1Q ) = X )
43	42	adantl 275	. . . . 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 2172	. . . 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 2180	. . 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 2208	. 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 148	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 103   = wceq 1331   e. wcel 1480   <.cop 3530   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   1stc1st 6036   2ndc2nd 6037   Q.cnq 7088   1Qc1q 7089   +Q cplq 7090   .Q cmq 7091   *Qcrq 7092   <Q cltq 7093   P.cnp 7099   +P. cpp 7101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-inp 7274  df-iplp 7276

This theorem is referenced by:  addlocprlemeq  7341  addlocprlemgt  7342  addclpr  7345