Theorem addnqprl 7545

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 7491	. . . . . 6 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		addnqprllem 7543	. . . . . 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 7491	. . . . . 6 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
6		addnqprllem 7543	. . . . . 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 7484	. . . . 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 7391	. . . . 5 |- ( ( r e. Q. /\ s e. Q. ) -> ( r +Q s ) e. Q. )
16	14, 15	genpprecll 7530	. . . 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 7497	. . . . . . . . 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 7497	. . . . . . . . 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 7391	. . . . . . 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 7408	. . . . . 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 7400	. . . . 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 1248	. . . 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 7392	. . . . . 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 7403	. . . . 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 1248	. . . 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 7399	. . . . . . . 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 7409	. . . . . . . 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 2221	. . . . . 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 5906	. . . . 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 7405	. . . . . 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 2221	. . . 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 2229	. . 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 2257	. 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 1363   e. wcel 2159   <.cop 3609   class class class wbr 4017   ` cfv 5230  (class class class)co 5890   1stc1st 6156   2ndc2nd 6157   Q.cnq 7296   1Qc1q 7297   +Q cplq 7298   .Q cmq 7299   *Qcrq 7300   <Q cltq 7301   P.cnp 7307   +P. cpp 7309

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-coll 4132  ax-sep 4135  ax-nul 4143  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-iinf 4601

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-ral 2472  df-rex 2473  df-reu 2474  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-tr 4116  df-eprel 4303  df-id 4307  df-iord 4380  df-on 4382  df-suc 4385  df-iom 4604  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-ov 5893  df-oprab 5894  df-mpo 5895  df-1st 6158  df-2nd 6159  df-recs 6323  df-irdg 6388  df-1o 6434  df-oadd 6438  df-omul 6439  df-er 6552  df-ec 6554  df-qs 6558  df-ni 7320  df-pli 7321  df-mi 7322  df-lti 7323  df-plpq 7360  df-mpq 7361  df-enq 7363  df-nqqs 7364  df-plqqs 7365  df-mqqs 7366  df-1nqqs 7367  df-rq 7368  df-ltnqqs 7369  df-inp 7482  df-iplp 7484

This theorem is referenced by:  addlocprlemlt  7547  addclpr  7553