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Theorem addnqprl 6512
Description: Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.)
Assertion
Ref Expression
addnqprl ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝑋 <Q (𝐺 +Q 𝐻) → 𝑋 (1st ‘(A +P B))))

Proof of Theorem addnqprl
Dummy variables x y 𝑟 𝑞 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 6457 . . . . . 6 (A P → ⟨(1stA), (2ndA)⟩ P)
2 addnqprllem 6510 . . . . . 6 (((⟨(1stA), (2ndA)⟩ P 𝐺 (1stA)) 𝑋 Q) → (𝑋 <Q (𝐺 +Q 𝐻) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) (1stA)))
31, 2sylanl1 382 . . . . 5 (((A P 𝐺 (1stA)) 𝑋 Q) → (𝑋 <Q (𝐺 +Q 𝐻) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) (1stA)))
43adantlr 446 . . . 4 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝑋 <Q (𝐺 +Q 𝐻) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) (1stA)))
5 prop 6457 . . . . . 6 (B P → ⟨(1stB), (2ndB)⟩ P)
6 addnqprllem 6510 . . . . . 6 (((⟨(1stB), (2ndB)⟩ P 𝐻 (1stB)) 𝑋 Q) → (𝑋 <Q (𝐺 +Q 𝐻) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) (1stB)))
75, 6sylanl1 382 . . . . 5 (((B P 𝐻 (1stB)) 𝑋 Q) → (𝑋 <Q (𝐺 +Q 𝐻) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) (1stB)))
87adantll 445 . . . 4 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝑋 <Q (𝐺 +Q 𝐻) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) (1stB)))
94, 8jcad 291 . . 3 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝑋 <Q (𝐺 +Q 𝐻) → (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) (1stA) ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) (1stB))))
10 simpl 102 . . . 4 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → ((A P 𝐺 (1stA)) (B P 𝐻 (1stB))))
11 simpl 102 . . . . 5 ((A P 𝐺 (1stA)) → A P)
12 simpl 102 . . . . 5 ((B P 𝐻 (1stB)) → B P)
1311, 12anim12i 321 . . . 4 (((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) → (A P B P))
14 df-iplp 6450 . . . . 5 +P = (x P, y P ↦ ⟨{𝑞 Q𝑟 Q 𝑠 Q (𝑟 (1stx) 𝑠 (1sty) 𝑞 = (𝑟 +Q 𝑠))}, {𝑞 Q𝑟 Q 𝑠 Q (𝑟 (2ndx) 𝑠 (2ndy) 𝑞 = (𝑟 +Q 𝑠))}⟩)
15 addclnq 6359 . . . . 5 ((𝑟 Q 𝑠 Q) → (𝑟 +Q 𝑠) Q)
1614, 15genpprecll 6496 . . . 4 ((A P B P) → ((((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) (1stA) ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) (1stB)) → (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)) (1st ‘(A +P B))))
1710, 13, 163syl 17 . . 3 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → ((((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) (1stA) ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) (1stB)) → (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)) (1st ‘(A +P B))))
189, 17syld 40 . 2 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝑋 <Q (𝐺 +Q 𝐻) → (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)) (1st ‘(A +P B))))
19 simpr 103 . . . . 5 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → 𝑋 Q)
20 elprnql 6463 . . . . . . . . 9 ((⟨(1stA), (2ndA)⟩ P 𝐺 (1stA)) → 𝐺 Q)
211, 20sylan 267 . . . . . . . 8 ((A P 𝐺 (1stA)) → 𝐺 Q)
2221ad2antrr 457 . . . . . . 7 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → 𝐺 Q)
23 elprnql 6463 . . . . . . . . 9 ((⟨(1stB), (2ndB)⟩ P 𝐻 (1stB)) → 𝐻 Q)
245, 23sylan 267 . . . . . . . 8 ((B P 𝐻 (1stB)) → 𝐻 Q)
2524ad2antlr 458 . . . . . . 7 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → 𝐻 Q)
26 addclnq 6359 . . . . . . 7 ((𝐺 Q 𝐻 Q) → (𝐺 +Q 𝐻) Q)
2722, 25, 26syl2anc 391 . . . . . 6 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝐺 +Q 𝐻) Q)
28 recclnq 6376 . . . . . 6 ((𝐺 +Q 𝐻) Q → (*Q‘(𝐺 +Q 𝐻)) Q)
2927, 28syl 14 . . . . 5 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (*Q‘(𝐺 +Q 𝐻)) Q)
30 mulassnqg 6368 . . . . 5 ((𝑋 Q (*Q‘(𝐺 +Q 𝐻)) Q (𝐺 +Q 𝐻) Q) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q (𝐺 +Q 𝐻)) = (𝑋 ·Q ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻))))
3119, 29, 27, 30syl3anc 1134 . . . 4 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q (𝐺 +Q 𝐻)) = (𝑋 ·Q ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻))))
32 mulclnq 6360 . . . . . 6 ((𝑋 Q (*Q‘(𝐺 +Q 𝐻)) Q) → (𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) Q)
3319, 29, 32syl2anc 391 . . . . 5 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) Q)
34 distrnqg 6371 . . . . 5 (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) Q 𝐺 Q 𝐻 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)))
3533, 22, 25, 34syl3anc 1134 . . . 4 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 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 6367 . . . . . . . 8 (((*Q‘(𝐺 +Q 𝐻)) Q (𝐺 +Q 𝐻) Q) → ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻)) = ((𝐺 +Q 𝐻) ·Q (*Q‘(𝐺 +Q 𝐻))))
3729, 27, 36syl2anc 391 . . . . . . 7 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻)) = ((𝐺 +Q 𝐻) ·Q (*Q‘(𝐺 +Q 𝐻))))
38 recidnq 6377 . . . . . . . 8 ((𝐺 +Q 𝐻) Q → ((𝐺 +Q 𝐻) ·Q (*Q‘(𝐺 +Q 𝐻))) = 1Q)
3927, 38syl 14 . . . . . . 7 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → ((𝐺 +Q 𝐻) ·Q (*Q‘(𝐺 +Q 𝐻))) = 1Q)
4037, 39eqtrd 2069 . . . . . 6 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻)) = 1Q)
4140oveq2d 5471 . . . . 5 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝑋 ·Q ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻))) = (𝑋 ·Q 1Q))
42 mulidnq 6373 . . . . . 6 (𝑋 Q → (𝑋 ·Q 1Q) = 𝑋)
4342adantl 262 . . . . 5 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝑋 ·Q 1Q) = 𝑋)
4441, 43eqtrd 2069 . . . 4 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝑋 ·Q ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻))) = 𝑋)
4531, 35, 443eqtr3d 2077 . . 3 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)) = 𝑋)
4645eleq1d 2103 . 2 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → ((((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)) (1st ‘(A +P B)) ↔ 𝑋 (1st ‘(A +P B))))
4718, 46sylibd 138 1 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝑋 <Q (𝐺 +Q 𝐻) → 𝑋 (1st ‘(A +P B))))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  cop 3370   class class class wbr 3755  cfv 4845  (class class class)co 5455  1st c1st 5707  2nd c2nd 5708  Qcnq 6264  1Qc1q 6265   +Q cplq 6266   ·Q cmq 6267  *Qcrq 6268   <Q cltq 6269  Pcnp 6275   +P cpp 6277
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-inp 6448  df-iplp 6450
This theorem is referenced by:  addlocprlemlt  6514  addclpr  6520
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