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Theorem addnqpru 6506
Description: Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.)
Assertion
Ref Expression
addnqpru ((((A P 𝐺 (2ndA)) (B P 𝐻 (2ndB))) 𝑋 Q) → ((𝐺 +Q 𝐻) <Q 𝑋𝑋 (2nd ‘(A +P B))))

Proof of Theorem addnqpru
Dummy variables x y 𝑟 𝑞 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 6450 . . . . . 6 (A P → ⟨(1stA), (2ndA)⟩ P)
2 addnqprulem 6504 . . . . . 6 (((⟨(1stA), (2ndA)⟩ P 𝐺 (2ndA)) 𝑋 Q) → ((𝐺 +Q 𝐻) <Q 𝑋 → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) (2ndA)))
31, 2sylanl1 382 . . . . 5 (((A P 𝐺 (2ndA)) 𝑋 Q) → ((𝐺 +Q 𝐻) <Q 𝑋 → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) (2ndA)))
43adantlr 446 . . . 4 ((((A P 𝐺 (2ndA)) (B P 𝐻 (2ndB))) 𝑋 Q) → ((𝐺 +Q 𝐻) <Q 𝑋 → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) (2ndA)))
5 prop 6450 . . . . . 6 (B P → ⟨(1stB), (2ndB)⟩ P)
6 addnqprulem 6504 . . . . . 6 (((⟨(1stB), (2ndB)⟩ P 𝐻 (2ndB)) 𝑋 Q) → ((𝐺 +Q 𝐻) <Q 𝑋 → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) (2ndB)))
75, 6sylanl1 382 . . . . 5 (((B P 𝐻 (2ndB)) 𝑋 Q) → ((𝐺 +Q 𝐻) <Q 𝑋 → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) (2ndB)))
87adantll 445 . . . 4 ((((A P 𝐺 (2ndA)) (B P 𝐻 (2ndB))) 𝑋 Q) → ((𝐺 +Q 𝐻) <Q 𝑋 → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) (2ndB)))
94, 8jcad 291 . . 3 ((((A P 𝐺 (2ndA)) (B P 𝐻 (2ndB))) 𝑋 Q) → ((𝐺 +Q 𝐻) <Q 𝑋 → (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) (2ndA) ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) (2ndB))))
10 simpl 102 . . . 4 ((((A P 𝐺 (2ndA)) (B P 𝐻 (2ndB))) 𝑋 Q) → ((A P 𝐺 (2ndA)) (B P 𝐻 (2ndB))))
11 simpl 102 . . . . 5 ((A P 𝐺 (2ndA)) → A P)
12 simpl 102 . . . . 5 ((B P 𝐻 (2ndB)) → B P)
1311, 12anim12i 321 . . . 4 (((A P 𝐺 (2ndA)) (B P 𝐻 (2ndB))) → (A P B P))
14 df-iplp 6443 . . . . 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 6352 . . . . 5 ((𝑟 Q 𝑠 Q) → (𝑟 +Q 𝑠) Q)
1614, 15genppreclu 6490 . . . 4 ((A P B P) → ((((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) (2ndA) ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) (2ndB)) → (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)) (2nd ‘(A +P B))))
1710, 13, 163syl 17 . . 3 ((((A P 𝐺 (2ndA)) (B P 𝐻 (2ndB))) 𝑋 Q) → ((((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) (2ndA) ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) (2ndB)) → (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)) (2nd ‘(A +P B))))
189, 17syld 40 . 2 ((((A P 𝐺 (2ndA)) (B P 𝐻 (2ndB))) 𝑋 Q) → ((𝐺 +Q 𝐻) <Q 𝑋 → (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)) (2nd ‘(A +P B))))
19 simpr 103 . . . . 5 ((((A P 𝐺 (2ndA)) (B P 𝐻 (2ndB))) 𝑋 Q) → 𝑋 Q)
20 elprnqu 6457 . . . . . . . . 9 ((⟨(1stA), (2ndA)⟩ P 𝐺 (2ndA)) → 𝐺 Q)
211, 20sylan 267 . . . . . . . 8 ((A P 𝐺 (2ndA)) → 𝐺 Q)
2221ad2antrr 457 . . . . . . 7 ((((A P 𝐺 (2ndA)) (B P 𝐻 (2ndB))) 𝑋 Q) → 𝐺 Q)
23 elprnqu 6457 . . . . . . . . 9 ((⟨(1stB), (2ndB)⟩ P 𝐻 (2ndB)) → 𝐻 Q)
245, 23sylan 267 . . . . . . . 8 ((B P 𝐻 (2ndB)) → 𝐻 Q)
2524ad2antlr 458 . . . . . . 7 ((((A P 𝐺 (2ndA)) (B P 𝐻 (2ndB))) 𝑋 Q) → 𝐻 Q)
26 addclnq 6352 . . . . . . 7 ((𝐺 Q 𝐻 Q) → (𝐺 +Q 𝐻) Q)
2722, 25, 26syl2anc 391 . . . . . 6 ((((A P 𝐺 (2ndA)) (B P 𝐻 (2ndB))) 𝑋 Q) → (𝐺 +Q 𝐻) Q)
28 recclnq 6369 . . . . . 6 ((𝐺 +Q 𝐻) Q → (*Q‘(𝐺 +Q 𝐻)) Q)
2927, 28syl 14 . . . . 5 ((((A P 𝐺 (2ndA)) (B P 𝐻 (2ndB))) 𝑋 Q) → (*Q‘(𝐺 +Q 𝐻)) Q)
30 mulassnqg 6361 . . . . 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 (2ndA)) (B P 𝐻 (2ndB))) 𝑋 Q) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q (𝐺 +Q 𝐻)) = (𝑋 ·Q ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻))))
32 mulclnq 6353 . . . . . 6 ((𝑋 Q (*Q‘(𝐺 +Q 𝐻)) Q) → (𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) Q)
3319, 29, 32syl2anc 391 . . . . 5 ((((A P 𝐺 (2ndA)) (B P 𝐻 (2ndB))) 𝑋 Q) → (𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) Q)
34 distrnqg 6364 . . . . 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 (2ndA)) (B P 𝐻 (2ndB))) 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 6360 . . . . . . . 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 (2ndA)) (B P 𝐻 (2ndB))) 𝑋 Q) → ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻)) = ((𝐺 +Q 𝐻) ·Q (*Q‘(𝐺 +Q 𝐻))))
38 recidnq 6370 . . . . . . . 8 ((𝐺 +Q 𝐻) Q → ((𝐺 +Q 𝐻) ·Q (*Q‘(𝐺 +Q 𝐻))) = 1Q)
3927, 38syl 14 . . . . . . 7 ((((A P 𝐺 (2ndA)) (B P 𝐻 (2ndB))) 𝑋 Q) → ((𝐺 +Q 𝐻) ·Q (*Q‘(𝐺 +Q 𝐻))) = 1Q)
4037, 39eqtrd 2069 . . . . . 6 ((((A P 𝐺 (2ndA)) (B P 𝐻 (2ndB))) 𝑋 Q) → ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻)) = 1Q)
4140oveq2d 5468 . . . . 5 ((((A P 𝐺 (2ndA)) (B P 𝐻 (2ndB))) 𝑋 Q) → (𝑋 ·Q ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻))) = (𝑋 ·Q 1Q))
42 mulidnq 6366 . . . . . 6 (𝑋 Q → (𝑋 ·Q 1Q) = 𝑋)
4342adantl 262 . . . . 5 ((((A P 𝐺 (2ndA)) (B P 𝐻 (2ndB))) 𝑋 Q) → (𝑋 ·Q 1Q) = 𝑋)
4441, 43eqtrd 2069 . . . 4 ((((A P 𝐺 (2ndA)) (B P 𝐻 (2ndB))) 𝑋 Q) → (𝑋 ·Q ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻))) = 𝑋)
4531, 35, 443eqtr3d 2077 . . 3 ((((A P 𝐺 (2ndA)) (B P 𝐻 (2ndB))) 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 (2ndA)) (B P 𝐻 (2ndB))) 𝑋 Q) → ((((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)) (2nd ‘(A +P B)) ↔ 𝑋 (2nd ‘(A +P B))))
4718, 46sylibd 138 1 ((((A P 𝐺 (2ndA)) (B P 𝐻 (2ndB))) 𝑋 Q) → ((𝐺 +Q 𝐻) <Q 𝑋𝑋 (2nd ‘(A +P B))))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  cop 3369   class class class wbr 3754  cfv 4844  (class class class)co 5452  1st c1st 5704  2nd c2nd 5705  Qcnq 6257  1Qc1q 6258   +Q cplq 6259   ·Q cmq 6260  *Qcrq 6261   <Q cltq 6262  Pcnp 6268   +P cpp 6270
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 3862  ax-sep 3865  ax-nul 3873  ax-pow 3917  ax-pr 3934  ax-un 4135  ax-setind 4219  ax-iinf 4253
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 3352  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-int 3606  df-iun 3649  df-br 3755  df-opab 3809  df-mpt 3810  df-tr 3845  df-eprel 4016  df-id 4020  df-iord 4068  df-on 4070  df-suc 4073  df-iom 4256  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-rn 4298  df-res 4299  df-ima 4300  df-iota 4809  df-fun 4846  df-fn 4847  df-f 4848  df-f1 4849  df-fo 4850  df-f1o 4851  df-fv 4852  df-ov 5455  df-oprab 5456  df-mpt2 5457  df-1st 5706  df-2nd 5707  df-recs 5858  df-irdg 5894  df-1o 5933  df-oadd 5937  df-omul 5938  df-er 6035  df-ec 6037  df-qs 6041  df-ni 6281  df-pli 6282  df-mi 6283  df-lti 6284  df-plpq 6321  df-mpq 6322  df-enq 6324  df-nqqs 6325  df-plqqs 6326  df-mqqs 6327  df-1nqqs 6328  df-rq 6329  df-ltnqqs 6330  df-inp 6441  df-iplp 6443
This theorem is referenced by:  addlocprlemeq  6509  addlocprlemgt  6510  addclpr  6513
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