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Theorem mulnqprl 6539
Description: Lemma to prove downward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.)
Assertion
Ref Expression
mulnqprl ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝑋 <Q (𝐺 ·Q 𝐻) → 𝑋 (1st ‘(A ·P B))))

Proof of Theorem mulnqprl
Dummy variables v w x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltmnqg 6378 . . . . . . 7 ((y Q z Q w Q) → (y <Q z ↔ (w ·Q y) <Q (w ·Q z)))
21adantl 262 . . . . . 6 (((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) (y Q z Q w Q)) → (y <Q z ↔ (w ·Q y) <Q (w ·Q z)))
3 simpr 103 . . . . . 6 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → 𝑋 Q)
4 prop 6450 . . . . . . . . 9 (A P → ⟨(1stA), (2ndA)⟩ P)
5 elprnql 6456 . . . . . . . . 9 ((⟨(1stA), (2ndA)⟩ P 𝐺 (1stA)) → 𝐺 Q)
64, 5sylan 267 . . . . . . . 8 ((A P 𝐺 (1stA)) → 𝐺 Q)
76ad2antrr 457 . . . . . . 7 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → 𝐺 Q)
8 prop 6450 . . . . . . . . 9 (B P → ⟨(1stB), (2ndB)⟩ P)
9 elprnql 6456 . . . . . . . . 9 ((⟨(1stB), (2ndB)⟩ P 𝐻 (1stB)) → 𝐻 Q)
108, 9sylan 267 . . . . . . . 8 ((B P 𝐻 (1stB)) → 𝐻 Q)
1110ad2antlr 458 . . . . . . 7 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → 𝐻 Q)
12 mulclnq 6353 . . . . . . 7 ((𝐺 Q 𝐻 Q) → (𝐺 ·Q 𝐻) Q)
137, 11, 12syl2anc 391 . . . . . 6 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝐺 ·Q 𝐻) Q)
14 recclnq 6369 . . . . . . 7 (𝐻 Q → (*Q𝐻) Q)
1511, 14syl 14 . . . . . 6 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (*Q𝐻) Q)
16 mulcomnqg 6360 . . . . . . 7 ((y Q z Q) → (y ·Q z) = (z ·Q y))
1716adantl 262 . . . . . 6 (((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) (y Q z Q)) → (y ·Q z) = (z ·Q y))
182, 3, 13, 15, 17caovord2d 5609 . . . . 5 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝑋 <Q (𝐺 ·Q 𝐻) ↔ (𝑋 ·Q (*Q𝐻)) <Q ((𝐺 ·Q 𝐻) ·Q (*Q𝐻))))
19 mulassnqg 6361 . . . . . . . 8 ((𝐺 Q 𝐻 Q (*Q𝐻) Q) → ((𝐺 ·Q 𝐻) ·Q (*Q𝐻)) = (𝐺 ·Q (𝐻 ·Q (*Q𝐻))))
207, 11, 15, 19syl3anc 1134 . . . . . . 7 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → ((𝐺 ·Q 𝐻) ·Q (*Q𝐻)) = (𝐺 ·Q (𝐻 ·Q (*Q𝐻))))
21 recidnq 6370 . . . . . . . . 9 (𝐻 Q → (𝐻 ·Q (*Q𝐻)) = 1Q)
2221oveq2d 5468 . . . . . . . 8 (𝐻 Q → (𝐺 ·Q (𝐻 ·Q (*Q𝐻))) = (𝐺 ·Q 1Q))
2311, 22syl 14 . . . . . . 7 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝐺 ·Q (𝐻 ·Q (*Q𝐻))) = (𝐺 ·Q 1Q))
24 mulidnq 6366 . . . . . . . 8 (𝐺 Q → (𝐺 ·Q 1Q) = 𝐺)
257, 24syl 14 . . . . . . 7 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝐺 ·Q 1Q) = 𝐺)
2620, 23, 253eqtrd 2073 . . . . . 6 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → ((𝐺 ·Q 𝐻) ·Q (*Q𝐻)) = 𝐺)
2726breq2d 3766 . . . . 5 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → ((𝑋 ·Q (*Q𝐻)) <Q ((𝐺 ·Q 𝐻) ·Q (*Q𝐻)) ↔ (𝑋 ·Q (*Q𝐻)) <Q 𝐺))
2818, 27bitrd 177 . . . 4 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝑋 <Q (𝐺 ·Q 𝐻) ↔ (𝑋 ·Q (*Q𝐻)) <Q 𝐺))
29 prcdnql 6459 . . . . . 6 ((⟨(1stA), (2ndA)⟩ P 𝐺 (1stA)) → ((𝑋 ·Q (*Q𝐻)) <Q 𝐺 → (𝑋 ·Q (*Q𝐻)) (1stA)))
304, 29sylan 267 . . . . 5 ((A P 𝐺 (1stA)) → ((𝑋 ·Q (*Q𝐻)) <Q 𝐺 → (𝑋 ·Q (*Q𝐻)) (1stA)))
3130ad2antrr 457 . . . 4 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → ((𝑋 ·Q (*Q𝐻)) <Q 𝐺 → (𝑋 ·Q (*Q𝐻)) (1stA)))
3228, 31sylbid 139 . . 3 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝑋 <Q (𝐺 ·Q 𝐻) → (𝑋 ·Q (*Q𝐻)) (1stA)))
33 df-imp 6444 . . . . . . . . 9 ·P = (w P, v P ↦ ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y ·Q z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y ·Q z))}⟩)
34 mulclnq 6353 . . . . . . . . 9 ((y Q z Q) → (y ·Q z) Q)
3533, 34genpprecll 6489 . . . . . . . 8 ((A P B P) → (((𝑋 ·Q (*Q𝐻)) (1stA) 𝐻 (1stB)) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) (1st ‘(A ·P B))))
3635exp4b 349 . . . . . . 7 (A P → (B P → ((𝑋 ·Q (*Q𝐻)) (1stA) → (𝐻 (1stB) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) (1st ‘(A ·P B))))))
3736com34 77 . . . . . 6 (A P → (B P → (𝐻 (1stB) → ((𝑋 ·Q (*Q𝐻)) (1stA) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) (1st ‘(A ·P B))))))
3837imp32 244 . . . . 5 ((A P (B P 𝐻 (1stB))) → ((𝑋 ·Q (*Q𝐻)) (1stA) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) (1st ‘(A ·P B))))
3938adantlr 446 . . . 4 (((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) → ((𝑋 ·Q (*Q𝐻)) (1stA) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) (1st ‘(A ·P B))))
4039adantr 261 . . 3 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → ((𝑋 ·Q (*Q𝐻)) (1stA) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) (1st ‘(A ·P B))))
4132, 40syld 40 . 2 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝑋 <Q (𝐺 ·Q 𝐻) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) (1st ‘(A ·P B))))
42 mulassnqg 6361 . . . . 5 ((𝑋 Q (*Q𝐻) Q 𝐻 Q) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) = (𝑋 ·Q ((*Q𝐻) ·Q 𝐻)))
433, 15, 11, 42syl3anc 1134 . . . 4 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) = (𝑋 ·Q ((*Q𝐻) ·Q 𝐻)))
44 mulcomnqg 6360 . . . . . . 7 (((*Q𝐻) Q 𝐻 Q) → ((*Q𝐻) ·Q 𝐻) = (𝐻 ·Q (*Q𝐻)))
4515, 11, 44syl2anc 391 . . . . . 6 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → ((*Q𝐻) ·Q 𝐻) = (𝐻 ·Q (*Q𝐻)))
4611, 21syl 14 . . . . . 6 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝐻 ·Q (*Q𝐻)) = 1Q)
4745, 46eqtrd 2069 . . . . 5 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → ((*Q𝐻) ·Q 𝐻) = 1Q)
4847oveq2d 5468 . . . 4 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝑋 ·Q ((*Q𝐻) ·Q 𝐻)) = (𝑋 ·Q 1Q))
49 mulidnq 6366 . . . . 5 (𝑋 Q → (𝑋 ·Q 1Q) = 𝑋)
5049adantl 262 . . . 4 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝑋 ·Q 1Q) = 𝑋)
5143, 48, 503eqtrd 2073 . . 3 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) = 𝑋)
5251eleq1d 2103 . 2 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) (1st ‘(A ·P B)) ↔ 𝑋 (1st ‘(A ·P B))))
5341, 52sylibd 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  wb 98   w3a 884   = 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 cmq 6260  *Qcrq 6261   <Q cltq 6262  Pcnp 6268   ·P cmp 6271
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-mi 6283  df-lti 6284  df-mpq 6322  df-enq 6324  df-nqqs 6325  df-mqqs 6327  df-1nqqs 6328  df-rq 6329  df-ltnqqs 6330  df-inp 6441  df-imp 6444
This theorem is referenced by:  mullocprlem  6541  mulclpr  6543
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