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Theorem mulnqprl 6809
Description: Lemma to prove downward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.)
Assertion
Ref Expression
mulnqprl ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (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 6642 . . . . . . 7 ((𝑦Q𝑧Q𝑤Q) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
21adantl 271 . . . . . 6 (((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) ∧ (𝑦Q𝑧Q𝑤Q)) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
3 simpr 108 . . . . . 6 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → 𝑋Q)
4 prop 6716 . . . . . . . . 9 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
5 elprnql 6722 . . . . . . . . 9 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝐺 ∈ (1st𝐴)) → 𝐺Q)
64, 5sylan 277 . . . . . . . 8 ((𝐴P𝐺 ∈ (1st𝐴)) → 𝐺Q)
76ad2antrr 472 . . . . . . 7 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → 𝐺Q)
8 prop 6716 . . . . . . . . 9 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
9 elprnql 6722 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐻 ∈ (1st𝐵)) → 𝐻Q)
108, 9sylan 277 . . . . . . . 8 ((𝐵P𝐻 ∈ (1st𝐵)) → 𝐻Q)
1110ad2antlr 473 . . . . . . 7 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → 𝐻Q)
12 mulclnq 6617 . . . . . . 7 ((𝐺Q𝐻Q) → (𝐺 ·Q 𝐻) ∈ Q)
137, 11, 12syl2anc 403 . . . . . 6 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝐺 ·Q 𝐻) ∈ Q)
14 recclnq 6633 . . . . . . 7 (𝐻Q → (*Q𝐻) ∈ Q)
1511, 14syl 14 . . . . . 6 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (*Q𝐻) ∈ Q)
16 mulcomnqg 6624 . . . . . . 7 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
1716adantl 271 . . . . . 6 (((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) ∧ (𝑦Q𝑧Q)) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
182, 3, 13, 15, 17caovord2d 5695 . . . . 5 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 <Q (𝐺 ·Q 𝐻) ↔ (𝑋 ·Q (*Q𝐻)) <Q ((𝐺 ·Q 𝐻) ·Q (*Q𝐻))))
19 mulassnqg 6625 . . . . . . . 8 ((𝐺Q𝐻Q ∧ (*Q𝐻) ∈ Q) → ((𝐺 ·Q 𝐻) ·Q (*Q𝐻)) = (𝐺 ·Q (𝐻 ·Q (*Q𝐻))))
207, 11, 15, 19syl3anc 1170 . . . . . . 7 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((𝐺 ·Q 𝐻) ·Q (*Q𝐻)) = (𝐺 ·Q (𝐻 ·Q (*Q𝐻))))
21 recidnq 6634 . . . . . . . . 9 (𝐻Q → (𝐻 ·Q (*Q𝐻)) = 1Q)
2221oveq2d 5553 . . . . . . . 8 (𝐻Q → (𝐺 ·Q (𝐻 ·Q (*Q𝐻))) = (𝐺 ·Q 1Q))
2311, 22syl 14 . . . . . . 7 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝐺 ·Q (𝐻 ·Q (*Q𝐻))) = (𝐺 ·Q 1Q))
24 mulidnq 6630 . . . . . . . 8 (𝐺Q → (𝐺 ·Q 1Q) = 𝐺)
257, 24syl 14 . . . . . . 7 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝐺 ·Q 1Q) = 𝐺)
2620, 23, 253eqtrd 2118 . . . . . 6 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((𝐺 ·Q 𝐻) ·Q (*Q𝐻)) = 𝐺)
2726breq2d 3799 . . . . 5 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((𝑋 ·Q (*Q𝐻)) <Q ((𝐺 ·Q 𝐻) ·Q (*Q𝐻)) ↔ (𝑋 ·Q (*Q𝐻)) <Q 𝐺))
2818, 27bitrd 186 . . . 4 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 <Q (𝐺 ·Q 𝐻) ↔ (𝑋 ·Q (*Q𝐻)) <Q 𝐺))
29 prcdnql 6725 . . . . . 6 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝐺 ∈ (1st𝐴)) → ((𝑋 ·Q (*Q𝐻)) <Q 𝐺 → (𝑋 ·Q (*Q𝐻)) ∈ (1st𝐴)))
304, 29sylan 277 . . . . 5 ((𝐴P𝐺 ∈ (1st𝐴)) → ((𝑋 ·Q (*Q𝐻)) <Q 𝐺 → (𝑋 ·Q (*Q𝐻)) ∈ (1st𝐴)))
3130ad2antrr 472 . . . 4 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((𝑋 ·Q (*Q𝐻)) <Q 𝐺 → (𝑋 ·Q (*Q𝐻)) ∈ (1st𝐴)))
3228, 31sylbid 148 . . 3 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 <Q (𝐺 ·Q 𝐻) → (𝑋 ·Q (*Q𝐻)) ∈ (1st𝐴)))
33 df-imp 6710 . . . . . . . . 9 ·P = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦 ·Q 𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦 ·Q 𝑧))}⟩)
34 mulclnq 6617 . . . . . . . . 9 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) ∈ Q)
3533, 34genpprecll 6755 . . . . . . . 8 ((𝐴P𝐵P) → (((𝑋 ·Q (*Q𝐻)) ∈ (1st𝐴) ∧ 𝐻 ∈ (1st𝐵)) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (1st ‘(𝐴 ·P 𝐵))))
3635exp4b 359 . . . . . . 7 (𝐴P → (𝐵P → ((𝑋 ·Q (*Q𝐻)) ∈ (1st𝐴) → (𝐻 ∈ (1st𝐵) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (1st ‘(𝐴 ·P 𝐵))))))
3736com34 82 . . . . . 6 (𝐴P → (𝐵P → (𝐻 ∈ (1st𝐵) → ((𝑋 ·Q (*Q𝐻)) ∈ (1st𝐴) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (1st ‘(𝐴 ·P 𝐵))))))
3837imp32 253 . . . . 5 ((𝐴P ∧ (𝐵P𝐻 ∈ (1st𝐵))) → ((𝑋 ·Q (*Q𝐻)) ∈ (1st𝐴) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (1st ‘(𝐴 ·P 𝐵))))
3938adantlr 461 . . . 4 (((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) → ((𝑋 ·Q (*Q𝐻)) ∈ (1st𝐴) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (1st ‘(𝐴 ·P 𝐵))))
4039adantr 270 . . 3 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((𝑋 ·Q (*Q𝐻)) ∈ (1st𝐴) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (1st ‘(𝐴 ·P 𝐵))))
4132, 40syld 44 . 2 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 <Q (𝐺 ·Q 𝐻) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (1st ‘(𝐴 ·P 𝐵))))
42 mulassnqg 6625 . . . . 5 ((𝑋Q ∧ (*Q𝐻) ∈ Q𝐻Q) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) = (𝑋 ·Q ((*Q𝐻) ·Q 𝐻)))
433, 15, 11, 42syl3anc 1170 . . . 4 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) = (𝑋 ·Q ((*Q𝐻) ·Q 𝐻)))
44 mulcomnqg 6624 . . . . . . 7 (((*Q𝐻) ∈ Q𝐻Q) → ((*Q𝐻) ·Q 𝐻) = (𝐻 ·Q (*Q𝐻)))
4515, 11, 44syl2anc 403 . . . . . 6 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((*Q𝐻) ·Q 𝐻) = (𝐻 ·Q (*Q𝐻)))
4611, 21syl 14 . . . . . 6 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝐻 ·Q (*Q𝐻)) = 1Q)
4745, 46eqtrd 2114 . . . . 5 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((*Q𝐻) ·Q 𝐻) = 1Q)
4847oveq2d 5553 . . . 4 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 ·Q ((*Q𝐻) ·Q 𝐻)) = (𝑋 ·Q 1Q))
49 mulidnq 6630 . . . . 5 (𝑋Q → (𝑋 ·Q 1Q) = 𝑋)
5049adantl 271 . . . 4 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 ·Q 1Q) = 𝑋)
5143, 48, 503eqtrd 2118 . . 3 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) = 𝑋)
5251eleq1d 2148 . 2 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ 𝑋 ∈ (1st ‘(𝐴 ·P 𝐵))))
5341, 52sylibd 147 1 ((((𝐴P𝐺 ∈ (1st𝐴)) ∧ (𝐵P𝐻 ∈ (1st𝐵))) ∧ 𝑋Q) → (𝑋 <Q (𝐺 ·Q 𝐻) → 𝑋 ∈ (1st ‘(𝐴 ·P 𝐵))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 920   = wceq 1285  wcel 1434  cop 3403   class class class wbr 3787  cfv 4926  (class class class)co 5537  1st c1st 5790  2nd c2nd 5791  Qcnq 6521  1Qc1q 6522   ·Q cmq 6524  *Qcrq 6525   <Q cltq 6526  Pcnp 6532   ·P cmp 6535
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3895  ax-sep 3898  ax-nul 3906  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-iinf 4331
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-iun 3682  df-br 3788  df-opab 3842  df-mpt 3843  df-tr 3878  df-eprel 4046  df-id 4050  df-iord 4123  df-on 4125  df-suc 4128  df-iom 4334  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933  df-fv 4934  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-irdg 6013  df-1o 6059  df-oadd 6063  df-omul 6064  df-er 6165  df-ec 6167  df-qs 6171  df-ni 6545  df-mi 6547  df-lti 6548  df-mpq 6586  df-enq 6588  df-nqqs 6589  df-mqqs 6591  df-1nqqs 6592  df-rq 6593  df-ltnqqs 6594  df-inp 6707  df-imp 6710
This theorem is referenced by:  mullocprlem  6811  mulclpr  6813
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