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Theorem mulnqpru 7586
Description: Lemma to prove upward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.)
Assertion
Ref Expression
mulnqpru ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝐺 ·Q 𝐻) <Q 𝑋𝑋 ∈ (2nd ‘(𝐴 ·P 𝐵))))

Proof of Theorem mulnqpru
Dummy variables 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltmnqg 7418 . . . . . . 7 ((𝑦Q𝑧Q𝑤Q) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
21adantl 277 . . . . . 6 (((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) ∧ (𝑦Q𝑧Q𝑤Q)) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
3 prop 7492 . . . . . . . . 9 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
4 elprnqu 7499 . . . . . . . . 9 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝐺 ∈ (2nd𝐴)) → 𝐺Q)
53, 4sylan 283 . . . . . . . 8 ((𝐴P𝐺 ∈ (2nd𝐴)) → 𝐺Q)
65ad2antrr 488 . . . . . . 7 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → 𝐺Q)
7 prop 7492 . . . . . . . . 9 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
8 elprnqu 7499 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐻 ∈ (2nd𝐵)) → 𝐻Q)
97, 8sylan 283 . . . . . . . 8 ((𝐵P𝐻 ∈ (2nd𝐵)) → 𝐻Q)
109ad2antlr 489 . . . . . . 7 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → 𝐻Q)
11 mulclnq 7393 . . . . . . 7 ((𝐺Q𝐻Q) → (𝐺 ·Q 𝐻) ∈ Q)
126, 10, 11syl2anc 411 . . . . . 6 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → (𝐺 ·Q 𝐻) ∈ Q)
13 simpr 110 . . . . . 6 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → 𝑋Q)
14 recclnq 7409 . . . . . . 7 (𝐻Q → (*Q𝐻) ∈ Q)
1510, 14syl 14 . . . . . 6 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → (*Q𝐻) ∈ Q)
16 mulcomnqg 7400 . . . . . . 7 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
1716adantl 277 . . . . . 6 (((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) ∧ (𝑦Q𝑧Q)) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
182, 12, 13, 15, 17caovord2d 6061 . . . . 5 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝐺 ·Q 𝐻) <Q 𝑋 ↔ ((𝐺 ·Q 𝐻) ·Q (*Q𝐻)) <Q (𝑋 ·Q (*Q𝐻))))
19 mulassnqg 7401 . . . . . . . 8 ((𝐺Q𝐻Q ∧ (*Q𝐻) ∈ Q) → ((𝐺 ·Q 𝐻) ·Q (*Q𝐻)) = (𝐺 ·Q (𝐻 ·Q (*Q𝐻))))
206, 10, 15, 19syl3anc 1249 . . . . . . 7 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝐺 ·Q 𝐻) ·Q (*Q𝐻)) = (𝐺 ·Q (𝐻 ·Q (*Q𝐻))))
21 recidnq 7410 . . . . . . . . 9 (𝐻Q → (𝐻 ·Q (*Q𝐻)) = 1Q)
2221oveq2d 5907 . . . . . . . 8 (𝐻Q → (𝐺 ·Q (𝐻 ·Q (*Q𝐻))) = (𝐺 ·Q 1Q))
2310, 22syl 14 . . . . . . 7 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → (𝐺 ·Q (𝐻 ·Q (*Q𝐻))) = (𝐺 ·Q 1Q))
24 mulidnq 7406 . . . . . . . 8 (𝐺Q → (𝐺 ·Q 1Q) = 𝐺)
256, 24syl 14 . . . . . . 7 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → (𝐺 ·Q 1Q) = 𝐺)
2620, 23, 253eqtrd 2226 . . . . . 6 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝐺 ·Q 𝐻) ·Q (*Q𝐻)) = 𝐺)
2726breq1d 4028 . . . . 5 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → (((𝐺 ·Q 𝐻) ·Q (*Q𝐻)) <Q (𝑋 ·Q (*Q𝐻)) ↔ 𝐺 <Q (𝑋 ·Q (*Q𝐻))))
2818, 27bitrd 188 . . . 4 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝐺 ·Q 𝐻) <Q 𝑋𝐺 <Q (𝑋 ·Q (*Q𝐻))))
29 prcunqu 7502 . . . . . 6 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝐺 ∈ (2nd𝐴)) → (𝐺 <Q (𝑋 ·Q (*Q𝐻)) → (𝑋 ·Q (*Q𝐻)) ∈ (2nd𝐴)))
303, 29sylan 283 . . . . 5 ((𝐴P𝐺 ∈ (2nd𝐴)) → (𝐺 <Q (𝑋 ·Q (*Q𝐻)) → (𝑋 ·Q (*Q𝐻)) ∈ (2nd𝐴)))
3130ad2antrr 488 . . . 4 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → (𝐺 <Q (𝑋 ·Q (*Q𝐻)) → (𝑋 ·Q (*Q𝐻)) ∈ (2nd𝐴)))
3228, 31sylbid 150 . . 3 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝐺 ·Q 𝐻) <Q 𝑋 → (𝑋 ·Q (*Q𝐻)) ∈ (2nd𝐴)))
33 df-imp 7486 . . . . . . . . 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 7393 . . . . . . . . 9 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) ∈ Q)
3533, 34genppreclu 7532 . . . . . . . 8 ((𝐴P𝐵P) → (((𝑋 ·Q (*Q𝐻)) ∈ (2nd𝐴) ∧ 𝐻 ∈ (2nd𝐵)) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (2nd ‘(𝐴 ·P 𝐵))))
3635exp4b 367 . . . . . . 7 (𝐴P → (𝐵P → ((𝑋 ·Q (*Q𝐻)) ∈ (2nd𝐴) → (𝐻 ∈ (2nd𝐵) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (2nd ‘(𝐴 ·P 𝐵))))))
3736com34 83 . . . . . 6 (𝐴P → (𝐵P → (𝐻 ∈ (2nd𝐵) → ((𝑋 ·Q (*Q𝐻)) ∈ (2nd𝐴) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (2nd ‘(𝐴 ·P 𝐵))))))
3837imp32 257 . . . . 5 ((𝐴P ∧ (𝐵P𝐻 ∈ (2nd𝐵))) → ((𝑋 ·Q (*Q𝐻)) ∈ (2nd𝐴) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (2nd ‘(𝐴 ·P 𝐵))))
3938adantlr 477 . . . 4 (((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) → ((𝑋 ·Q (*Q𝐻)) ∈ (2nd𝐴) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (2nd ‘(𝐴 ·P 𝐵))))
4039adantr 276 . . 3 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝑋 ·Q (*Q𝐻)) ∈ (2nd𝐴) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (2nd ‘(𝐴 ·P 𝐵))))
4132, 40syld 45 . 2 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝐺 ·Q 𝐻) <Q 𝑋 → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (2nd ‘(𝐴 ·P 𝐵))))
42 mulassnqg 7401 . . . . 5 ((𝑋Q ∧ (*Q𝐻) ∈ Q𝐻Q) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) = (𝑋 ·Q ((*Q𝐻) ·Q 𝐻)))
4313, 15, 10, 42syl3anc 1249 . . . 4 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) = (𝑋 ·Q ((*Q𝐻) ·Q 𝐻)))
44 mulcomnqg 7400 . . . . . . 7 (((*Q𝐻) ∈ Q𝐻Q) → ((*Q𝐻) ·Q 𝐻) = (𝐻 ·Q (*Q𝐻)))
4515, 10, 44syl2anc 411 . . . . . 6 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((*Q𝐻) ·Q 𝐻) = (𝐻 ·Q (*Q𝐻)))
4610, 21syl 14 . . . . . 6 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → (𝐻 ·Q (*Q𝐻)) = 1Q)
4745, 46eqtrd 2222 . . . . 5 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((*Q𝐻) ·Q 𝐻) = 1Q)
4847oveq2d 5907 . . . 4 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → (𝑋 ·Q ((*Q𝐻) ·Q 𝐻)) = (𝑋 ·Q 1Q))
49 mulidnq 7406 . . . . 5 (𝑋Q → (𝑋 ·Q 1Q) = 𝑋)
5049adantl 277 . . . 4 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → (𝑋 ·Q 1Q) = 𝑋)
5143, 48, 503eqtrd 2226 . . 3 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) = 𝑋)
5251eleq1d 2258 . 2 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → (((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ 𝑋 ∈ (2nd ‘(𝐴 ·P 𝐵))))
5341, 52sylibd 149 1 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝐺 ·Q 𝐻) <Q 𝑋𝑋 ∈ (2nd ‘(𝐴 ·P 𝐵))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 980   = wceq 1364  wcel 2160  cop 3610   class class class wbr 4018  cfv 5231  (class class class)co 5891  1st c1st 6157  2nd c2nd 6158  Qcnq 7297  1Qc1q 7298   ·Q cmq 7300  *Qcrq 7301   <Q cltq 7302  Pcnp 7308   ·P cmp 7311
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4304  df-id 4308  df-iord 4381  df-on 4383  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-irdg 6389  df-1o 6435  df-oadd 6439  df-omul 6440  df-er 6553  df-ec 6555  df-qs 6559  df-ni 7321  df-mi 7323  df-lti 7324  df-mpq 7362  df-enq 7364  df-nqqs 7365  df-mqqs 7367  df-1nqqs 7368  df-rq 7369  df-ltnqqs 7370  df-inp 7483  df-imp 7486
This theorem is referenced by:  mullocprlem  7587  mulclpr  7589
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