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Theorem mulnqpru 7401
 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 7233 . . . . . . 7 ((𝑦Q𝑧Q𝑤Q) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
21adantl 275 . . . . . 6 (((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) ∧ (𝑦Q𝑧Q𝑤Q)) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
3 prop 7307 . . . . . . . . 9 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
4 elprnqu 7314 . . . . . . . . 9 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝐺 ∈ (2nd𝐴)) → 𝐺Q)
53, 4sylan 281 . . . . . . . 8 ((𝐴P𝐺 ∈ (2nd𝐴)) → 𝐺Q)
65ad2antrr 480 . . . . . . 7 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → 𝐺Q)
7 prop 7307 . . . . . . . . 9 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
8 elprnqu 7314 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐻 ∈ (2nd𝐵)) → 𝐻Q)
97, 8sylan 281 . . . . . . . 8 ((𝐵P𝐻 ∈ (2nd𝐵)) → 𝐻Q)
109ad2antlr 481 . . . . . . 7 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → 𝐻Q)
11 mulclnq 7208 . . . . . . 7 ((𝐺Q𝐻Q) → (𝐺 ·Q 𝐻) ∈ Q)
126, 10, 11syl2anc 409 . . . . . 6 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → (𝐺 ·Q 𝐻) ∈ Q)
13 simpr 109 . . . . . 6 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → 𝑋Q)
14 recclnq 7224 . . . . . . 7 (𝐻Q → (*Q𝐻) ∈ Q)
1510, 14syl 14 . . . . . 6 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → (*Q𝐻) ∈ Q)
16 mulcomnqg 7215 . . . . . . 7 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
1716adantl 275 . . . . . 6 (((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) ∧ (𝑦Q𝑧Q)) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
182, 12, 13, 15, 17caovord2d 5948 . . . . 5 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝐺 ·Q 𝐻) <Q 𝑋 ↔ ((𝐺 ·Q 𝐻) ·Q (*Q𝐻)) <Q (𝑋 ·Q (*Q𝐻))))
19 mulassnqg 7216 . . . . . . . 8 ((𝐺Q𝐻Q ∧ (*Q𝐻) ∈ Q) → ((𝐺 ·Q 𝐻) ·Q (*Q𝐻)) = (𝐺 ·Q (𝐻 ·Q (*Q𝐻))))
206, 10, 15, 19syl3anc 1217 . . . . . . 7 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝐺 ·Q 𝐻) ·Q (*Q𝐻)) = (𝐺 ·Q (𝐻 ·Q (*Q𝐻))))
21 recidnq 7225 . . . . . . . . 9 (𝐻Q → (𝐻 ·Q (*Q𝐻)) = 1Q)
2221oveq2d 5798 . . . . . . . 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 7221 . . . . . . . 8 (𝐺Q → (𝐺 ·Q 1Q) = 𝐺)
256, 24syl 14 . . . . . . 7 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → (𝐺 ·Q 1Q) = 𝐺)
2620, 23, 253eqtrd 2177 . . . . . 6 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝐺 ·Q 𝐻) ·Q (*Q𝐻)) = 𝐺)
2726breq1d 3947 . . . . 5 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → (((𝐺 ·Q 𝐻) ·Q (*Q𝐻)) <Q (𝑋 ·Q (*Q𝐻)) ↔ 𝐺 <Q (𝑋 ·Q (*Q𝐻))))
2818, 27bitrd 187 . . . 4 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝐺 ·Q 𝐻) <Q 𝑋𝐺 <Q (𝑋 ·Q (*Q𝐻))))
29 prcunqu 7317 . . . . . 6 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝐺 ∈ (2nd𝐴)) → (𝐺 <Q (𝑋 ·Q (*Q𝐻)) → (𝑋 ·Q (*Q𝐻)) ∈ (2nd𝐴)))
303, 29sylan 281 . . . . 5 ((𝐴P𝐺 ∈ (2nd𝐴)) → (𝐺 <Q (𝑋 ·Q (*Q𝐻)) → (𝑋 ·Q (*Q𝐻)) ∈ (2nd𝐴)))
3130ad2antrr 480 . . . 4 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → (𝐺 <Q (𝑋 ·Q (*Q𝐻)) → (𝑋 ·Q (*Q𝐻)) ∈ (2nd𝐴)))
3228, 31sylbid 149 . . 3 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝐺 ·Q 𝐻) <Q 𝑋 → (𝑋 ·Q (*Q𝐻)) ∈ (2nd𝐴)))
33 df-imp 7301 . . . . . . . . 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 7208 . . . . . . . . 9 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) ∈ Q)
3533, 34genppreclu 7347 . . . . . . . 8 ((𝐴P𝐵P) → (((𝑋 ·Q (*Q𝐻)) ∈ (2nd𝐴) ∧ 𝐻 ∈ (2nd𝐵)) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (2nd ‘(𝐴 ·P 𝐵))))
3635exp4b 365 . . . . . . 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 255 . . . . 5 ((𝐴P ∧ (𝐵P𝐻 ∈ (2nd𝐵))) → ((𝑋 ·Q (*Q𝐻)) ∈ (2nd𝐴) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (2nd ‘(𝐴 ·P 𝐵))))
3938adantlr 469 . . . 4 (((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) → ((𝑋 ·Q (*Q𝐻)) ∈ (2nd𝐴) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (2nd ‘(𝐴 ·P 𝐵))))
4039adantr 274 . . 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 7216 . . . . 5 ((𝑋Q ∧ (*Q𝐻) ∈ Q𝐻Q) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) = (𝑋 ·Q ((*Q𝐻) ·Q 𝐻)))
4313, 15, 10, 42syl3anc 1217 . . . 4 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) = (𝑋 ·Q ((*Q𝐻) ·Q 𝐻)))
44 mulcomnqg 7215 . . . . . . 7 (((*Q𝐻) ∈ Q𝐻Q) → ((*Q𝐻) ·Q 𝐻) = (𝐻 ·Q (*Q𝐻)))
4515, 10, 44syl2anc 409 . . . . . 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 2173 . . . . 5 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((*Q𝐻) ·Q 𝐻) = 1Q)
4847oveq2d 5798 . . . 4 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → (𝑋 ·Q ((*Q𝐻) ·Q 𝐻)) = (𝑋 ·Q 1Q))
49 mulidnq 7221 . . . . 5 (𝑋Q → (𝑋 ·Q 1Q) = 𝑋)
5049adantl 275 . . . 4 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → (𝑋 ·Q 1Q) = 𝑋)
5143, 48, 503eqtrd 2177 . . 3 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → ((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) = 𝑋)
5251eleq1d 2209 . 2 ((((𝐴P𝐺 ∈ (2nd𝐴)) ∧ (𝐵P𝐻 ∈ (2nd𝐵))) ∧ 𝑋Q) → (((𝑋 ·Q (*Q𝐻)) ·Q 𝐻) ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ 𝑋 ∈ (2nd ‘(𝐴 ·P 𝐵))))
5341, 52sylibd 148 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 103   ↔ wb 104   ∧ w3a 963   = wceq 1332   ∈ wcel 1481  ⟨cop 3535   class class class wbr 3937  ‘cfv 5131  (class class class)co 5782  1st c1st 6044  2nd c2nd 6045  Qcnq 7112  1Qc1q 7113   ·Q cmq 7115  *Qcrq 7116
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