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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
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