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Theorem mulnqprl 6549
 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 6385 . . . . . . 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 6458 . . . . . . . . 9 (A P → ⟨(1stA), (2ndA)⟩ P)
5 elprnql 6464 . . . . . . . . 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 6458 . . . . . . . . 9 (B P → ⟨(1stB), (2ndB)⟩ P)
9 elprnql 6464 . . . . . . . . 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 6360 . . . . . . 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 6376 . . . . . . 7 (𝐻 Q → (*Q𝐻) Q)
1511, 14syl 14 . . . . . 6 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (*Q𝐻) Q)
16 mulcomnqg 6367 . . . . . . 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 5612 . . . . 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 6368 . . . . . . . 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 6377 . . . . . . . . 9 (𝐻 Q → (𝐻 ·Q (*Q𝐻)) = 1Q)
2221oveq2d 5471 . . . . . . . 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 6373 . . . . . . . 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 3767 . . . . 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 6467 . . . . . 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 6452 . . . . . . . . 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 6360 . . . . . . . . 9 ((y Q z Q) → (y ·Q z) Q)
3533, 34genpprecll 6497 . . . . . . . 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 6368 . . . . 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 6367 . . . . . . 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 5471 . . . 4 ((((A P 𝐺 (1stA)) (B P 𝐻 (1stB))) 𝑋 Q) → (𝑋 ·Q ((*Q𝐻) ·Q 𝐻)) = (𝑋 ·Q 1Q))
49 mulidnq 6373 . . . . 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 3370   class class class wbr 3755  ‘cfv 4845  (class class class)co 5455  1st c1st 5707  2nd c2nd 5708  Qcnq 6264  1Qc1q 6265   ·Q cmq 6267  *Qcrq 6268
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