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Theorem genprndl 6376
 Description: The lower cut produced by addition or multiplication on positive reals is rounded. (Contributed by Jim Kingdon, 7-Oct-2019.)
Hypotheses
Ref Expression
genpelvl.1 𝐹 = (w P, v P ↦ ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))}⟩)
genpelvl.2 ((y Q z Q) → (y𝐺z) Q)
genprndl.ord ((x Q y Q z Q) → (x <Q y ↔ (z𝐺x) <Q (z𝐺y)))
genprndl.com ((x Q y Q) → (x𝐺y) = (y𝐺x))
genprndl.lower ((((A P g (1stA)) (B P (1stB))) x Q) → (x <Q (g𝐺) → x (1st ‘(A𝐹B))))
Assertion
Ref Expression
genprndl ((A P B P) → 𝑞 Q (𝑞 (1st ‘(A𝐹B)) ↔ 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B)))))
Distinct variable groups:   x,y,z,g,,w,v,𝑞,A   x,B,y,z,g,,w,v,𝑞   x,𝐺,y,z,g,,w,v,𝑞   g,𝐹,𝑞   A,𝑟,𝑞,v,w,x,y,z   B,𝑟,g,   ,𝐹,𝑟,v,w,x,y,z   𝐺,𝑟

Proof of Theorem genprndl
Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 genpelvl.1 . . . . . . . . . 10 𝐹 = (w P, v P ↦ ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))}⟩)
2 genpelvl.2 . . . . . . . . . 10 ((y Q z Q) → (y𝐺z) Q)
31, 2genpelvl 6366 . . . . . . . . 9 ((A P B P) → (𝑞 (1st ‘(A𝐹B)) ↔ 𝑎 (1stA)𝑏 (1stB)𝑞 = (𝑎𝐺𝑏)))
4 r2ex 2322 . . . . . . . . 9 (𝑎 (1stA)𝑏 (1stB)𝑞 = (𝑎𝐺𝑏) ↔ 𝑎𝑏((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏)))
53, 4syl6bb 185 . . . . . . . 8 ((A P B P) → (𝑞 (1st ‘(A𝐹B)) ↔ 𝑎𝑏((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏))))
65biimpa 280 . . . . . . 7 (((A P B P) 𝑞 (1st ‘(A𝐹B))) → 𝑎𝑏((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏)))
76adantrl 450 . . . . . 6 (((A P B P) (𝑞 Q 𝑞 (1st ‘(A𝐹B)))) → 𝑎𝑏((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏)))
8 prop 6329 . . . . . . . . . . . . . . . 16 (A P → ⟨(1stA), (2ndA)⟩ P)
9 prnmaxl 6342 . . . . . . . . . . . . . . . 16 ((⟨(1stA), (2ndA)⟩ P 𝑎 (1stA)) → 𝑐 (1stA)𝑎 <Q 𝑐)
108, 9sylan 267 . . . . . . . . . . . . . . 15 ((A P 𝑎 (1stA)) → 𝑐 (1stA)𝑎 <Q 𝑐)
11 prop 6329 . . . . . . . . . . . . . . . 16 (B P → ⟨(1stB), (2ndB)⟩ P)
12 prnmaxl 6342 . . . . . . . . . . . . . . . 16 ((⟨(1stB), (2ndB)⟩ P 𝑏 (1stB)) → 𝑑 (1stB)𝑏 <Q 𝑑)
1311, 12sylan 267 . . . . . . . . . . . . . . 15 ((B P 𝑏 (1stB)) → 𝑑 (1stB)𝑏 <Q 𝑑)
1410, 13anim12i 321 . . . . . . . . . . . . . 14 (((A P 𝑎 (1stA)) (B P 𝑏 (1stB))) → (𝑐 (1stA)𝑎 <Q 𝑐 𝑑 (1stB)𝑏 <Q 𝑑))
1514an4s 509 . . . . . . . . . . . . 13 (((A P B P) (𝑎 (1stA) 𝑏 (1stB))) → (𝑐 (1stA)𝑎 <Q 𝑐 𝑑 (1stB)𝑏 <Q 𝑑))
16 reeanv 2457 . . . . . . . . . . . . 13 (𝑐 (1stA)𝑑 (1stB)(𝑎 <Q 𝑐 𝑏 <Q 𝑑) ↔ (𝑐 (1stA)𝑎 <Q 𝑐 𝑑 (1stB)𝑏 <Q 𝑑))
1715, 16sylibr 137 . . . . . . . . . . . 12 (((A P B P) (𝑎 (1stA) 𝑏 (1stB))) → 𝑐 (1stA)𝑑 (1stB)(𝑎 <Q 𝑐 𝑏 <Q 𝑑))
18 genprndl.ord . . . . . . . . . . . . . . 15 ((x Q y Q z Q) → (x <Q y ↔ (z𝐺x) <Q (z𝐺y)))
19 genprndl.com . . . . . . . . . . . . . . 15 ((x Q y Q) → (x𝐺y) = (y𝐺x))
2018, 19genplt2i 6364 . . . . . . . . . . . . . 14 ((𝑎 <Q 𝑐 𝑏 <Q 𝑑) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
2120reximi 2394 . . . . . . . . . . . . 13 (𝑑 (1stB)(𝑎 <Q 𝑐 𝑏 <Q 𝑑) → 𝑑 (1stB)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
2221reximi 2394 . . . . . . . . . . . 12 (𝑐 (1stA)𝑑 (1stB)(𝑎 <Q 𝑐 𝑏 <Q 𝑑) → 𝑐 (1stA)𝑑 (1stB)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
2317, 22syl 14 . . . . . . . . . . 11 (((A P B P) (𝑎 (1stA) 𝑏 (1stB))) → 𝑐 (1stA)𝑑 (1stB)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
2423adantrr 451 . . . . . . . . . 10 (((A P B P) ((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏))) → 𝑐 (1stA)𝑑 (1stB)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
25 breq1 3741 . . . . . . . . . . . . . 14 (𝑞 = (𝑎𝐺𝑏) → (𝑞 <Q (𝑐𝐺𝑑) ↔ (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑)))
2625biimprd 147 . . . . . . . . . . . . 13 (𝑞 = (𝑎𝐺𝑏) → ((𝑎𝐺𝑏) <Q (𝑐𝐺𝑑) → 𝑞 <Q (𝑐𝐺𝑑)))
2726reximdv 2398 . . . . . . . . . . . 12 (𝑞 = (𝑎𝐺𝑏) → (𝑑 (1stB)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑) → 𝑑 (1stB)𝑞 <Q (𝑐𝐺𝑑)))
2827reximdv 2398 . . . . . . . . . . 11 (𝑞 = (𝑎𝐺𝑏) → (𝑐 (1stA)𝑑 (1stB)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑) → 𝑐 (1stA)𝑑 (1stB)𝑞 <Q (𝑐𝐺𝑑)))
2928ad2antll 464 . . . . . . . . . 10 (((A P B P) ((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏))) → (𝑐 (1stA)𝑑 (1stB)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑) → 𝑐 (1stA)𝑑 (1stB)𝑞 <Q (𝑐𝐺𝑑)))
3024, 29mpd 13 . . . . . . . . 9 (((A P B P) ((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏))) → 𝑐 (1stA)𝑑 (1stB)𝑞 <Q (𝑐𝐺𝑑))
3130ex 108 . . . . . . . 8 ((A P B P) → (((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏)) → 𝑐 (1stA)𝑑 (1stB)𝑞 <Q (𝑐𝐺𝑑)))
3231exlimdvv 1759 . . . . . . 7 ((A P B P) → (𝑎𝑏((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏)) → 𝑐 (1stA)𝑑 (1stB)𝑞 <Q (𝑐𝐺𝑑)))
3332adantr 261 . . . . . 6 (((A P B P) (𝑞 Q 𝑞 (1st ‘(A𝐹B)))) → (𝑎𝑏((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏)) → 𝑐 (1stA)𝑑 (1stB)𝑞 <Q (𝑐𝐺𝑑)))
347, 33mpd 13 . . . . 5 (((A P B P) (𝑞 Q 𝑞 (1st ‘(A𝐹B)))) → 𝑐 (1stA)𝑑 (1stB)𝑞 <Q (𝑐𝐺𝑑))
351, 2genpprecll 6368 . . . . . . . . 9 ((A P B P) → ((𝑐 (1stA) 𝑑 (1stB)) → (𝑐𝐺𝑑) (1st ‘(A𝐹B))))
3635imp 115 . . . . . . . 8 (((A P B P) (𝑐 (1stA) 𝑑 (1stB))) → (𝑐𝐺𝑑) (1st ‘(A𝐹B)))
37 elprnql 6335 . . . . . . . . . . . . 13 ((⟨(1stA), (2ndA)⟩ P 𝑐 (1stA)) → 𝑐 Q)
388, 37sylan 267 . . . . . . . . . . . 12 ((A P 𝑐 (1stA)) → 𝑐 Q)
39 elprnql 6335 . . . . . . . . . . . . 13 ((⟨(1stB), (2ndB)⟩ P 𝑑 (1stB)) → 𝑑 Q)
4011, 39sylan 267 . . . . . . . . . . . 12 ((B P 𝑑 (1stB)) → 𝑑 Q)
4138, 40anim12i 321 . . . . . . . . . . 11 (((A P 𝑐 (1stA)) (B P 𝑑 (1stB))) → (𝑐 Q 𝑑 Q))
4241an4s 509 . . . . . . . . . 10 (((A P B P) (𝑐 (1stA) 𝑑 (1stB))) → (𝑐 Q 𝑑 Q))
432caovcl 5578 . . . . . . . . . 10 ((𝑐 Q 𝑑 Q) → (𝑐𝐺𝑑) Q)
4442, 43syl 14 . . . . . . . . 9 (((A P B P) (𝑐 (1stA) 𝑑 (1stB))) → (𝑐𝐺𝑑) Q)
45 breq2 3742 . . . . . . . . . . 11 (𝑟 = (𝑐𝐺𝑑) → (𝑞 <Q 𝑟𝑞 <Q (𝑐𝐺𝑑)))
46 eleq1 2082 . . . . . . . . . . 11 (𝑟 = (𝑐𝐺𝑑) → (𝑟 (1st ‘(A𝐹B)) ↔ (𝑐𝐺𝑑) (1st ‘(A𝐹B))))
4745, 46anbi12d 445 . . . . . . . . . 10 (𝑟 = (𝑐𝐺𝑑) → ((𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B))) ↔ (𝑞 <Q (𝑐𝐺𝑑) (𝑐𝐺𝑑) (1st ‘(A𝐹B)))))
4847adantl 262 . . . . . . . . 9 ((((A P B P) (𝑐 (1stA) 𝑑 (1stB))) 𝑟 = (𝑐𝐺𝑑)) → ((𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B))) ↔ (𝑞 <Q (𝑐𝐺𝑑) (𝑐𝐺𝑑) (1st ‘(A𝐹B)))))
4944, 48rspcedv 2637 . . . . . . . 8 (((A P B P) (𝑐 (1stA) 𝑑 (1stB))) → ((𝑞 <Q (𝑐𝐺𝑑) (𝑐𝐺𝑑) (1st ‘(A𝐹B))) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B)))))
5036, 49mpan2d 406 . . . . . . 7 (((A P B P) (𝑐 (1stA) 𝑑 (1stB))) → (𝑞 <Q (𝑐𝐺𝑑) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B)))))
5150rexlimdvva 2418 . . . . . 6 ((A P B P) → (𝑐 (1stA)𝑑 (1stB)𝑞 <Q (𝑐𝐺𝑑) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B)))))
5251adantr 261 . . . . 5 (((A P B P) (𝑞 Q 𝑞 (1st ‘(A𝐹B)))) → (𝑐 (1stA)𝑑 (1stB)𝑞 <Q (𝑐𝐺𝑑) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B)))))
5334, 52mpd 13 . . . 4 (((A P B P) (𝑞 Q 𝑞 (1st ‘(A𝐹B)))) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B))))
5453expr 357 . . 3 (((A P B P) 𝑞 Q) → (𝑞 (1st ‘(A𝐹B)) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B)))))
55 genprndl.lower . . . . . . . . . . 11 ((((A P g (1stA)) (B P (1stB))) x Q) → (x <Q (g𝐺) → x (1st ‘(A𝐹B))))
561, 2, 55genpcdl 6374 . . . . . . . . . 10 ((A P B P) → (𝑟 (1st ‘(A𝐹B)) → (x <Q 𝑟x (1st ‘(A𝐹B)))))
5756alrimdv 1738 . . . . . . . . 9 ((A P B P) → (𝑟 (1st ‘(A𝐹B)) → x(x <Q 𝑟x (1st ‘(A𝐹B)))))
58 breq1 3741 . . . . . . . . . . 11 (x = 𝑞 → (x <Q 𝑟𝑞 <Q 𝑟))
59 eleq1 2082 . . . . . . . . . . 11 (x = 𝑞 → (x (1st ‘(A𝐹B)) ↔ 𝑞 (1st ‘(A𝐹B))))
6058, 59imbi12d 223 . . . . . . . . . 10 (x = 𝑞 → ((x <Q 𝑟x (1st ‘(A𝐹B))) ↔ (𝑞 <Q 𝑟𝑞 (1st ‘(A𝐹B)))))
6160cbvalv 1776 . . . . . . . . 9 (x(x <Q 𝑟x (1st ‘(A𝐹B))) ↔ 𝑞(𝑞 <Q 𝑟𝑞 (1st ‘(A𝐹B))))
6257, 61syl6ib 150 . . . . . . . 8 ((A P B P) → (𝑟 (1st ‘(A𝐹B)) → 𝑞(𝑞 <Q 𝑟𝑞 (1st ‘(A𝐹B)))))
63 sp 1382 . . . . . . . 8 (𝑞(𝑞 <Q 𝑟𝑞 (1st ‘(A𝐹B))) → (𝑞 <Q 𝑟𝑞 (1st ‘(A𝐹B))))
6462, 63syl6 29 . . . . . . 7 ((A P B P) → (𝑟 (1st ‘(A𝐹B)) → (𝑞 <Q 𝑟𝑞 (1st ‘(A𝐹B)))))
6564impd 242 . . . . . 6 ((A P B P) → ((𝑟 (1st ‘(A𝐹B)) 𝑞 <Q 𝑟) → 𝑞 (1st ‘(A𝐹B))))
6665ancomsd 256 . . . . 5 ((A P B P) → ((𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B))) → 𝑞 (1st ‘(A𝐹B))))
6766ad2antrr 460 . . . 4 ((((A P B P) 𝑞 Q) 𝑟 Q) → ((𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B))) → 𝑞 (1st ‘(A𝐹B))))
6867rexlimdva 2411 . . 3 (((A P B P) 𝑞 Q) → (𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B))) → 𝑞 (1st ‘(A𝐹B))))
6954, 68impbid 120 . 2 (((A P B P) 𝑞 Q) → (𝑞 (1st ‘(A𝐹B)) ↔ 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B)))))
7069ralrimiva 2370 1 ((A P B P) → 𝑞 Q (𝑞 (1st ‘(A𝐹B)) ↔ 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B)))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 873  ∀wal 1226   = wceq 1228  ∃wex 1362   ∈ wcel 1374  ∀wral 2284  ∃wrex 2285  {crab 2288  ⟨cop 3353   class class class wbr 3738  ‘cfv 4829  (class class class)co 5436   ↦ cmpt2 5438  1st c1st 5688  2nd c2nd 5689  Qcnq 6138
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