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Theorem genprndl 6365
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 6355 . . . . . . . . 9 ((A P B P) → (𝑞 (1st ‘(A𝐹B)) ↔ 𝑎 (1stA)𝑏 (1stB)𝑞 = (𝑎𝐺𝑏)))
4 r2ex 2316 . . . . . . . . 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 447 . . . . . 6 (((A P B P) (𝑞 Q 𝑞 (1st ‘(A𝐹B)))) → 𝑎𝑏((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏)))
8 prop 6318 . . . . . . . . . . . . . . . 16 (A P → ⟨(1stA), (2ndA)⟩ P)
9 prnmaxl 6331 . . . . . . . . . . . . . . . 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 6318 . . . . . . . . . . . . . . . 16 (B P → ⟨(1stB), (2ndB)⟩ P)
12 prnmaxl 6331 . . . . . . . . . . . . . . . 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 507 . . . . . . . . . . . . 13 (((A P B P) (𝑎 (1stA) 𝑏 (1stB))) → (𝑐 (1stA)𝑎 <Q 𝑐 𝑑 (1stB)𝑏 <Q 𝑑))
16 reeanv 2451 . . . . . . . . . . . . 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 6353 . . . . . . . . . . . . . 14 ((𝑎 <Q 𝑐 𝑏 <Q 𝑑) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
2120reximi 2388 . . . . . . . . . . . . 13 (𝑑 (1stB)(𝑎 <Q 𝑐 𝑏 <Q 𝑑) → 𝑑 (1stB)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
2221reximi 2388 . . . . . . . . . . . 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 448 . . . . . . . . . 10 (((A P B P) ((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏))) → 𝑐 (1stA)𝑑 (1stB)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
25 breq1 3733 . . . . . . . . . . . . . 14 (𝑞 = (𝑎𝐺𝑏) → (𝑞 <Q (𝑐𝐺𝑑) ↔ (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑)))
2625biimprd 147 . . . . . . . . . . . . 13 (𝑞 = (𝑎𝐺𝑏) → ((𝑎𝐺𝑏) <Q (𝑐𝐺𝑑) → 𝑞 <Q (𝑐𝐺𝑑)))
2726reximdv 2392 . . . . . . . . . . . 12 (𝑞 = (𝑎𝐺𝑏) → (𝑑 (1stB)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑) → 𝑑 (1stB)𝑞 <Q (𝑐𝐺𝑑)))
2827reximdv 2392 . . . . . . . . . . 11 (𝑞 = (𝑎𝐺𝑏) → (𝑐 (1stA)𝑑 (1stB)(𝑎𝐺𝑏) <Q (𝑐𝐺𝑑) → 𝑐 (1stA)𝑑 (1stB)𝑞 <Q (𝑐𝐺𝑑)))
2928ad2antll 461 . . . . . . . . . 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 1753 . . . . . . 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 6357 . . . . . . . . 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 6324 . . . . . . . . . . . . 13 ((⟨(1stA), (2ndA)⟩ P 𝑐 (1stA)) → 𝑐 Q)
388, 37sylan 267 . . . . . . . . . . . 12 ((A P 𝑐 (1stA)) → 𝑐 Q)
39 elprnql 6324 . . . . . . . . . . . . 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 507 . . . . . . . . . 10 (((A P B P) (𝑐 (1stA) 𝑑 (1stB))) → (𝑐 Q 𝑑 Q))
432caovcl 5569 . . . . . . . . . 10 ((𝑐 Q 𝑑 Q) → (𝑐𝐺𝑑) Q)
4442, 43syl 14 . . . . . . . . 9 (((A P B P) (𝑐 (1stA) 𝑑 (1stB))) → (𝑐𝐺𝑑) Q)
45 breq2 3734 . . . . . . . . . . 11 (𝑟 = (𝑐𝐺𝑑) → (𝑞 <Q 𝑟𝑞 <Q (𝑐𝐺𝑑)))
46 eleq1 2076 . . . . . . . . . . 11 (𝑟 = (𝑐𝐺𝑑) → (𝑟 (1st ‘(A𝐹B)) ↔ (𝑐𝐺𝑑) (1st ‘(A𝐹B))))
4745, 46anbi12d 442 . . . . . . . . . 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 2631 . . . . . . . 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 404 . . . . . . 7 (((A P B P) (𝑐 (1stA) 𝑑 (1stB))) → (𝑞 <Q (𝑐𝐺𝑑) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B)))))
5150rexlimdvva 2412 . . . . . 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 6363 . . . . . . . . . 10 ((A P B P) → (𝑟 (1st ‘(A𝐹B)) → (x <Q 𝑟x (1st ‘(A𝐹B)))))
5756alrimdv 1732 . . . . . . . . 9 ((A P B P) → (𝑟 (1st ‘(A𝐹B)) → x(x <Q 𝑟x (1st ‘(A𝐹B)))))
58 breq1 3733 . . . . . . . . . . 11 (x = 𝑞 → (x <Q 𝑟𝑞 <Q 𝑟))
59 eleq1 2076 . . . . . . . . . . 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 1770 . . . . . . . . 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 1377 . . . . . . . 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 457 . . . 4 ((((A P B P) 𝑞 Q) 𝑟 Q) → ((𝑞 <Q 𝑟 𝑟 (1st ‘(A𝐹B))) → 𝑞 (1st ‘(A𝐹B))))
6867rexlimdva 2405 . . 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 2364 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 869  wal 1224   = wceq 1226  wex 1357   wcel 1369  wral 2278  wrex 2279  {crab 2282  cop 3345   class class class wbr 3730  cfv 4820  (class class class)co 5427  cmpt2 5429  1st c1st 5679  2nd c2nd 5680  Qcnq 6129   <Q cltq 6134  Pcnp 6140
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 529  ax-in2 530  ax-io 614  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1358  ax-ie2 1359  ax-8 1371  ax-10 1372  ax-11 1373  ax-i12 1374  ax-bnd 1375  ax-4 1376  ax-13 1380  ax-14 1381  ax-17 1395  ax-i9 1399  ax-ial 1403  ax-i5r 1404  ax-ext 1998  ax-coll 3838  ax-sep 3841  ax-nul 3849  ax-pow 3893  ax-pr 3910  ax-un 4111  ax-setind 4195  ax-iinf 4229
This theorem depends on definitions:  df-bi 110  df-dc 727  df-3or 870  df-3an 871  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1622  df-eu 1879  df-mo 1880  df-clab 2003  df-cleq 2009  df-clel 2012  df-nfc 2143  df-ne 2182  df-ral 2283  df-rex 2284  df-reu 2285  df-rab 2287  df-v 2531  df-sbc 2736  df-csb 2824  df-dif 2891  df-un 2893  df-in 2895  df-ss 2902  df-nul 3196  df-pw 3328  df-sn 3348  df-pr 3349  df-op 3351  df-uni 3547  df-int 3582  df-iun 3625  df-br 3731  df-opab 3785  df-mpt 3786  df-tr 3821  df-eprel 3992  df-id 3996  df-po 3999  df-iso 4000  df-iord 4044  df-on 4046  df-suc 4049  df-iom 4232  df-xp 4269  df-rel 4270  df-cnv 4271  df-co 4272  df-dm 4273  df-rn 4274  df-res 4275  df-ima 4276  df-iota 4785  df-fun 4822  df-fn 4823  df-f 4824  df-f1 4825  df-fo 4826  df-f1o 4827  df-fv 4828  df-ov 5430  df-oprab 5431  df-mpt2 5432  df-1st 5681  df-2nd 5682  df-recs 5833  df-irdg 5869  df-oadd 5911  df-omul 5912  df-er 6008  df-ec 6010  df-qs 6014  df-ni 6153  df-mi 6155  df-lti 6156  df-enq 6195  df-nqqs 6196  df-ltnqqs 6201  df-inp 6309
This theorem is referenced by:  addclpr  6381  mulclpr  6405
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