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Theorem ltexprlemupu 6435
Description: The upper cut of our constructed difference is upper. Lemma for ltexpri 6444. (Contributed by Jim Kingdon, 21-Dec-2019.)
Hypothesis
Ref Expression
ltexprlem.1 𝐶 = ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩
Assertion
Ref Expression
ltexprlemupu ((A<P B 𝑟 Q) → (𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd𝐶)) → 𝑟 (2nd𝐶)))
Distinct variable groups:   x,y,𝑞,𝑟,A   x,B,y,𝑞,𝑟   x,𝐶,y,𝑞,𝑟

Proof of Theorem ltexprlemupu
StepHypRef Expression
1 simplr 470 . . . . . 6 (((A<P B 𝑟 Q) (𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB))))) → 𝑟 Q)
2 simprrr 480 . . . . . . 7 (((A<P B 𝑟 Q) (𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB))))) → (y (1stA) (y +Q 𝑞) (2ndB)))
32simpld 105 . . . . . 6 (((A<P B 𝑟 Q) (𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB))))) → y (1stA))
4 simprl 471 . . . . . . . 8 (((A<P B 𝑟 Q) (𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB))))) → 𝑞 <Q 𝑟)
5 simpll 469 . . . . . . . . 9 (((A<P B 𝑟 Q) (𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB))))) → A<P B)
6 simprrl 479 . . . . . . . . . 10 ((𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB)))) → y (1stA))
76adantl 262 . . . . . . . . 9 (((A<P B 𝑟 Q) (𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB))))) → y (1stA))
8 ltrelpr 6353 . . . . . . . . . . . . 13 <P ⊆ (P × P)
98brel 4315 . . . . . . . . . . . 12 (A<P B → (A P B P))
109simpld 105 . . . . . . . . . . 11 (A<P BA P)
11 prop 6323 . . . . . . . . . . 11 (A P → ⟨(1stA), (2ndA)⟩ P)
1210, 11syl 14 . . . . . . . . . 10 (A<P B → ⟨(1stA), (2ndA)⟩ P)
13 elprnql 6329 . . . . . . . . . 10 ((⟨(1stA), (2ndA)⟩ P y (1stA)) → y Q)
1412, 13sylan 267 . . . . . . . . 9 ((A<P B y (1stA)) → y Q)
155, 7, 14syl2anc 393 . . . . . . . 8 (((A<P B 𝑟 Q) (𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB))))) → y Q)
16 ltanqi 6255 . . . . . . . 8 ((𝑞 <Q 𝑟 y Q) → (y +Q 𝑞) <Q (y +Q 𝑟))
174, 15, 16syl2anc 393 . . . . . . 7 (((A<P B 𝑟 Q) (𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB))))) → (y +Q 𝑞) <Q (y +Q 𝑟))
189simprd 107 . . . . . . . . 9 (A<P BB P)
195, 18syl 14 . . . . . . . 8 (((A<P B 𝑟 Q) (𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB))))) → B P)
202simprd 107 . . . . . . . 8 (((A<P B 𝑟 Q) (𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB))))) → (y +Q 𝑞) (2ndB))
21 prop 6323 . . . . . . . . 9 (B P → ⟨(1stB), (2ndB)⟩ P)
22 prcunqu 6333 . . . . . . . . 9 ((⟨(1stB), (2ndB)⟩ P (y +Q 𝑞) (2ndB)) → ((y +Q 𝑞) <Q (y +Q 𝑟) → (y +Q 𝑟) (2ndB)))
2321, 22sylan 267 . . . . . . . 8 ((B P (y +Q 𝑞) (2ndB)) → ((y +Q 𝑞) <Q (y +Q 𝑟) → (y +Q 𝑟) (2ndB)))
2419, 20, 23syl2anc 393 . . . . . . 7 (((A<P B 𝑟 Q) (𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB))))) → ((y +Q 𝑞) <Q (y +Q 𝑟) → (y +Q 𝑟) (2ndB)))
2517, 24mpd 13 . . . . . 6 (((A<P B 𝑟 Q) (𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB))))) → (y +Q 𝑟) (2ndB))
261, 3, 25jca32 293 . . . . 5 (((A<P B 𝑟 Q) (𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB))))) → (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB))))
2726eximi 1469 . . . 4 (y((A<P B 𝑟 Q) (𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB))))) → y(𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB))))
28 ltexprlem.1 . . . . . . . . . 10 𝐶 = ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩
2928ltexprlemelu 6430 . . . . . . . . 9 (𝑞 (2nd𝐶) ↔ (𝑞 Q y(y (1stA) (y +Q 𝑞) (2ndB))))
30 19.42v 1764 . . . . . . . . 9 (y(𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB))) ↔ (𝑞 Q y(y (1stA) (y +Q 𝑞) (2ndB))))
3129, 30bitr4i 176 . . . . . . . 8 (𝑞 (2nd𝐶) ↔ y(𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB))))
3231anbi2i 433 . . . . . . 7 ((𝑞 <Q 𝑟 𝑞 (2nd𝐶)) ↔ (𝑞 <Q 𝑟 y(𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB)))))
33 19.42v 1764 . . . . . . 7 (y(𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB)))) ↔ (𝑞 <Q 𝑟 y(𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB)))))
3432, 33bitr4i 176 . . . . . 6 ((𝑞 <Q 𝑟 𝑞 (2nd𝐶)) ↔ y(𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB)))))
3534anbi2i 433 . . . . 5 (((A<P B 𝑟 Q) (𝑞 <Q 𝑟 𝑞 (2nd𝐶))) ↔ ((A<P B 𝑟 Q) y(𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB))))))
36 19.42v 1764 . . . . 5 (y((A<P B 𝑟 Q) (𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB))))) ↔ ((A<P B 𝑟 Q) y(𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB))))))
3735, 36bitr4i 176 . . . 4 (((A<P B 𝑟 Q) (𝑞 <Q 𝑟 𝑞 (2nd𝐶))) ↔ y((A<P B 𝑟 Q) (𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB))))))
3828ltexprlemelu 6430 . . . . 5 (𝑟 (2nd𝐶) ↔ (𝑟 Q y(y (1stA) (y +Q 𝑟) (2ndB))))
39 19.42v 1764 . . . . 5 (y(𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB))) ↔ (𝑟 Q y(y (1stA) (y +Q 𝑟) (2ndB))))
4038, 39bitr4i 176 . . . 4 (𝑟 (2nd𝐶) ↔ y(𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB))))
4127, 37, 403imtr4i 190 . . 3 (((A<P B 𝑟 Q) (𝑞 <Q 𝑟 𝑞 (2nd𝐶))) → 𝑟 (2nd𝐶))
4241ex 108 . 2 ((A<P B 𝑟 Q) → ((𝑞 <Q 𝑟 𝑞 (2nd𝐶)) → 𝑟 (2nd𝐶)))
4342rexlimdvw 2410 1 ((A<P B 𝑟 Q) → (𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd𝐶)) → 𝑟 (2nd𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1226  wex 1358   wcel 1370  wrex 2281  {crab 2284  cop 3349   class class class wbr 3734  cfv 4825  (class class class)co 5432  1st c1st 5684  2nd c2nd 5685  Qcnq 6134   +Q cplq 6136   <Q cltq 6139  Pcnp 6145  <P cltp 6149
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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-coll 3842  ax-sep 3845  ax-nul 3853  ax-pow 3897  ax-pr 3914  ax-un 4116  ax-setind 4200  ax-iinf 4234
This theorem depends on definitions:  df-bi 110  df-dc 731  df-3or 872  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ne 2184  df-ral 2285  df-rex 2286  df-reu 2287  df-rab 2289  df-v 2533  df-sbc 2738  df-csb 2826  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-nul 3198  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-int 3586  df-iun 3629  df-br 3735  df-opab 3789  df-mpt 3790  df-tr 3825  df-eprel 3996  df-id 4000  df-iord 4048  df-on 4050  df-suc 4053  df-iom 4237  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832  df-fv 4833  df-ov 5435  df-oprab 5436  df-mpt2 5437  df-1st 5686  df-2nd 5687  df-recs 5838  df-irdg 5874  df-oadd 5916  df-omul 5917  df-er 6013  df-ec 6015  df-qs 6019  df-ni 6158  df-pli 6159  df-mi 6160  df-lti 6161  df-plpq 6197  df-enq 6200  df-nqqs 6201  df-plqqs 6202  df-ltnqqs 6206  df-inp 6314  df-iltp 6318
This theorem is referenced by:  ltexprlemrnd  6436
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