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Theorem ltexprlemfu 6583
Description: Lemma for ltexpri 6585. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-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
ltexprlemfu (A<P B → (2nd ‘(A +P 𝐶)) ⊆ (2ndB))
Distinct variable groups:   x,y,A   x,B,y   x,𝐶,y

Proof of Theorem ltexprlemfu
Dummy variables z w u f g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelpr 6487 . . . . . 6 <P ⊆ (P × P)
21brel 4335 . . . . 5 (A<P B → (A P B P))
32simpld 105 . . . 4 (A<P BA P)
4 ltexprlem.1 . . . . 5 𝐶 = ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩
54ltexprlempr 6580 . . . 4 (A<P B𝐶 P)
6 df-iplp 6450 . . . . 5 +P = (z P, y P ↦ ⟨{f Qg Q Q (g (1stz) (1sty) f = (g +Q ))}, {f Qg Q Q (g (2ndz) (2ndy) f = (g +Q ))}⟩)
7 addclnq 6359 . . . . 5 ((g Q Q) → (g +Q ) Q)
86, 7genpelvu 6495 . . . 4 ((A P 𝐶 P) → (z (2nd ‘(A +P 𝐶)) ↔ w (2ndA)u (2nd𝐶)z = (w +Q u)))
93, 5, 8syl2anc 391 . . 3 (A<P B → (z (2nd ‘(A +P 𝐶)) ↔ w (2ndA)u (2nd𝐶)z = (w +Q u)))
10 simprr 484 . . . . . 6 ((A<P B ((w (2ndA) u (2nd𝐶)) z = (w +Q u))) → z = (w +Q u))
114ltexprlemelu 6571 . . . . . . . . . . 11 (u (2nd𝐶) ↔ (u Q y(y (1stA) (y +Q u) (2ndB))))
1211biimpi 113 . . . . . . . . . 10 (u (2nd𝐶) → (u Q y(y (1stA) (y +Q u) (2ndB))))
1312ad2antlr 458 . . . . . . . . 9 (((w (2ndA) u (2nd𝐶)) z = (w +Q u)) → (u Q y(y (1stA) (y +Q u) (2ndB))))
1413simprd 107 . . . . . . . 8 (((w (2ndA) u (2nd𝐶)) z = (w +Q u)) → y(y (1stA) (y +Q u) (2ndB)))
1514adantl 262 . . . . . . 7 ((A<P B ((w (2ndA) u (2nd𝐶)) z = (w +Q u))) → y(y (1stA) (y +Q u) (2ndB)))
16 prop 6457 . . . . . . . . . . . . . . 15 (A P → ⟨(1stA), (2ndA)⟩ P)
173, 16syl 14 . . . . . . . . . . . . . 14 (A<P B → ⟨(1stA), (2ndA)⟩ P)
18 prltlu 6469 . . . . . . . . . . . . . 14 ((⟨(1stA), (2ndA)⟩ P y (1stA) w (2ndA)) → y <Q w)
1917, 18syl3an1 1167 . . . . . . . . . . . . 13 ((A<P B y (1stA) w (2ndA)) → y <Q w)
20193com23 1109 . . . . . . . . . . . 12 ((A<P B w (2ndA) y (1stA)) → y <Q w)
21203adant2r 1129 . . . . . . . . . . 11 ((A<P B (w (2ndA) u (2nd𝐶)) y (1stA)) → y <Q w)
22213adant2r 1129 . . . . . . . . . 10 ((A<P B ((w (2ndA) u (2nd𝐶)) z = (w +Q u)) y (1stA)) → y <Q w)
23223adant3r 1131 . . . . . . . . 9 ((A<P B ((w (2ndA) u (2nd𝐶)) z = (w +Q u)) (y (1stA) (y +Q u) (2ndB))) → y <Q w)
24 ltanqg 6384 . . . . . . . . . . . 12 ((f Q g Q Q) → (f <Q g ↔ ( +Q f) <Q ( +Q g)))
2524adantl 262 . . . . . . . . . . 11 (((A<P B ((w (2ndA) u (2nd𝐶)) z = (w +Q u)) (y (1stA) (y +Q u) (2ndB))) (f Q g Q Q)) → (f <Q g ↔ ( +Q f) <Q ( +Q g)))
26 elprnql 6463 . . . . . . . . . . . . . 14 ((⟨(1stA), (2ndA)⟩ P y (1stA)) → y Q)
2717, 26sylan 267 . . . . . . . . . . . . 13 ((A<P B y (1stA)) → y Q)
2827adantrr 448 . . . . . . . . . . . 12 ((A<P B (y (1stA) (y +Q u) (2ndB))) → y Q)
29283adant2 922 . . . . . . . . . . 11 ((A<P B ((w (2ndA) u (2nd𝐶)) z = (w +Q u)) (y (1stA) (y +Q u) (2ndB))) → y Q)
30 elprnqu 6464 . . . . . . . . . . . . . . 15 ((⟨(1stA), (2ndA)⟩ P w (2ndA)) → w Q)
3117, 30sylan 267 . . . . . . . . . . . . . 14 ((A<P B w (2ndA)) → w Q)
3231adantrr 448 . . . . . . . . . . . . 13 ((A<P B (w (2ndA) u (2nd𝐶))) → w Q)
3332adantrr 448 . . . . . . . . . . . 12 ((A<P B ((w (2ndA) u (2nd𝐶)) z = (w +Q u))) → w Q)
34333adant3 923 . . . . . . . . . . 11 ((A<P B ((w (2ndA) u (2nd𝐶)) z = (w +Q u)) (y (1stA) (y +Q u) (2ndB))) → w Q)
35 prop 6457 . . . . . . . . . . . . . . . 16 (𝐶 P → ⟨(1st𝐶), (2nd𝐶)⟩ P)
365, 35syl 14 . . . . . . . . . . . . . . 15 (A<P B → ⟨(1st𝐶), (2nd𝐶)⟩ P)
37 elprnqu 6464 . . . . . . . . . . . . . . 15 ((⟨(1st𝐶), (2nd𝐶)⟩ P u (2nd𝐶)) → u Q)
3836, 37sylan 267 . . . . . . . . . . . . . 14 ((A<P B u (2nd𝐶)) → u Q)
3938adantrl 447 . . . . . . . . . . . . 13 ((A<P B (w (2ndA) u (2nd𝐶))) → u Q)
4039adantrr 448 . . . . . . . . . . . 12 ((A<P B ((w (2ndA) u (2nd𝐶)) z = (w +Q u))) → u Q)
41403adant3 923 . . . . . . . . . . 11 ((A<P B ((w (2ndA) u (2nd𝐶)) z = (w +Q u)) (y (1stA) (y +Q u) (2ndB))) → u Q)
42 addcomnqg 6365 . . . . . . . . . . . 12 ((f Q g Q) → (f +Q g) = (g +Q f))
4342adantl 262 . . . . . . . . . . 11 (((A<P B ((w (2ndA) u (2nd𝐶)) z = (w +Q u)) (y (1stA) (y +Q u) (2ndB))) (f Q g Q)) → (f +Q g) = (g +Q f))
4425, 29, 34, 41, 43caovord2d 5612 . . . . . . . . . 10 ((A<P B ((w (2ndA) u (2nd𝐶)) z = (w +Q u)) (y (1stA) (y +Q u) (2ndB))) → (y <Q w ↔ (y +Q u) <Q (w +Q u)))
452simprd 107 . . . . . . . . . . . . . 14 (A<P BB P)
46 prop 6457 . . . . . . . . . . . . . 14 (B P → ⟨(1stB), (2ndB)⟩ P)
4745, 46syl 14 . . . . . . . . . . . . 13 (A<P B → ⟨(1stB), (2ndB)⟩ P)
48 prcunqu 6467 . . . . . . . . . . . . 13 ((⟨(1stB), (2ndB)⟩ P (y +Q u) (2ndB)) → ((y +Q u) <Q (w +Q u) → (w +Q u) (2ndB)))
4947, 48sylan 267 . . . . . . . . . . . 12 ((A<P B (y +Q u) (2ndB)) → ((y +Q u) <Q (w +Q u) → (w +Q u) (2ndB)))
5049adantrl 447 . . . . . . . . . . 11 ((A<P B (y (1stA) (y +Q u) (2ndB))) → ((y +Q u) <Q (w +Q u) → (w +Q u) (2ndB)))
51503adant2 922 . . . . . . . . . 10 ((A<P B ((w (2ndA) u (2nd𝐶)) z = (w +Q u)) (y (1stA) (y +Q u) (2ndB))) → ((y +Q u) <Q (w +Q u) → (w +Q u) (2ndB)))
5244, 51sylbid 139 . . . . . . . . 9 ((A<P B ((w (2ndA) u (2nd𝐶)) z = (w +Q u)) (y (1stA) (y +Q u) (2ndB))) → (y <Q w → (w +Q u) (2ndB)))
5323, 52mpd 13 . . . . . . . 8 ((A<P B ((w (2ndA) u (2nd𝐶)) z = (w +Q u)) (y (1stA) (y +Q u) (2ndB))) → (w +Q u) (2ndB))
54533expa 1103 . . . . . . 7 (((A<P B ((w (2ndA) u (2nd𝐶)) z = (w +Q u))) (y (1stA) (y +Q u) (2ndB))) → (w +Q u) (2ndB))
5515, 54exlimddv 1775 . . . . . 6 ((A<P B ((w (2ndA) u (2nd𝐶)) z = (w +Q u))) → (w +Q u) (2ndB))
5610, 55eqeltrd 2111 . . . . 5 ((A<P B ((w (2ndA) u (2nd𝐶)) z = (w +Q u))) → z (2ndB))
5756expr 357 . . . 4 ((A<P B (w (2ndA) u (2nd𝐶))) → (z = (w +Q u) → z (2ndB)))
5857rexlimdvva 2434 . . 3 (A<P B → (w (2ndA)u (2nd𝐶)z = (w +Q u) → z (2ndB)))
599, 58sylbid 139 . 2 (A<P B → (z (2nd ‘(A +P 𝐶)) → z (2ndB)))
6059ssrdv 2945 1 (A<P B → (2nd ‘(A +P 𝐶)) ⊆ (2ndB))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242  wex 1378   wcel 1390  wrex 2301  {crab 2304  wss 2911  cop 3370   class class class wbr 3755  cfv 4845  (class class class)co 5455  1st c1st 5707  2nd c2nd 5708  Qcnq 6264   +Q cplq 6266   <Q cltq 6269  Pcnp 6275   +P cpp 6277  <P cltp 6279
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6406  df-nq0 6407  df-0nq0 6408  df-plq0 6409  df-mq0 6410  df-inp 6448  df-iplp 6450  df-iltp 6452
This theorem is referenced by:  ltexpri  6585
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