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Theorem ltexprlemfl 6573
Description: Lemma for ltexpri 6577. One directon of our result for lower 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
ltexprlemfl (A<P B → (1st ‘(A +P 𝐶)) ⊆ (1stB))
Distinct variable groups:   x,y,A   x,B,y   x,𝐶,y

Proof of Theorem ltexprlemfl
Dummy variables z w u f g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelpr 6480 . . . . . 6 <P ⊆ (P × P)
21brel 4334 . . . . 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 6572 . . . 4 (A<P B𝐶 P)
6 df-iplp 6443 . . . . 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 6352 . . . . 5 ((g Q Q) → (g +Q ) Q)
86, 7genpelvl 6487 . . . 4 ((A P 𝐶 P) → (z (1st ‘(A +P 𝐶)) ↔ w (1stA)u (1st𝐶)z = (w +Q u)))
93, 5, 8syl2anc 391 . . 3 (A<P B → (z (1st ‘(A +P 𝐶)) ↔ w (1stA)u (1st𝐶)z = (w +Q u)))
10 simprr 484 . . . . . 6 ((A<P B ((w (1stA) u (1st𝐶)) z = (w +Q u))) → z = (w +Q u))
114ltexprlemell 6562 . . . . . . . . . . 11 (u (1st𝐶) ↔ (u Q y(y (2ndA) (y +Q u) (1stB))))
1211biimpi 113 . . . . . . . . . 10 (u (1st𝐶) → (u Q y(y (2ndA) (y +Q u) (1stB))))
1312ad2antlr 458 . . . . . . . . 9 (((w (1stA) u (1st𝐶)) z = (w +Q u)) → (u Q y(y (2ndA) (y +Q u) (1stB))))
1413adantl 262 . . . . . . . 8 ((A<P B ((w (1stA) u (1st𝐶)) z = (w +Q u))) → (u Q y(y (2ndA) (y +Q u) (1stB))))
1514simprd 107 . . . . . . 7 ((A<P B ((w (1stA) u (1st𝐶)) z = (w +Q u))) → y(y (2ndA) (y +Q u) (1stB)))
16 prop 6450 . . . . . . . . . . . . . 14 (A P → ⟨(1stA), (2ndA)⟩ P)
173, 16syl 14 . . . . . . . . . . . . 13 (A<P B → ⟨(1stA), (2ndA)⟩ P)
18 prltlu 6462 . . . . . . . . . . . . 13 ((⟨(1stA), (2ndA)⟩ P w (1stA) y (2ndA)) → w <Q y)
1917, 18syl3an1 1167 . . . . . . . . . . . 12 ((A<P B w (1stA) y (2ndA)) → w <Q y)
20193adant2r 1129 . . . . . . . . . . 11 ((A<P B (w (1stA) u (1st𝐶)) y (2ndA)) → w <Q y)
21203adant2r 1129 . . . . . . . . . 10 ((A<P B ((w (1stA) u (1st𝐶)) z = (w +Q u)) y (2ndA)) → w <Q y)
22213adant3r 1131 . . . . . . . . 9 ((A<P B ((w (1stA) u (1st𝐶)) z = (w +Q u)) (y (2ndA) (y +Q u) (1stB))) → w <Q y)
23 ltanqg 6377 . . . . . . . . . . . 12 ((f Q g Q Q) → (f <Q g ↔ ( +Q f) <Q ( +Q g)))
2423adantl 262 . . . . . . . . . . 11 (((A<P B ((w (1stA) u (1st𝐶)) z = (w +Q u)) (y (2ndA) (y +Q u) (1stB))) (f Q g Q Q)) → (f <Q g ↔ ( +Q f) <Q ( +Q g)))
25 ltrelnq 6342 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
2625brel 4334 . . . . . . . . . . . . 13 (w <Q y → (w Q y Q))
2722, 26syl 14 . . . . . . . . . . . 12 ((A<P B ((w (1stA) u (1st𝐶)) z = (w +Q u)) (y (2ndA) (y +Q u) (1stB))) → (w Q y Q))
2827simpld 105 . . . . . . . . . . 11 ((A<P B ((w (1stA) u (1st𝐶)) z = (w +Q u)) (y (2ndA) (y +Q u) (1stB))) → w Q)
2927simprd 107 . . . . . . . . . . 11 ((A<P B ((w (1stA) u (1st𝐶)) z = (w +Q u)) (y (2ndA) (y +Q u) (1stB))) → y Q)
30 prop 6450 . . . . . . . . . . . . . . . 16 (𝐶 P → ⟨(1st𝐶), (2nd𝐶)⟩ P)
315, 30syl 14 . . . . . . . . . . . . . . 15 (A<P B → ⟨(1st𝐶), (2nd𝐶)⟩ P)
32 elprnql 6456 . . . . . . . . . . . . . . 15 ((⟨(1st𝐶), (2nd𝐶)⟩ P u (1st𝐶)) → u Q)
3331, 32sylan 267 . . . . . . . . . . . . . 14 ((A<P B u (1st𝐶)) → u Q)
3433adantrl 447 . . . . . . . . . . . . 13 ((A<P B (w (1stA) u (1st𝐶))) → u Q)
3534adantrr 448 . . . . . . . . . . . 12 ((A<P B ((w (1stA) u (1st𝐶)) z = (w +Q u))) → u Q)
36353adant3 923 . . . . . . . . . . 11 ((A<P B ((w (1stA) u (1st𝐶)) z = (w +Q u)) (y (2ndA) (y +Q u) (1stB))) → u Q)
37 addcomnqg 6358 . . . . . . . . . . . 12 ((f Q g Q) → (f +Q g) = (g +Q f))
3837adantl 262 . . . . . . . . . . 11 (((A<P B ((w (1stA) u (1st𝐶)) z = (w +Q u)) (y (2ndA) (y +Q u) (1stB))) (f Q g Q)) → (f +Q g) = (g +Q f))
3924, 28, 29, 36, 38caovord2d 5609 . . . . . . . . . 10 ((A<P B ((w (1stA) u (1st𝐶)) z = (w +Q u)) (y (2ndA) (y +Q u) (1stB))) → (w <Q y ↔ (w +Q u) <Q (y +Q u)))
402simprd 107 . . . . . . . . . . . . . 14 (A<P BB P)
41 prop 6450 . . . . . . . . . . . . . 14 (B P → ⟨(1stB), (2ndB)⟩ P)
4240, 41syl 14 . . . . . . . . . . . . 13 (A<P B → ⟨(1stB), (2ndB)⟩ P)
43 prcdnql 6459 . . . . . . . . . . . . 13 ((⟨(1stB), (2ndB)⟩ P (y +Q u) (1stB)) → ((w +Q u) <Q (y +Q u) → (w +Q u) (1stB)))
4442, 43sylan 267 . . . . . . . . . . . 12 ((A<P B (y +Q u) (1stB)) → ((w +Q u) <Q (y +Q u) → (w +Q u) (1stB)))
4544adantrl 447 . . . . . . . . . . 11 ((A<P B (y (2ndA) (y +Q u) (1stB))) → ((w +Q u) <Q (y +Q u) → (w +Q u) (1stB)))
46453adant2 922 . . . . . . . . . 10 ((A<P B ((w (1stA) u (1st𝐶)) z = (w +Q u)) (y (2ndA) (y +Q u) (1stB))) → ((w +Q u) <Q (y +Q u) → (w +Q u) (1stB)))
4739, 46sylbid 139 . . . . . . . . 9 ((A<P B ((w (1stA) u (1st𝐶)) z = (w +Q u)) (y (2ndA) (y +Q u) (1stB))) → (w <Q y → (w +Q u) (1stB)))
4822, 47mpd 13 . . . . . . . 8 ((A<P B ((w (1stA) u (1st𝐶)) z = (w +Q u)) (y (2ndA) (y +Q u) (1stB))) → (w +Q u) (1stB))
49483expa 1103 . . . . . . 7 (((A<P B ((w (1stA) u (1st𝐶)) z = (w +Q u))) (y (2ndA) (y +Q u) (1stB))) → (w +Q u) (1stB))
5015, 49exlimddv 1775 . . . . . 6 ((A<P B ((w (1stA) u (1st𝐶)) z = (w +Q u))) → (w +Q u) (1stB))
5110, 50eqeltrd 2111 . . . . 5 ((A<P B ((w (1stA) u (1st𝐶)) z = (w +Q u))) → z (1stB))
5251expr 357 . . . 4 ((A<P B (w (1stA) u (1st𝐶))) → (z = (w +Q u) → z (1stB)))
5352rexlimdvva 2434 . . 3 (A<P B → (w (1stA)u (1st𝐶)z = (w +Q u) → z (1stB)))
549, 53sylbid 139 . 2 (A<P B → (z (1st ‘(A +P 𝐶)) → z (1stB)))
5554ssrdv 2945 1 (A<P B → (1st ‘(A +P 𝐶)) ⊆ (1stB))
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 3369   class class class wbr 3754  cfv 4844  (class class class)co 5452  1st c1st 5704  2nd c2nd 5705  Qcnq 6257   +Q cplq 6259   <Q cltq 6262  Pcnp 6268   +P cpp 6270  <P cltp 6272
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 3862  ax-sep 3865  ax-nul 3873  ax-pow 3917  ax-pr 3934  ax-un 4135  ax-setind 4219  ax-iinf 4253
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 3352  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-int 3606  df-iun 3649  df-br 3755  df-opab 3809  df-mpt 3810  df-tr 3845  df-eprel 4016  df-id 4020  df-po 4023  df-iso 4024  df-iord 4068  df-on 4070  df-suc 4073  df-iom 4256  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-rn 4298  df-res 4299  df-ima 4300  df-iota 4809  df-fun 4846  df-fn 4847  df-f 4848  df-f1 4849  df-fo 4850  df-f1o 4851  df-fv 4852  df-ov 5455  df-oprab 5456  df-mpt2 5457  df-1st 5706  df-2nd 5707  df-recs 5858  df-irdg 5894  df-1o 5933  df-2o 5934  df-oadd 5937  df-omul 5938  df-er 6035  df-ec 6037  df-qs 6041  df-ni 6281  df-pli 6282  df-mi 6283  df-lti 6284  df-plpq 6321  df-mpq 6322  df-enq 6324  df-nqqs 6325  df-plqqs 6326  df-mqqs 6327  df-1nqqs 6328  df-rq 6329  df-ltnqqs 6330  df-enq0 6399  df-nq0 6400  df-0nq0 6401  df-plq0 6402  df-mq0 6403  df-inp 6441  df-iplp 6443  df-iltp 6445
This theorem is referenced by:  ltexpri  6577
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