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Theorem ltexprlemfl 6440
Description: Lemma for ltexpri 6444. 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 6353 . . . . . 6 <P ⊆ (P × P)
21brel 4315 . . . . 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 6439 . . . 4 (A<P B𝐶 P)
6 df-iplp 6316 . . . . 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 6228 . . . . 5 ((g Q Q) → (g +Q ) Q)
86, 7genpelvl 6360 . . . 4 ((A P 𝐶 P) → (z (1st ‘(A +P 𝐶)) ↔ w (1stA)u (1st𝐶)z = (w +Q u)))
93, 5, 8syl2anc 393 . . 3 (A<P B → (z (1st ‘(A +P 𝐶)) ↔ w (1stA)u (1st𝐶)z = (w +Q u)))
10 simprr 472 . . . . . 6 ((A<P B ((w (1stA) u (1st𝐶)) z = (w +Q u))) → z = (w +Q u))
114ltexprlemell 6429 . . . . . . . . . . 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 462 . . . . . . . . 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 6323 . . . . . . . . . . . . . 14 (A P → ⟨(1stA), (2ndA)⟩ P)
173, 16syl 14 . . . . . . . . . . . . 13 (A<P B → ⟨(1stA), (2ndA)⟩ P)
18 prltlu 6335 . . . . . . . . . . . . 13 ((⟨(1stA), (2ndA)⟩ P w (1stA) y (2ndA)) → w <Q y)
1917, 18syl3an1 1152 . . . . . . . . . . . 12 ((A<P B w (1stA) y (2ndA)) → w <Q y)
20193adant2r 1114 . . . . . . . . . . 11 ((A<P B (w (1stA) u (1st𝐶)) y (2ndA)) → w <Q y)
21203adant2r 1114 . . . . . . . . . 10 ((A<P B ((w (1stA) u (1st𝐶)) z = (w +Q u)) y (2ndA)) → w <Q y)
22213adant3r 1116 . . . . . . . . 9 ((A<P B ((w (1stA) u (1st𝐶)) z = (w +Q u)) (y (2ndA) (y +Q u) (1stB))) → w <Q y)
23 ltanqg 6253 . . . . . . . . . . . 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 6218 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
2625brel 4315 . . . . . . . . . . . . 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 6323 . . . . . . . . . . . . . . . 16 (𝐶 P → ⟨(1st𝐶), (2nd𝐶)⟩ P)
315, 30syl 14 . . . . . . . . . . . . . . 15 (A<P B → ⟨(1st𝐶), (2nd𝐶)⟩ P)
32 elprnql 6329 . . . . . . . . . . . . . . 15 ((⟨(1st𝐶), (2nd𝐶)⟩ P u (1st𝐶)) → u Q)
3331, 32sylan 267 . . . . . . . . . . . . . 14 ((A<P B u (1st𝐶)) → u Q)
3433adantrl 450 . . . . . . . . . . . . 13 ((A<P B (w (1stA) u (1st𝐶))) → u Q)
3534adantrr 451 . . . . . . . . . . . 12 ((A<P B ((w (1stA) u (1st𝐶)) z = (w +Q u))) → u Q)
36353adant3 910 . . . . . . . . . . 11 ((A<P B ((w (1stA) u (1st𝐶)) z = (w +Q u)) (y (2ndA) (y +Q u) (1stB))) → u Q)
37 addcomnqg 6234 . . . . . . . . . . . 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 5589 . . . . . . . . . 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 6323 . . . . . . . . . . . . . 14 (B P → ⟨(1stB), (2ndB)⟩ P)
4240, 41syl 14 . . . . . . . . . . . . 13 (A<P B → ⟨(1stB), (2ndB)⟩ P)
43 prcdnql 6332 . . . . . . . . . . . . 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 450 . . . . . . . . . . 11 ((A<P B (y (2ndA) (y +Q u) (1stB))) → ((w +Q u) <Q (y +Q u) → (w +Q u) (1stB)))
46453adant2 909 . . . . . . . . . 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 1088 . . . . . . 7 (((A<P B ((w (1stA) u (1st𝐶)) z = (w +Q u))) (y (2ndA) (y +Q u) (1stB))) → (w +Q u) (1stB))
5015, 49exlimddv 1756 . . . . . 6 ((A<P B ((w (1stA) u (1st𝐶)) z = (w +Q u))) → (w +Q u) (1stB))
5110, 50eqeltrd 2092 . . . . 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 2414 . . 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 2924 1 (A<P B → (1st ‘(A +P 𝐶)) ⊆ (1stB))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 871   = wceq 1226  wex 1358   wcel 1370  wrex 2281  {crab 2284  wss 2890  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 cpp 6147  <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-po 4003  df-iso 4004  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-1o 5912  df-2o 5913  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-mpq 6198  df-enq 6200  df-nqqs 6201  df-plqqs 6202  df-mqqs 6203  df-1nqqs 6204  df-rq 6205  df-ltnqqs 6206  df-enq0 6273  df-nq0 6274  df-0nq0 6275  df-plq0 6276  df-mq0 6277  df-inp 6314  df-iplp 6316  df-iltp 6318
This theorem is referenced by:  ltexpri  6444
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