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Theorem ltexprlemfu 6585
 Description: Lemma for ltexpri 6587. 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 6488 . . . . . 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 6582 . . . 4 (A<P B𝐶 P)
6 df-iplp 6451 . . . . 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 6496 . . . 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 6573 . . . . . . . . . . 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 6458 . . . . . . . . . . . . . . 15 (A P → ⟨(1stA), (2ndA)⟩ P)
173, 16syl 14 . . . . . . . . . . . . . 14 (A<P B → ⟨(1stA), (2ndA)⟩ P)
18 prltlu 6470 . . . . . . . . . . . . . 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 6464 . . . . . . . . . . . . . 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 6465 . . . . . . . . . . . . . . 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 6458 . . . . . . . . . . . . . . . 16 (𝐶 P → ⟨(1st𝐶), (2nd𝐶)⟩ P)
365, 35syl 14 . . . . . . . . . . . . . . 15 (A<P B → ⟨(1st𝐶), (2nd𝐶)⟩ P)
37 elprnqu 6465 . . . . . . . . . . . . . . 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 6458 . . . . . . . . . . . . . 14 (B P → ⟨(1stB), (2ndB)⟩ P)
4745, 46syl 14 . . . . . . . . . . . . 13 (A<P B → ⟨(1stB), (2ndB)⟩ P)
48 prcunqu 6468 . . . . . . . . . . . . 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
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