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Theorem ltexprlemfl 6581
 Description: Lemma for ltexpri 6585. 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 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, 7genpelvl 6494 . . . 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 6570 . . . . . . . . . . 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 6457 . . . . . . . . . . . . . 14 (A P → ⟨(1stA), (2ndA)⟩ P)
173, 16syl 14 . . . . . . . . . . . . 13 (A<P B → ⟨(1stA), (2ndA)⟩ P)
18 prltlu 6469 . . . . . . . . . . . . 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 6384 . . . . . . . . . . . 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 6349 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
2625brel 4335 . . . . . . . . . . . . 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 6457 . . . . . . . . . . . . . . . 16 (𝐶 P → ⟨(1st𝐶), (2nd𝐶)⟩ P)
315, 30syl 14 . . . . . . . . . . . . . . 15 (A<P B → ⟨(1st𝐶), (2nd𝐶)⟩ P)
32 elprnql 6463 . . . . . . . . . . . . . . 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 6365 . . . . . . . . . . . 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 5612 . . . . . . . . . 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 6457 . . . . . . . . . . . . . 14 (B P → ⟨(1stB), (2ndB)⟩ P)
4240, 41syl 14 . . . . . . . . . . . . 13 (A<P B → ⟨(1stB), (2ndB)⟩ P)
43 prcdnql 6466 . . . . . . . . . . . . 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 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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