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Theorem ltexpri 6587
 Description: Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 13-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.)
Assertion
Ref Expression
ltexpri (A<P Bx P (A +P x) = B)
Distinct variable groups:   x,A   x,B

Proof of Theorem ltexpri
Dummy variables y z u v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 103 . . . . . . . 8 ((y = u z = v) → z = v)
21eleq1d 2103 . . . . . . 7 ((y = u z = v) → (z (2ndA) ↔ v (2ndA)))
3 simpl 102 . . . . . . . . 9 ((y = u z = v) → y = u)
41, 3oveq12d 5473 . . . . . . . 8 ((y = u z = v) → (z +Q y) = (v +Q u))
54eleq1d 2103 . . . . . . 7 ((y = u z = v) → ((z +Q y) (1stB) ↔ (v +Q u) (1stB)))
62, 5anbi12d 442 . . . . . 6 ((y = u z = v) → ((z (2ndA) (z +Q y) (1stB)) ↔ (v (2ndA) (v +Q u) (1stB))))
76cbvexdva 1801 . . . . 5 (y = u → (z(z (2ndA) (z +Q y) (1stB)) ↔ v(v (2ndA) (v +Q u) (1stB))))
87cbvrabv 2550 . . . 4 {y Qz(z (2ndA) (z +Q y) (1stB))} = {u Qv(v (2ndA) (v +Q u) (1stB))}
91eleq1d 2103 . . . . . . 7 ((y = u z = v) → (z (1stA) ↔ v (1stA)))
104eleq1d 2103 . . . . . . 7 ((y = u z = v) → ((z +Q y) (2ndB) ↔ (v +Q u) (2ndB)))
119, 10anbi12d 442 . . . . . 6 ((y = u z = v) → ((z (1stA) (z +Q y) (2ndB)) ↔ (v (1stA) (v +Q u) (2ndB))))
1211cbvexdva 1801 . . . . 5 (y = u → (z(z (1stA) (z +Q y) (2ndB)) ↔ v(v (1stA) (v +Q u) (2ndB))))
1312cbvrabv 2550 . . . 4 {y Qz(z (1stA) (z +Q y) (2ndB))} = {u Qv(v (1stA) (v +Q u) (2ndB))}
148, 13opeq12i 3545 . . 3 ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩ = ⟨{u Qv(v (2ndA) (v +Q u) (1stB))}, {u Qv(v (1stA) (v +Q u) (2ndB))}⟩
1514ltexprlempr 6582 . 2 (A<P B → ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩ P)
1614ltexprlemfl 6583 . . . 4 (A<P B → (1st ‘(A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩)) ⊆ (1stB))
1714ltexprlemrl 6584 . . . 4 (A<P B → (1stB) ⊆ (1st ‘(A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩)))
1816, 17eqssd 2956 . . 3 (A<P B → (1st ‘(A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩)) = (1stB))
1914ltexprlemfu 6585 . . . 4 (A<P B → (2nd ‘(A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩)) ⊆ (2ndB))
2014ltexprlemru 6586 . . . 4 (A<P B → (2ndB) ⊆ (2nd ‘(A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩)))
2119, 20eqssd 2956 . . 3 (A<P B → (2nd ‘(A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩)) = (2ndB))
22 ltrelpr 6488 . . . . . . 7 <P ⊆ (P × P)
2322brel 4335 . . . . . 6 (A<P B → (A P B P))
2423simpld 105 . . . . 5 (A<P BA P)
25 addclpr 6520 . . . . 5 ((A P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩ P) → (A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩) P)
2624, 15, 25syl2anc 391 . . . 4 (A<P B → (A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩) P)
2723simprd 107 . . . 4 (A<P BB P)
28 preqlu 6455 . . . 4 (((A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩) P B P) → ((A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩) = B ↔ ((1st ‘(A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩)) = (1stB) (2nd ‘(A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩)) = (2ndB))))
2926, 27, 28syl2anc 391 . . 3 (A<P B → ((A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩) = B ↔ ((1st ‘(A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩)) = (1stB) (2nd ‘(A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩)) = (2ndB))))
3018, 21, 29mpbir2and 850 . 2 (A<P B → (A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩) = B)
31 oveq2 5463 . . . 4 (x = ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩ → (A +P x) = (A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩))
3231eqeq1d 2045 . . 3 (x = ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩ → ((A +P x) = B ↔ (A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩) = B))
3332rspcev 2650 . 2 ((⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩ P (A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩) = B) → x P (A +P x) = B)
3415, 30, 33syl2anc 391 1 (A<P Bx P (A +P x) = B)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃wrex 2301  {crab 2304  ⟨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  Pcnp 6275   +P cpp 6277
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