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Theorem addsrpr 6673
Description: Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
Assertion
Ref Expression
addsrpr (((A P B P) (𝐶 P 𝐷 P)) → ([⟨A, B⟩] ~R +R [⟨𝐶, 𝐷⟩] ~R ) = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R )

Proof of Theorem addsrpr
Dummy variables x y z w v u 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opelxpi 4319 . . . 4 ((A P B P) → ⟨A, B (P × P))
2 enrex 6665 . . . . 5 ~R V
32ecelqsi 6096 . . . 4 (⟨A, B (P × P) → [⟨A, B⟩] ~R ((P × P) / ~R ))
41, 3syl 14 . . 3 ((A P B P) → [⟨A, B⟩] ~R ((P × P) / ~R ))
5 opelxpi 4319 . . . 4 ((𝐶 P 𝐷 P) → ⟨𝐶, 𝐷 (P × P))
62ecelqsi 6096 . . . 4 (⟨𝐶, 𝐷 (P × P) → [⟨𝐶, 𝐷⟩] ~R ((P × P) / ~R ))
75, 6syl 14 . . 3 ((𝐶 P 𝐷 P) → [⟨𝐶, 𝐷⟩] ~R ((P × P) / ~R ))
84, 7anim12i 321 . 2 (((A P B P) (𝐶 P 𝐷 P)) → ([⟨A, B⟩] ~R ((P × P) / ~R ) [⟨𝐶, 𝐷⟩] ~R ((P × P) / ~R )))
9 eqid 2037 . . . 4 [⟨A, B⟩] ~R = [⟨A, B⟩] ~R
10 eqid 2037 . . . 4 [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R
119, 10pm3.2i 257 . . 3 ([⟨A, B⟩] ~R = [⟨A, B⟩] ~R [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R )
12 eqid 2037 . . 3 [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R
13 opeq12 3542 . . . . . . . . 9 ((w = A v = B) → ⟨w, v⟩ = ⟨A, B⟩)
1413eceq1d 6078 . . . . . . . 8 ((w = A v = B) → [⟨w, v⟩] ~R = [⟨A, B⟩] ~R )
1514eqeq2d 2048 . . . . . . 7 ((w = A v = B) → ([⟨A, B⟩] ~R = [⟨w, v⟩] ~R ↔ [⟨A, B⟩] ~R = [⟨A, B⟩] ~R ))
1615anbi1d 438 . . . . . 6 ((w = A v = B) → (([⟨A, B⟩] ~R = [⟨w, v⟩] ~R [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) ↔ ([⟨A, B⟩] ~R = [⟨A, B⟩] ~R [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R )))
17 simpl 102 . . . . . . . . . 10 ((w = A v = B) → w = A)
1817oveq1d 5470 . . . . . . . . 9 ((w = A v = B) → (w +P 𝐶) = (A +P 𝐶))
19 simpr 103 . . . . . . . . . 10 ((w = A v = B) → v = B)
2019oveq1d 5470 . . . . . . . . 9 ((w = A v = B) → (v +P 𝐷) = (B +P 𝐷))
2118, 20opeq12d 3548 . . . . . . . 8 ((w = A v = B) → ⟨(w +P 𝐶), (v +P 𝐷)⟩ = ⟨(A +P 𝐶), (B +P 𝐷)⟩)
2221eceq1d 6078 . . . . . . 7 ((w = A v = B) → [⟨(w +P 𝐶), (v +P 𝐷)⟩] ~R = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R )
2322eqeq2d 2048 . . . . . 6 ((w = A v = B) → ([⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P 𝐶), (v +P 𝐷)⟩] ~R ↔ [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R ))
2416, 23anbi12d 442 . . . . 5 ((w = A v = B) → ((([⟨A, B⟩] ~R = [⟨w, v⟩] ~R [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P 𝐶), (v +P 𝐷)⟩] ~R ) ↔ (([⟨A, B⟩] ~R = [⟨A, B⟩] ~R [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R )))
2524spc2egv 2636 . . . 4 ((A P B P) → ((([⟨A, B⟩] ~R = [⟨A, B⟩] ~R [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R ) → wv(([⟨A, B⟩] ~R = [⟨w, v⟩] ~R [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P 𝐶), (v +P 𝐷)⟩] ~R )))
26 opeq12 3542 . . . . . . . . . 10 ((u = 𝐶 𝑡 = 𝐷) → ⟨u, 𝑡⟩ = ⟨𝐶, 𝐷⟩)
2726eceq1d 6078 . . . . . . . . 9 ((u = 𝐶 𝑡 = 𝐷) → [⟨u, 𝑡⟩] ~R = [⟨𝐶, 𝐷⟩] ~R )
2827eqeq2d 2048 . . . . . . . 8 ((u = 𝐶 𝑡 = 𝐷) → ([⟨𝐶, 𝐷⟩] ~R = [⟨u, 𝑡⟩] ~R ↔ [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ))
2928anbi2d 437 . . . . . . 7 ((u = 𝐶 𝑡 = 𝐷) → (([⟨A, B⟩] ~R = [⟨w, v⟩] ~R [⟨𝐶, 𝐷⟩] ~R = [⟨u, 𝑡⟩] ~R ) ↔ ([⟨A, B⟩] ~R = [⟨w, v⟩] ~R [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R )))
30 simpl 102 . . . . . . . . . . 11 ((u = 𝐶 𝑡 = 𝐷) → u = 𝐶)
3130oveq2d 5471 . . . . . . . . . 10 ((u = 𝐶 𝑡 = 𝐷) → (w +P u) = (w +P 𝐶))
32 simpr 103 . . . . . . . . . . 11 ((u = 𝐶 𝑡 = 𝐷) → 𝑡 = 𝐷)
3332oveq2d 5471 . . . . . . . . . 10 ((u = 𝐶 𝑡 = 𝐷) → (v +P 𝑡) = (v +P 𝐷))
3431, 33opeq12d 3548 . . . . . . . . 9 ((u = 𝐶 𝑡 = 𝐷) → ⟨(w +P u), (v +P 𝑡)⟩ = ⟨(w +P 𝐶), (v +P 𝐷)⟩)
3534eceq1d 6078 . . . . . . . 8 ((u = 𝐶 𝑡 = 𝐷) → [⟨(w +P u), (v +P 𝑡)⟩] ~R = [⟨(w +P 𝐶), (v +P 𝐷)⟩] ~R )
3635eqeq2d 2048 . . . . . . 7 ((u = 𝐶 𝑡 = 𝐷) → ([⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P u), (v +P 𝑡)⟩] ~R ↔ [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P 𝐶), (v +P 𝐷)⟩] ~R ))
3729, 36anbi12d 442 . . . . . 6 ((u = 𝐶 𝑡 = 𝐷) → ((([⟨A, B⟩] ~R = [⟨w, v⟩] ~R [⟨𝐶, 𝐷⟩] ~R = [⟨u, 𝑡⟩] ~R ) [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) ↔ (([⟨A, B⟩] ~R = [⟨w, v⟩] ~R [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P 𝐶), (v +P 𝐷)⟩] ~R )))
3837spc2egv 2636 . . . . 5 ((𝐶 P 𝐷 P) → ((([⟨A, B⟩] ~R = [⟨w, v⟩] ~R [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P 𝐶), (v +P 𝐷)⟩] ~R ) → u𝑡(([⟨A, B⟩] ~R = [⟨w, v⟩] ~R [⟨𝐶, 𝐷⟩] ~R = [⟨u, 𝑡⟩] ~R ) [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P u), (v +P 𝑡)⟩] ~R )))
39382eximdv 1759 . . . 4 ((𝐶 P 𝐷 P) → (wv(([⟨A, B⟩] ~R = [⟨w, v⟩] ~R [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P 𝐶), (v +P 𝐷)⟩] ~R ) → wvu𝑡(([⟨A, B⟩] ~R = [⟨w, v⟩] ~R [⟨𝐶, 𝐷⟩] ~R = [⟨u, 𝑡⟩] ~R ) [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P u), (v +P 𝑡)⟩] ~R )))
4025, 39sylan9 389 . . 3 (((A P B P) (𝐶 P 𝐷 P)) → ((([⟨A, B⟩] ~R = [⟨A, B⟩] ~R [⟨𝐶, 𝐷⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ) [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R ) → wvu𝑡(([⟨A, B⟩] ~R = [⟨w, v⟩] ~R [⟨𝐶, 𝐷⟩] ~R = [⟨u, 𝑡⟩] ~R ) [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P u), (v +P 𝑡)⟩] ~R )))
4111, 12, 40mp2ani 408 . 2 (((A P B P) (𝐶 P 𝐷 P)) → wvu𝑡(([⟨A, B⟩] ~R = [⟨w, v⟩] ~R [⟨𝐶, 𝐷⟩] ~R = [⟨u, 𝑡⟩] ~R ) [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P u), (v +P 𝑡)⟩] ~R ))
42 ecexg 6046 . . . 4 ( ~R V → [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R V)
432, 42ax-mp 7 . . 3 [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R V
44 simp1 903 . . . . . . . 8 ((x = [⟨A, B⟩] ~R y = [⟨𝐶, 𝐷⟩] ~R z = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R ) → x = [⟨A, B⟩] ~R )
4544eqeq1d 2045 . . . . . . 7 ((x = [⟨A, B⟩] ~R y = [⟨𝐶, 𝐷⟩] ~R z = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R ) → (x = [⟨w, v⟩] ~R ↔ [⟨A, B⟩] ~R = [⟨w, v⟩] ~R ))
46 simp2 904 . . . . . . . 8 ((x = [⟨A, B⟩] ~R y = [⟨𝐶, 𝐷⟩] ~R z = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R ) → y = [⟨𝐶, 𝐷⟩] ~R )
4746eqeq1d 2045 . . . . . . 7 ((x = [⟨A, B⟩] ~R y = [⟨𝐶, 𝐷⟩] ~R z = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R ) → (y = [⟨u, 𝑡⟩] ~R ↔ [⟨𝐶, 𝐷⟩] ~R = [⟨u, 𝑡⟩] ~R ))
4845, 47anbi12d 442 . . . . . 6 ((x = [⟨A, B⟩] ~R y = [⟨𝐶, 𝐷⟩] ~R z = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R ) → ((x = [⟨w, v⟩] ~R y = [⟨u, 𝑡⟩] ~R ) ↔ ([⟨A, B⟩] ~R = [⟨w, v⟩] ~R [⟨𝐶, 𝐷⟩] ~R = [⟨u, 𝑡⟩] ~R )))
49 simp3 905 . . . . . . 7 ((x = [⟨A, B⟩] ~R y = [⟨𝐶, 𝐷⟩] ~R z = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R ) → z = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R )
5049eqeq1d 2045 . . . . . 6 ((x = [⟨A, B⟩] ~R y = [⟨𝐶, 𝐷⟩] ~R z = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R ) → (z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ↔ [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P u), (v +P 𝑡)⟩] ~R ))
5148, 50anbi12d 442 . . . . 5 ((x = [⟨A, B⟩] ~R y = [⟨𝐶, 𝐷⟩] ~R z = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R ) → (((x = [⟨w, v⟩] ~R y = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) ↔ (([⟨A, B⟩] ~R = [⟨w, v⟩] ~R [⟨𝐶, 𝐷⟩] ~R = [⟨u, 𝑡⟩] ~R ) [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P u), (v +P 𝑡)⟩] ~R )))
52514exbidv 1747 . . . 4 ((x = [⟨A, B⟩] ~R y = [⟨𝐶, 𝐷⟩] ~R z = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R ) → (wvu𝑡((x = [⟨w, v⟩] ~R y = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) ↔ wvu𝑡(([⟨A, B⟩] ~R = [⟨w, v⟩] ~R [⟨𝐶, 𝐷⟩] ~R = [⟨u, 𝑡⟩] ~R ) [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P u), (v +P 𝑡)⟩] ~R )))
53 addsrmo 6671 . . . 4 ((x ((P × P) / ~R ) y ((P × P) / ~R )) → ∃*zwvu𝑡((x = [⟨w, v⟩] ~R y = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ))
54 df-plr 6656 . . . . 5 +R = {⟨⟨x, y⟩, z⟩ ∣ ((x R y R) wvu𝑡((x = [⟨w, v⟩] ~R y = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ))}
55 df-nr 6655 . . . . . . . . 9 R = ((P × P) / ~R )
5655eleq2i 2101 . . . . . . . 8 (x Rx ((P × P) / ~R ))
5755eleq2i 2101 . . . . . . . 8 (y Ry ((P × P) / ~R ))
5856, 57anbi12i 433 . . . . . . 7 ((x R y R) ↔ (x ((P × P) / ~R ) y ((P × P) / ~R )))
5958anbi1i 431 . . . . . 6 (((x R y R) wvu𝑡((x = [⟨w, v⟩] ~R y = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R )) ↔ ((x ((P × P) / ~R ) y ((P × P) / ~R )) wvu𝑡((x = [⟨w, v⟩] ~R y = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R )))
6059oprabbii 5502 . . . . 5 {⟨⟨x, y⟩, z⟩ ∣ ((x R y R) wvu𝑡((x = [⟨w, v⟩] ~R y = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ))} = {⟨⟨x, y⟩, z⟩ ∣ ((x ((P × P) / ~R ) y ((P × P) / ~R )) wvu𝑡((x = [⟨w, v⟩] ~R y = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ))}
6154, 60eqtri 2057 . . . 4 +R = {⟨⟨x, y⟩, z⟩ ∣ ((x ((P × P) / ~R ) y ((P × P) / ~R )) wvu𝑡((x = [⟨w, v⟩] ~R y = [⟨u, 𝑡⟩] ~R ) z = [⟨(w +P u), (v +P 𝑡)⟩] ~R ))}
6252, 53, 61ovig 5564 . . 3 (([⟨A, B⟩] ~R ((P × P) / ~R ) [⟨𝐶, 𝐷⟩] ~R ((P × P) / ~R ) [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R V) → (wvu𝑡(([⟨A, B⟩] ~R = [⟨w, v⟩] ~R [⟨𝐶, 𝐷⟩] ~R = [⟨u, 𝑡⟩] ~R ) [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) → ([⟨A, B⟩] ~R +R [⟨𝐶, 𝐷⟩] ~R ) = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R ))
6343, 62mp3an3 1220 . 2 (([⟨A, B⟩] ~R ((P × P) / ~R ) [⟨𝐶, 𝐷⟩] ~R ((P × P) / ~R )) → (wvu𝑡(([⟨A, B⟩] ~R = [⟨w, v⟩] ~R [⟨𝐶, 𝐷⟩] ~R = [⟨u, 𝑡⟩] ~R ) [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R = [⟨(w +P u), (v +P 𝑡)⟩] ~R ) → ([⟨A, B⟩] ~R +R [⟨𝐶, 𝐷⟩] ~R ) = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R ))
648, 41, 63sylc 56 1 (((A P B P) (𝐶 P 𝐷 P)) → ([⟨A, B⟩] ~R +R [⟨𝐶, 𝐷⟩] ~R ) = [⟨(A +P 𝐶), (B +P 𝐷)⟩] ~R )
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551  cop 3370   × cxp 4286  (class class class)co 5455  {coprab 5456  [cec 6040   / cqs 6041  Pcnp 6275   +P cpp 6277   ~R cer 6280  Rcnr 6281   +R cplr 6285
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 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
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 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6407  df-nq0 6408  df-0nq0 6409  df-plq0 6410  df-mq0 6411  df-inp 6449  df-iplp 6451  df-enr 6654  df-nr 6655  df-plr 6656
This theorem is referenced by:  addclsr  6681  addcomsrg  6683  addasssrg  6684  distrsrg  6687  m1p1sr  6688  0idsr  6695  ltasrg  6698  pitonnlem2  6743
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