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Theorem recexsrlem 11128
Description: The reciprocal of a positive signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
Assertion
Ref Expression
recexsrlem (0R <R 𝐴 → ∃𝑥R (𝐴 ·R 𝑥) = 1R)
Distinct variable group:   𝑥,𝐴

Proof of Theorem recexsrlem
Dummy variables 𝑦 𝑧 𝑤 𝑣 𝑢 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelsr 11093 . . . 4 <R ⊆ (R × R)
21brel 5743 . . 3 (0R <R 𝐴 → (0RR𝐴R))
32simprd 494 . 2 (0R <R 𝐴𝐴R)
4 df-nr 11081 . . 3 R = ((P × P) / ~R )
5 breq2 5153 . . . 4 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → (0R <R [⟨𝑦, 𝑧⟩] ~R ↔ 0R <R 𝐴))
6 oveq1 7426 . . . . . 6 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = (𝐴 ·R 𝑥))
76eqeq1d 2727 . . . . 5 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → (([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R ↔ (𝐴 ·R 𝑥) = 1R))
87rexbidv 3168 . . . 4 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → (∃𝑥R ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R ↔ ∃𝑥R (𝐴 ·R 𝑥) = 1R))
95, 8imbi12d 343 . . 3 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → ((0R <R [⟨𝑦, 𝑧⟩] ~R → ∃𝑥R ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R) ↔ (0R <R 𝐴 → ∃𝑥R (𝐴 ·R 𝑥) = 1R)))
10 gt0srpr 11103 . . . . 5 (0R <R [⟨𝑦, 𝑧⟩] ~R𝑧<P 𝑦)
11 ltexpri 11068 . . . . 5 (𝑧<P 𝑦 → ∃𝑤P (𝑧 +P 𝑤) = 𝑦)
1210, 11sylbi 216 . . . 4 (0R <R [⟨𝑦, 𝑧⟩] ~R → ∃𝑤P (𝑧 +P 𝑤) = 𝑦)
13 recexpr 11076 . . . . . 6 (𝑤P → ∃𝑣P (𝑤 ·P 𝑣) = 1P)
14 1pr 11040 . . . . . . . . . . . 12 1PP
15 addclpr 11043 . . . . . . . . . . . 12 ((𝑣P ∧ 1PP) → (𝑣 +P 1P) ∈ P)
1614, 15mpan2 689 . . . . . . . . . . 11 (𝑣P → (𝑣 +P 1P) ∈ P)
17 enrex 11092 . . . . . . . . . . . 12 ~R ∈ V
1817, 4ecopqsi 8793 . . . . . . . . . . 11 (((𝑣 +P 1P) ∈ P ∧ 1PP) → [⟨(𝑣 +P 1P), 1P⟩] ~RR)
1916, 14, 18sylancl 584 . . . . . . . . . 10 (𝑣P → [⟨(𝑣 +P 1P), 1P⟩] ~RR)
2019ad2antlr 725 . . . . . . . . 9 ((((𝑦P𝑧P) ∧ 𝑣P) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → [⟨(𝑣 +P 1P), 1P⟩] ~RR)
2116, 14jctir 519 . . . . . . . . . . . . . 14 (𝑣P → ((𝑣 +P 1P) ∈ P ∧ 1PP))
2221anim2i 615 . . . . . . . . . . . . 13 (((𝑦P𝑧P) ∧ 𝑣P) → ((𝑦P𝑧P) ∧ ((𝑣 +P 1P) ∈ P ∧ 1PP)))
2322adantr 479 . . . . . . . . . . . 12 ((((𝑦P𝑧P) ∧ 𝑣P) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ((𝑦P𝑧P) ∧ ((𝑣 +P 1P) ∈ P ∧ 1PP)))
24 mulsrpr 11101 . . . . . . . . . . . 12 (((𝑦P𝑧P) ∧ ((𝑣 +P 1P) ∈ P ∧ 1PP)) → ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ) = [⟨((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)), ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P)))⟩] ~R )
2523, 24syl 17 . . . . . . . . . . 11 ((((𝑦P𝑧P) ∧ 𝑣P) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ) = [⟨((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)), ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P)))⟩] ~R )
26 oveq1 7426 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 +P 𝑤) = 𝑦 → ((𝑧 +P 𝑤) ·P 𝑣) = (𝑦 ·P 𝑣))
2726eqcomd 2731 . . . . . . . . . . . . . . . . . . 19 ((𝑧 +P 𝑤) = 𝑦 → (𝑦 ·P 𝑣) = ((𝑧 +P 𝑤) ·P 𝑣))
28 vex 3465 . . . . . . . . . . . . . . . . . . . . 21 𝑧 ∈ V
29 vex 3465 . . . . . . . . . . . . . . . . . . . . 21 𝑤 ∈ V
30 vex 3465 . . . . . . . . . . . . . . . . . . . . 21 𝑣 ∈ V
31 mulcompr 11048 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 ·P 𝑓) = (𝑓 ·P 𝑢)
32 distrpr 11053 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 ·P (𝑓 +P 𝑥)) = ((𝑢 ·P 𝑓) +P (𝑢 ·P 𝑥))
3328, 29, 30, 31, 32caovdir 7655 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 +P 𝑤) ·P 𝑣) = ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑣))
34 oveq2 7427 . . . . . . . . . . . . . . . . . . . 20 ((𝑤 ·P 𝑣) = 1P → ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑣)) = ((𝑧 ·P 𝑣) +P 1P))
3533, 34eqtrid 2777 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ·P 𝑣) = 1P → ((𝑧 +P 𝑤) ·P 𝑣) = ((𝑧 ·P 𝑣) +P 1P))
3627, 35sylan9eqr 2787 . . . . . . . . . . . . . . . . . 18 (((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦) → (𝑦 ·P 𝑣) = ((𝑧 ·P 𝑣) +P 1P))
3736oveq1d 7434 . . . . . . . . . . . . . . . . 17 (((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦) → ((𝑦 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) = (((𝑧 ·P 𝑣) +P 1P) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))))
38 ovex 7452 . . . . . . . . . . . . . . . . . 18 (𝑧 ·P 𝑣) ∈ V
3914elexi 3482 . . . . . . . . . . . . . . . . . 18 1P ∈ V
40 ovex 7452 . . . . . . . . . . . . . . . . . 18 ((𝑦 ·P 1P) +P (𝑧 ·P 1P)) ∈ V
41 addcompr 11046 . . . . . . . . . . . . . . . . . 18 (𝑢 +P 𝑓) = (𝑓 +P 𝑢)
42 addasspr 11047 . . . . . . . . . . . . . . . . . 18 ((𝑢 +P 𝑓) +P 𝑥) = (𝑢 +P (𝑓 +P 𝑥))
4338, 39, 40, 41, 42caov32 7648 . . . . . . . . . . . . . . . . 17 (((𝑧 ·P 𝑣) +P 1P) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) = (((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) +P 1P)
4437, 43eqtrdi 2781 . . . . . . . . . . . . . . . 16 (((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦) → ((𝑦 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) = (((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) +P 1P))
4544oveq1d 7434 . . . . . . . . . . . . . . 15 (((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦) → (((𝑦 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) +P 1P) = ((((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) +P 1P) +P 1P))
46 addasspr 11047 . . . . . . . . . . . . . . 15 ((((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) +P 1P) +P 1P) = (((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) +P (1P +P 1P))
4745, 46eqtrdi 2781 . . . . . . . . . . . . . 14 (((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦) → (((𝑦 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) +P 1P) = (((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) +P (1P +P 1P)))
48 distrpr 11053 . . . . . . . . . . . . . . . . 17 (𝑦 ·P (𝑣 +P 1P)) = ((𝑦 ·P 𝑣) +P (𝑦 ·P 1P))
4948oveq1i 7429 . . . . . . . . . . . . . . . 16 ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) = (((𝑦 ·P 𝑣) +P (𝑦 ·P 1P)) +P (𝑧 ·P 1P))
50 addasspr 11047 . . . . . . . . . . . . . . . 16 (((𝑦 ·P 𝑣) +P (𝑦 ·P 1P)) +P (𝑧 ·P 1P)) = ((𝑦 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P)))
5149, 50eqtri 2753 . . . . . . . . . . . . . . 15 ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) = ((𝑦 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P)))
5251oveq1i 7429 . . . . . . . . . . . . . 14 (((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) +P 1P) = (((𝑦 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) +P 1P)
53 distrpr 11053 . . . . . . . . . . . . . . . . 17 (𝑧 ·P (𝑣 +P 1P)) = ((𝑧 ·P 𝑣) +P (𝑧 ·P 1P))
5453oveq2i 7430 . . . . . . . . . . . . . . . 16 ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) = ((𝑦 ·P 1P) +P ((𝑧 ·P 𝑣) +P (𝑧 ·P 1P)))
55 ovex 7452 . . . . . . . . . . . . . . . . 17 (𝑦 ·P 1P) ∈ V
56 ovex 7452 . . . . . . . . . . . . . . . . 17 (𝑧 ·P 1P) ∈ V
5755, 38, 56, 41, 42caov12 7649 . . . . . . . . . . . . . . . 16 ((𝑦 ·P 1P) +P ((𝑧 ·P 𝑣) +P (𝑧 ·P 1P))) = ((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P)))
5854, 57eqtri 2753 . . . . . . . . . . . . . . 15 ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) = ((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P)))
5958oveq1i 7429 . . . . . . . . . . . . . 14 (((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) +P (1P +P 1P)) = (((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) +P (1P +P 1P))
6047, 52, 593eqtr4g 2790 . . . . . . . . . . . . 13 (((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦) → (((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) +P 1P) = (((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) +P (1P +P 1P)))
61 mulclpr 11045 . . . . . . . . . . . . . . . . . 18 ((𝑦P ∧ (𝑣 +P 1P) ∈ P) → (𝑦 ·P (𝑣 +P 1P)) ∈ P)
6216, 61sylan2 591 . . . . . . . . . . . . . . . . 17 ((𝑦P𝑣P) → (𝑦 ·P (𝑣 +P 1P)) ∈ P)
63 mulclpr 11045 . . . . . . . . . . . . . . . . . 18 ((𝑧P ∧ 1PP) → (𝑧 ·P 1P) ∈ P)
6414, 63mpan2 689 . . . . . . . . . . . . . . . . 17 (𝑧P → (𝑧 ·P 1P) ∈ P)
65 addclpr 11043 . . . . . . . . . . . . . . . . 17 (((𝑦 ·P (𝑣 +P 1P)) ∈ P ∧ (𝑧 ·P 1P) ∈ P) → ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) ∈ P)
6662, 64, 65syl2an 594 . . . . . . . . . . . . . . . 16 (((𝑦P𝑣P) ∧ 𝑧P) → ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) ∈ P)
6766an32s 650 . . . . . . . . . . . . . . 15 (((𝑦P𝑧P) ∧ 𝑣P) → ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) ∈ P)
68 mulclpr 11045 . . . . . . . . . . . . . . . . . 18 ((𝑦P ∧ 1PP) → (𝑦 ·P 1P) ∈ P)
6914, 68mpan2 689 . . . . . . . . . . . . . . . . 17 (𝑦P → (𝑦 ·P 1P) ∈ P)
70 mulclpr 11045 . . . . . . . . . . . . . . . . . 18 ((𝑧P ∧ (𝑣 +P 1P) ∈ P) → (𝑧 ·P (𝑣 +P 1P)) ∈ P)
7116, 70sylan2 591 . . . . . . . . . . . . . . . . 17 ((𝑧P𝑣P) → (𝑧 ·P (𝑣 +P 1P)) ∈ P)
72 addclpr 11043 . . . . . . . . . . . . . . . . 17 (((𝑦 ·P 1P) ∈ P ∧ (𝑧 ·P (𝑣 +P 1P)) ∈ P) → ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) ∈ P)
7369, 71, 72syl2an 594 . . . . . . . . . . . . . . . 16 ((𝑦P ∧ (𝑧P𝑣P)) → ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) ∈ P)
7473anassrs 466 . . . . . . . . . . . . . . 15 (((𝑦P𝑧P) ∧ 𝑣P) → ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) ∈ P)
7567, 74jca 510 . . . . . . . . . . . . . 14 (((𝑦P𝑧P) ∧ 𝑣P) → (((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) ∈ P ∧ ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) ∈ P))
76 addclpr 11043 . . . . . . . . . . . . . . . 16 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
7714, 14, 76mp2an 690 . . . . . . . . . . . . . . 15 (1P +P 1P) ∈ P
7877, 14pm3.2i 469 . . . . . . . . . . . . . 14 ((1P +P 1P) ∈ P ∧ 1PP)
79 enreceq 11091 . . . . . . . . . . . . . 14 (((((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) ∈ P ∧ ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) ∈ P) ∧ ((1P +P 1P) ∈ P ∧ 1PP)) → ([⟨((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)), ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P)))⟩] ~R = [⟨(1P +P 1P), 1P⟩] ~R ↔ (((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) +P 1P) = (((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) +P (1P +P 1P))))
8075, 78, 79sylancl 584 . . . . . . . . . . . . 13 (((𝑦P𝑧P) ∧ 𝑣P) → ([⟨((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)), ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P)))⟩] ~R = [⟨(1P +P 1P), 1P⟩] ~R ↔ (((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) +P 1P) = (((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) +P (1P +P 1P))))
8160, 80imbitrrid 245 . . . . . . . . . . . 12 (((𝑦P𝑧P) ∧ 𝑣P) → (((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦) → [⟨((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)), ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P)))⟩] ~R = [⟨(1P +P 1P), 1P⟩] ~R ))
8281imp 405 . . . . . . . . . . 11 ((((𝑦P𝑧P) ∧ 𝑣P) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → [⟨((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)), ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P)))⟩] ~R = [⟨(1P +P 1P), 1P⟩] ~R )
8325, 82eqtrd 2765 . . . . . . . . . 10 ((((𝑦P𝑧P) ∧ 𝑣P) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ) = [⟨(1P +P 1P), 1P⟩] ~R )
84 df-1r 11086 . . . . . . . . . 10 1R = [⟨(1P +P 1P), 1P⟩] ~R
8583, 84eqtr4di 2783 . . . . . . . . 9 ((((𝑦P𝑧P) ∧ 𝑣P) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ) = 1R)
86 oveq2 7427 . . . . . . . . . . 11 (𝑥 = [⟨(𝑣 +P 1P), 1P⟩] ~R → ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ))
8786eqeq1d 2727 . . . . . . . . . 10 (𝑥 = [⟨(𝑣 +P 1P), 1P⟩] ~R → (([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R ↔ ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ) = 1R))
8887rspcev 3606 . . . . . . . . 9 (([⟨(𝑣 +P 1P), 1P⟩] ~RR ∧ ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ) = 1R) → ∃𝑥R ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R)
8920, 85, 88syl2anc 582 . . . . . . . 8 ((((𝑦P𝑧P) ∧ 𝑣P) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ∃𝑥R ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R)
9089exp43 435 . . . . . . 7 ((𝑦P𝑧P) → (𝑣P → ((𝑤 ·P 𝑣) = 1P → ((𝑧 +P 𝑤) = 𝑦 → ∃𝑥R ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R))))
9190rexlimdv 3142 . . . . . 6 ((𝑦P𝑧P) → (∃𝑣P (𝑤 ·P 𝑣) = 1P → ((𝑧 +P 𝑤) = 𝑦 → ∃𝑥R ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R)))
9213, 91syl5 34 . . . . 5 ((𝑦P𝑧P) → (𝑤P → ((𝑧 +P 𝑤) = 𝑦 → ∃𝑥R ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R)))
9392rexlimdv 3142 . . . 4 ((𝑦P𝑧P) → (∃𝑤P (𝑧 +P 𝑤) = 𝑦 → ∃𝑥R ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R))
9412, 93syl5 34 . . 3 ((𝑦P𝑧P) → (0R <R [⟨𝑦, 𝑧⟩] ~R → ∃𝑥R ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R))
954, 9, 94ecoptocl 8826 . 2 (𝐴R → (0R <R 𝐴 → ∃𝑥R (𝐴 ·R 𝑥) = 1R))
963, 95mpcom 38 1 (0R <R 𝐴 → ∃𝑥R (𝐴 ·R 𝑥) = 1R)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wrex 3059  cop 4636   class class class wbr 5149  (class class class)co 7419  [cec 8723  Pcnp 10884  1Pc1p 10885   +P cpp 10886   ·P cmp 10887  <P cltp 10888   ~R cer 10889  Rcnr 10890  0Rc0r 10891  1Rc1r 10892   ·R cmr 10895   <R cltr 10896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9666
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-oadd 8491  df-omul 8492  df-er 8725  df-ec 8727  df-qs 8731  df-ni 10897  df-pli 10898  df-mi 10899  df-lti 10900  df-plpq 10933  df-mpq 10934  df-ltpq 10935  df-enq 10936  df-nq 10937  df-erq 10938  df-plq 10939  df-mq 10940  df-1nq 10941  df-rq 10942  df-ltnq 10943  df-np 11006  df-1p 11007  df-plp 11008  df-mp 11009  df-ltp 11010  df-enr 11080  df-nr 11081  df-mr 11083  df-ltr 11084  df-0r 11085  df-1r 11086
This theorem is referenced by:  recexsr  11132
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