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Theorem recexsrlem 11141
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 11106 . . . 4 <R ⊆ (R × R)
21brel 5754 . . 3 (0R <R 𝐴 → (0RR𝐴R))
32simprd 495 . 2 (0R <R 𝐴𝐴R)
4 df-nr 11094 . . 3 R = ((P × P) / ~R )
5 breq2 5152 . . . 4 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → (0R <R [⟨𝑦, 𝑧⟩] ~R ↔ 0R <R 𝐴))
6 oveq1 7438 . . . . . 6 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = (𝐴 ·R 𝑥))
76eqeq1d 2737 . . . . 5 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → (([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R ↔ (𝐴 ·R 𝑥) = 1R))
87rexbidv 3177 . . . 4 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → (∃𝑥R ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R ↔ ∃𝑥R (𝐴 ·R 𝑥) = 1R))
95, 8imbi12d 344 . . 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 11116 . . . . 5 (0R <R [⟨𝑦, 𝑧⟩] ~R𝑧<P 𝑦)
11 ltexpri 11081 . . . . 5 (𝑧<P 𝑦 → ∃𝑤P (𝑧 +P 𝑤) = 𝑦)
1210, 11sylbi 217 . . . 4 (0R <R [⟨𝑦, 𝑧⟩] ~R → ∃𝑤P (𝑧 +P 𝑤) = 𝑦)
13 recexpr 11089 . . . . . 6 (𝑤P → ∃𝑣P (𝑤 ·P 𝑣) = 1P)
14 1pr 11053 . . . . . . . . . . . 12 1PP
15 addclpr 11056 . . . . . . . . . . . 12 ((𝑣P ∧ 1PP) → (𝑣 +P 1P) ∈ P)
1614, 15mpan2 691 . . . . . . . . . . 11 (𝑣P → (𝑣 +P 1P) ∈ P)
17 enrex 11105 . . . . . . . . . . . 12 ~R ∈ V
1817, 4ecopqsi 8813 . . . . . . . . . . 11 (((𝑣 +P 1P) ∈ P ∧ 1PP) → [⟨(𝑣 +P 1P), 1P⟩] ~RR)
1916, 14, 18sylancl 586 . . . . . . . . . 10 (𝑣P → [⟨(𝑣 +P 1P), 1P⟩] ~RR)
2019ad2antlr 727 . . . . . . . . 9 ((((𝑦P𝑧P) ∧ 𝑣P) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → [⟨(𝑣 +P 1P), 1P⟩] ~RR)
2116, 14jctir 520 . . . . . . . . . . . . . 14 (𝑣P → ((𝑣 +P 1P) ∈ P ∧ 1PP))
2221anim2i 617 . . . . . . . . . . . . 13 (((𝑦P𝑧P) ∧ 𝑣P) → ((𝑦P𝑧P) ∧ ((𝑣 +P 1P) ∈ P ∧ 1PP)))
2322adantr 480 . . . . . . . . . . . 12 ((((𝑦P𝑧P) ∧ 𝑣P) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ((𝑦P𝑧P) ∧ ((𝑣 +P 1P) ∈ P ∧ 1PP)))
24 mulsrpr 11114 . . . . . . . . . . . 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 7438 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 +P 𝑤) = 𝑦 → ((𝑧 +P 𝑤) ·P 𝑣) = (𝑦 ·P 𝑣))
2726eqcomd 2741 . . . . . . . . . . . . . . . . . . 19 ((𝑧 +P 𝑤) = 𝑦 → (𝑦 ·P 𝑣) = ((𝑧 +P 𝑤) ·P 𝑣))
28 vex 3482 . . . . . . . . . . . . . . . . . . . . 21 𝑧 ∈ V
29 vex 3482 . . . . . . . . . . . . . . . . . . . . 21 𝑤 ∈ V
30 vex 3482 . . . . . . . . . . . . . . . . . . . . 21 𝑣 ∈ V
31 mulcompr 11061 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 ·P 𝑓) = (𝑓 ·P 𝑢)
32 distrpr 11066 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 ·P (𝑓 +P 𝑥)) = ((𝑢 ·P 𝑓) +P (𝑢 ·P 𝑥))
3328, 29, 30, 31, 32caovdir 7667 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 +P 𝑤) ·P 𝑣) = ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑣))
34 oveq2 7439 . . . . . . . . . . . . . . . . . . . 20 ((𝑤 ·P 𝑣) = 1P → ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑣)) = ((𝑧 ·P 𝑣) +P 1P))
3533, 34eqtrid 2787 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ·P 𝑣) = 1P → ((𝑧 +P 𝑤) ·P 𝑣) = ((𝑧 ·P 𝑣) +P 1P))
3627, 35sylan9eqr 2797 . . . . . . . . . . . . . . . . . 18 (((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦) → (𝑦 ·P 𝑣) = ((𝑧 ·P 𝑣) +P 1P))
3736oveq1d 7446 . . . . . . . . . . . . . . . . 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 7464 . . . . . . . . . . . . . . . . . 18 (𝑧 ·P 𝑣) ∈ V
3914elexi 3501 . . . . . . . . . . . . . . . . . 18 1P ∈ V
40 ovex 7464 . . . . . . . . . . . . . . . . . 18 ((𝑦 ·P 1P) +P (𝑧 ·P 1P)) ∈ V
41 addcompr 11059 . . . . . . . . . . . . . . . . . 18 (𝑢 +P 𝑓) = (𝑓 +P 𝑢)
42 addasspr 11060 . . . . . . . . . . . . . . . . . 18 ((𝑢 +P 𝑓) +P 𝑥) = (𝑢 +P (𝑓 +P 𝑥))
4338, 39, 40, 41, 42caov32 7660 . . . . . . . . . . . . . . . . 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 2791 . . . . . . . . . . . . . . . 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 7446 . . . . . . . . . . . . . . 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 11060 . . . . . . . . . . . . . . 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 2791 . . . . . . . . . . . . . 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 11066 . . . . . . . . . . . . . . . . 17 (𝑦 ·P (𝑣 +P 1P)) = ((𝑦 ·P 𝑣) +P (𝑦 ·P 1P))
4948oveq1i 7441 . . . . . . . . . . . . . . . 16 ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) = (((𝑦 ·P 𝑣) +P (𝑦 ·P 1P)) +P (𝑧 ·P 1P))
50 addasspr 11060 . . . . . . . . . . . . . . . 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 2763 . . . . . . . . . . . . . . 15 ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) = ((𝑦 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P)))
5251oveq1i 7441 . . . . . . . . . . . . . 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 11066 . . . . . . . . . . . . . . . . 17 (𝑧 ·P (𝑣 +P 1P)) = ((𝑧 ·P 𝑣) +P (𝑧 ·P 1P))
5453oveq2i 7442 . . . . . . . . . . . . . . . 16 ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) = ((𝑦 ·P 1P) +P ((𝑧 ·P 𝑣) +P (𝑧 ·P 1P)))
55 ovex 7464 . . . . . . . . . . . . . . . . 17 (𝑦 ·P 1P) ∈ V
56 ovex 7464 . . . . . . . . . . . . . . . . 17 (𝑧 ·P 1P) ∈ V
5755, 38, 56, 41, 42caov12 7661 . . . . . . . . . . . . . . . 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 2763 . . . . . . . . . . . . . . 15 ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) = ((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P)))
5958oveq1i 7441 . . . . . . . . . . . . . 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 2800 . . . . . . . . . . . . 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 11058 . . . . . . . . . . . . . . . . . 18 ((𝑦P ∧ (𝑣 +P 1P) ∈ P) → (𝑦 ·P (𝑣 +P 1P)) ∈ P)
6216, 61sylan2 593 . . . . . . . . . . . . . . . . 17 ((𝑦P𝑣P) → (𝑦 ·P (𝑣 +P 1P)) ∈ P)
63 mulclpr 11058 . . . . . . . . . . . . . . . . . 18 ((𝑧P ∧ 1PP) → (𝑧 ·P 1P) ∈ P)
6414, 63mpan2 691 . . . . . . . . . . . . . . . . 17 (𝑧P → (𝑧 ·P 1P) ∈ P)
65 addclpr 11056 . . . . . . . . . . . . . . . . 17 (((𝑦 ·P (𝑣 +P 1P)) ∈ P ∧ (𝑧 ·P 1P) ∈ P) → ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) ∈ P)
6662, 64, 65syl2an 596 . . . . . . . . . . . . . . . 16 (((𝑦P𝑣P) ∧ 𝑧P) → ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) ∈ P)
6766an32s 652 . . . . . . . . . . . . . . 15 (((𝑦P𝑧P) ∧ 𝑣P) → ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) ∈ P)
68 mulclpr 11058 . . . . . . . . . . . . . . . . . 18 ((𝑦P ∧ 1PP) → (𝑦 ·P 1P) ∈ P)
6914, 68mpan2 691 . . . . . . . . . . . . . . . . 17 (𝑦P → (𝑦 ·P 1P) ∈ P)
70 mulclpr 11058 . . . . . . . . . . . . . . . . . 18 ((𝑧P ∧ (𝑣 +P 1P) ∈ P) → (𝑧 ·P (𝑣 +P 1P)) ∈ P)
7116, 70sylan2 593 . . . . . . . . . . . . . . . . 17 ((𝑧P𝑣P) → (𝑧 ·P (𝑣 +P 1P)) ∈ P)
72 addclpr 11056 . . . . . . . . . . . . . . . . 17 (((𝑦 ·P 1P) ∈ P ∧ (𝑧 ·P (𝑣 +P 1P)) ∈ P) → ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) ∈ P)
7369, 71, 72syl2an 596 . . . . . . . . . . . . . . . 16 ((𝑦P ∧ (𝑧P𝑣P)) → ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) ∈ P)
7473anassrs 467 . . . . . . . . . . . . . . 15 (((𝑦P𝑧P) ∧ 𝑣P) → ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) ∈ P)
7567, 74jca 511 . . . . . . . . . . . . . 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 11056 . . . . . . . . . . . . . . . 16 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
7714, 14, 76mp2an 692 . . . . . . . . . . . . . . 15 (1P +P 1P) ∈ P
7877, 14pm3.2i 470 . . . . . . . . . . . . . 14 ((1P +P 1P) ∈ P ∧ 1PP)
79 enreceq 11104 . . . . . . . . . . . . . 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 586 . . . . . . . . . . . . 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 246 . . . . . . . . . . . 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 406 . . . . . . . . . . 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 2775 . . . . . . . . . 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 11099 . . . . . . . . . 10 1R = [⟨(1P +P 1P), 1P⟩] ~R
8583, 84eqtr4di 2793 . . . . . . . . 9 ((((𝑦P𝑧P) ∧ 𝑣P) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ) = 1R)
86 oveq2 7439 . . . . . . . . . . 11 (𝑥 = [⟨(𝑣 +P 1P), 1P⟩] ~R → ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ))
8786eqeq1d 2737 . . . . . . . . . 10 (𝑥 = [⟨(𝑣 +P 1P), 1P⟩] ~R → (([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R ↔ ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ) = 1R))
8887rspcev 3622 . . . . . . . . 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 584 . . . . . . . 8 ((((𝑦P𝑧P) ∧ 𝑣P) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ∃𝑥R ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R)
9089exp43 436 . . . . . . 7 ((𝑦P𝑧P) → (𝑣P → ((𝑤 ·P 𝑣) = 1P → ((𝑧 +P 𝑤) = 𝑦 → ∃𝑥R ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R))))
9190rexlimdv 3151 . . . . . 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 3151 . . . 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 8846 . 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 206  wa 395   = wceq 1537  wcel 2106  wrex 3068  cop 4637   class class class wbr 5148  (class class class)co 7431  [cec 8742  Pcnp 10897  1Pc1p 10898   +P cpp 10899   ·P cmp 10900  <P cltp 10901   ~R cer 10902  Rcnr 10903  0Rc0r 10904  1Rc1r 10905   ·R cmr 10908   <R cltr 10909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-omul 8510  df-er 8744  df-ec 8746  df-qs 8750  df-ni 10910  df-pli 10911  df-mi 10912  df-lti 10913  df-plpq 10946  df-mpq 10947  df-ltpq 10948  df-enq 10949  df-nq 10950  df-erq 10951  df-plq 10952  df-mq 10953  df-1nq 10954  df-rq 10955  df-ltnq 10956  df-np 11019  df-1p 11020  df-plp 11021  df-mp 11022  df-ltp 11023  df-enr 11093  df-nr 11094  df-mr 11096  df-ltr 11097  df-0r 11098  df-1r 11099
This theorem is referenced by:  recexsr  11145
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