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Theorem recexsrlem 9962
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 9927 . . . 4 <R ⊆ (R × R)
21brel 5202 . . 3 (0R <R 𝐴 → (0RR𝐴R))
32simprd 478 . 2 (0R <R 𝐴𝐴R)
4 df-nr 9916 . . 3 R = ((P × P) / ~R )
5 breq2 4689 . . . 4 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → (0R <R [⟨𝑦, 𝑧⟩] ~R ↔ 0R <R 𝐴))
6 oveq1 6697 . . . . . 6 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = (𝐴 ·R 𝑥))
76eqeq1d 2653 . . . . 5 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → (([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R ↔ (𝐴 ·R 𝑥) = 1R))
87rexbidv 3081 . . . 4 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → (∃𝑥R ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R ↔ ∃𝑥R (𝐴 ·R 𝑥) = 1R))
95, 8imbi12d 333 . . 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 9937 . . . . 5 (0R <R [⟨𝑦, 𝑧⟩] ~R𝑧<P 𝑦)
11 ltexpri 9903 . . . . 5 (𝑧<P 𝑦 → ∃𝑤P (𝑧 +P 𝑤) = 𝑦)
1210, 11sylbi 207 . . . 4 (0R <R [⟨𝑦, 𝑧⟩] ~R → ∃𝑤P (𝑧 +P 𝑤) = 𝑦)
13 recexpr 9911 . . . . . 6 (𝑤P → ∃𝑣P (𝑤 ·P 𝑣) = 1P)
14 1pr 9875 . . . . . . . . . . . 12 1PP
15 addclpr 9878 . . . . . . . . . . . 12 ((𝑣P ∧ 1PP) → (𝑣 +P 1P) ∈ P)
1614, 15mpan2 707 . . . . . . . . . . 11 (𝑣P → (𝑣 +P 1P) ∈ P)
17 enrex 9926 . . . . . . . . . . . 12 ~R ∈ V
1817, 4ecopqsi 7847 . . . . . . . . . . 11 (((𝑣 +P 1P) ∈ P ∧ 1PP) → [⟨(𝑣 +P 1P), 1P⟩] ~RR)
1916, 14, 18sylancl 695 . . . . . . . . . 10 (𝑣P → [⟨(𝑣 +P 1P), 1P⟩] ~RR)
2019ad2antlr 763 . . . . . . . . 9 ((((𝑦P𝑧P) ∧ 𝑣P) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → [⟨(𝑣 +P 1P), 1P⟩] ~RR)
2116, 14jctir 560 . . . . . . . . . . . . . 14 (𝑣P → ((𝑣 +P 1P) ∈ P ∧ 1PP))
2221anim2i 592 . . . . . . . . . . . . 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 9935 . . . . . . . . . . . 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 6697 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 +P 𝑤) = 𝑦 → ((𝑧 +P 𝑤) ·P 𝑣) = (𝑦 ·P 𝑣))
2726eqcomd 2657 . . . . . . . . . . . . . . . . . . 19 ((𝑧 +P 𝑤) = 𝑦 → (𝑦 ·P 𝑣) = ((𝑧 +P 𝑤) ·P 𝑣))
28 vex 3234 . . . . . . . . . . . . . . . . . . . . 21 𝑧 ∈ V
29 vex 3234 . . . . . . . . . . . . . . . . . . . . 21 𝑤 ∈ V
30 vex 3234 . . . . . . . . . . . . . . . . . . . . 21 𝑣 ∈ V
31 mulcompr 9883 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 ·P 𝑓) = (𝑓 ·P 𝑢)
32 distrpr 9888 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 ·P (𝑓 +P 𝑥)) = ((𝑢 ·P 𝑓) +P (𝑢 ·P 𝑥))
3328, 29, 30, 31, 32caovdir 6910 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 +P 𝑤) ·P 𝑣) = ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑣))
34 oveq2 6698 . . . . . . . . . . . . . . . . . . . 20 ((𝑤 ·P 𝑣) = 1P → ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑣)) = ((𝑧 ·P 𝑣) +P 1P))
3533, 34syl5eq 2697 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ·P 𝑣) = 1P → ((𝑧 +P 𝑤) ·P 𝑣) = ((𝑧 ·P 𝑣) +P 1P))
3627, 35sylan9eqr 2707 . . . . . . . . . . . . . . . . . 18 (((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦) → (𝑦 ·P 𝑣) = ((𝑧 ·P 𝑣) +P 1P))
3736oveq1d 6705 . . . . . . . . . . . . . . . . 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 6718 . . . . . . . . . . . . . . . . . 18 (𝑧 ·P 𝑣) ∈ V
3914elexi 3244 . . . . . . . . . . . . . . . . . 18 1P ∈ V
40 ovex 6718 . . . . . . . . . . . . . . . . . 18 ((𝑦 ·P 1P) +P (𝑧 ·P 1P)) ∈ V
41 addcompr 9881 . . . . . . . . . . . . . . . . . 18 (𝑢 +P 𝑓) = (𝑓 +P 𝑢)
42 addasspr 9882 . . . . . . . . . . . . . . . . . 18 ((𝑢 +P 𝑓) +P 𝑥) = (𝑢 +P (𝑓 +P 𝑥))
4338, 39, 40, 41, 42caov32 6903 . . . . . . . . . . . . . . . . 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, 43syl6eq 2701 . . . . . . . . . . . . . . . 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 6705 . . . . . . . . . . . . . . 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 9882 . . . . . . . . . . . . . . 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, 46syl6eq 2701 . . . . . . . . . . . . . 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 9888 . . . . . . . . . . . . . . . . 17 (𝑦 ·P (𝑣 +P 1P)) = ((𝑦 ·P 𝑣) +P (𝑦 ·P 1P))
4948oveq1i 6700 . . . . . . . . . . . . . . . 16 ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) = (((𝑦 ·P 𝑣) +P (𝑦 ·P 1P)) +P (𝑧 ·P 1P))
50 addasspr 9882 . . . . . . . . . . . . . . . 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 2673 . . . . . . . . . . . . . . 15 ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) = ((𝑦 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P)))
5251oveq1i 6700 . . . . . . . . . . . . . 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 9888 . . . . . . . . . . . . . . . . 17 (𝑧 ·P (𝑣 +P 1P)) = ((𝑧 ·P 𝑣) +P (𝑧 ·P 1P))
5453oveq2i 6701 . . . . . . . . . . . . . . . 16 ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) = ((𝑦 ·P 1P) +P ((𝑧 ·P 𝑣) +P (𝑧 ·P 1P)))
55 ovex 6718 . . . . . . . . . . . . . . . . 17 (𝑦 ·P 1P) ∈ V
56 ovex 6718 . . . . . . . . . . . . . . . . 17 (𝑧 ·P 1P) ∈ V
5755, 38, 56, 41, 42caov12 6904 . . . . . . . . . . . . . . . 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 2673 . . . . . . . . . . . . . . 15 ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) = ((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P)))
5958oveq1i 6700 . . . . . . . . . . . . . 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 2710 . . . . . . . . . . . . 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 9880 . . . . . . . . . . . . . . . . . 18 ((𝑦P ∧ (𝑣 +P 1P) ∈ P) → (𝑦 ·P (𝑣 +P 1P)) ∈ P)
6216, 61sylan2 490 . . . . . . . . . . . . . . . . 17 ((𝑦P𝑣P) → (𝑦 ·P (𝑣 +P 1P)) ∈ P)
63 mulclpr 9880 . . . . . . . . . . . . . . . . . 18 ((𝑧P ∧ 1PP) → (𝑧 ·P 1P) ∈ P)
6414, 63mpan2 707 . . . . . . . . . . . . . . . . 17 (𝑧P → (𝑧 ·P 1P) ∈ P)
65 addclpr 9878 . . . . . . . . . . . . . . . . 17 (((𝑦 ·P (𝑣 +P 1P)) ∈ P ∧ (𝑧 ·P 1P) ∈ P) → ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) ∈ P)
6662, 64, 65syl2an 493 . . . . . . . . . . . . . . . 16 (((𝑦P𝑣P) ∧ 𝑧P) → ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) ∈ P)
6766an32s 863 . . . . . . . . . . . . . . 15 (((𝑦P𝑧P) ∧ 𝑣P) → ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) ∈ P)
68 mulclpr 9880 . . . . . . . . . . . . . . . . . 18 ((𝑦P ∧ 1PP) → (𝑦 ·P 1P) ∈ P)
6914, 68mpan2 707 . . . . . . . . . . . . . . . . 17 (𝑦P → (𝑦 ·P 1P) ∈ P)
70 mulclpr 9880 . . . . . . . . . . . . . . . . . 18 ((𝑧P ∧ (𝑣 +P 1P) ∈ P) → (𝑧 ·P (𝑣 +P 1P)) ∈ P)
7116, 70sylan2 490 . . . . . . . . . . . . . . . . 17 ((𝑧P𝑣P) → (𝑧 ·P (𝑣 +P 1P)) ∈ P)
72 addclpr 9878 . . . . . . . . . . . . . . . . 17 (((𝑦 ·P 1P) ∈ P ∧ (𝑧 ·P (𝑣 +P 1P)) ∈ P) → ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) ∈ P)
7369, 71, 72syl2an 493 . . . . . . . . . . . . . . . 16 ((𝑦P ∧ (𝑧P𝑣P)) → ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) ∈ P)
7473anassrs 681 . . . . . . . . . . . . . . 15 (((𝑦P𝑧P) ∧ 𝑣P) → ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) ∈ P)
7567, 74jca 553 . . . . . . . . . . . . . 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 9878 . . . . . . . . . . . . . . . 16 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
7714, 14, 76mp2an 708 . . . . . . . . . . . . . . 15 (1P +P 1P) ∈ P
7877, 14pm3.2i 470 . . . . . . . . . . . . . 14 ((1P +P 1P) ∈ P ∧ 1PP)
79 enreceq 9925 . . . . . . . . . . . . . 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 695 . . . . . . . . . . . . 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, 80syl5ibr 236 . . . . . . . . . . . 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 444 . . . . . . . . . . 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 2685 . . . . . . . . . 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 9921 . . . . . . . . . 10 1R = [⟨(1P +P 1P), 1P⟩] ~R
8583, 84syl6eqr 2703 . . . . . . . . 9 ((((𝑦P𝑧P) ∧ 𝑣P) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ) = 1R)
86 oveq2 6698 . . . . . . . . . . 11 (𝑥 = [⟨(𝑣 +P 1P), 1P⟩] ~R → ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ))
8786eqeq1d 2653 . . . . . . . . . 10 (𝑥 = [⟨(𝑣 +P 1P), 1P⟩] ~R → (([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R ↔ ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ) = 1R))
8887rspcev 3340 . . . . . . . . 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 694 . . . . . . . 8 ((((𝑦P𝑧P) ∧ 𝑣P) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ∃𝑥R ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R)
9089exp43 639 . . . . . . 7 ((𝑦P𝑧P) → (𝑣P → ((𝑤 ·P 𝑣) = 1P → ((𝑧 +P 𝑤) = 𝑦 → ∃𝑥R ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R))))
9190rexlimdv 3059 . . . . . 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 3059 . . . 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 7880 . 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 196  wa 383   = wceq 1523  wcel 2030  wrex 2942  cop 4216   class class class wbr 4685  (class class class)co 6690  [cec 7785  Pcnp 9719  1Pc1p 9720   +P cpp 9721   ·P cmp 9722  <P cltp 9723   ~R cer 9724  Rcnr 9725  0Rc0r 9726  1Rc1r 9727   ·R cmr 9730   <R cltr 9731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-er 7787  df-ec 7789  df-qs 7793  df-ni 9732  df-pli 9733  df-mi 9734  df-lti 9735  df-plpq 9768  df-mpq 9769  df-ltpq 9770  df-enq 9771  df-nq 9772  df-erq 9773  df-plq 9774  df-mq 9775  df-1nq 9776  df-rq 9777  df-ltnq 9778  df-np 9841  df-1p 9842  df-plp 9843  df-mp 9844  df-ltp 9845  df-enr 9915  df-nr 9916  df-mr 9918  df-ltr 9919  df-0r 9920  df-1r 9921
This theorem is referenced by:  recexsr  9966
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