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Theorem recexsrlem 10513
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 10478 . . . 4 <R ⊆ (R × R)
21brel 5610 . . 3 (0R <R 𝐴 → (0RR𝐴R))
32simprd 496 . 2 (0R <R 𝐴𝐴R)
4 df-nr 10466 . . 3 R = ((P × P) / ~R )
5 breq2 5061 . . . 4 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → (0R <R [⟨𝑦, 𝑧⟩] ~R ↔ 0R <R 𝐴))
6 oveq1 7152 . . . . . 6 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = (𝐴 ·R 𝑥))
76eqeq1d 2820 . . . . 5 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → (([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R ↔ (𝐴 ·R 𝑥) = 1R))
87rexbidv 3294 . . . 4 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → (∃𝑥R ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R ↔ ∃𝑥R (𝐴 ·R 𝑥) = 1R))
95, 8imbi12d 346 . . 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 10488 . . . . 5 (0R <R [⟨𝑦, 𝑧⟩] ~R𝑧<P 𝑦)
11 ltexpri 10453 . . . . 5 (𝑧<P 𝑦 → ∃𝑤P (𝑧 +P 𝑤) = 𝑦)
1210, 11sylbi 218 . . . 4 (0R <R [⟨𝑦, 𝑧⟩] ~R → ∃𝑤P (𝑧 +P 𝑤) = 𝑦)
13 recexpr 10461 . . . . . 6 (𝑤P → ∃𝑣P (𝑤 ·P 𝑣) = 1P)
14 1pr 10425 . . . . . . . . . . . 12 1PP
15 addclpr 10428 . . . . . . . . . . . 12 ((𝑣P ∧ 1PP) → (𝑣 +P 1P) ∈ P)
1614, 15mpan2 687 . . . . . . . . . . 11 (𝑣P → (𝑣 +P 1P) ∈ P)
17 enrex 10477 . . . . . . . . . . . 12 ~R ∈ V
1817, 4ecopqsi 8343 . . . . . . . . . . 11 (((𝑣 +P 1P) ∈ P ∧ 1PP) → [⟨(𝑣 +P 1P), 1P⟩] ~RR)
1916, 14, 18sylancl 586 . . . . . . . . . 10 (𝑣P → [⟨(𝑣 +P 1P), 1P⟩] ~RR)
2019ad2antlr 723 . . . . . . . . 9 ((((𝑦P𝑧P) ∧ 𝑣P) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → [⟨(𝑣 +P 1P), 1P⟩] ~RR)
2116, 14jctir 521 . . . . . . . . . . . . . 14 (𝑣P → ((𝑣 +P 1P) ∈ P ∧ 1PP))
2221anim2i 616 . . . . . . . . . . . . 13 (((𝑦P𝑧P) ∧ 𝑣P) → ((𝑦P𝑧P) ∧ ((𝑣 +P 1P) ∈ P ∧ 1PP)))
2322adantr 481 . . . . . . . . . . . 12 ((((𝑦P𝑧P) ∧ 𝑣P) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ((𝑦P𝑧P) ∧ ((𝑣 +P 1P) ∈ P ∧ 1PP)))
24 mulsrpr 10486 . . . . . . . . . . . 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 7152 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 +P 𝑤) = 𝑦 → ((𝑧 +P 𝑤) ·P 𝑣) = (𝑦 ·P 𝑣))
2726eqcomd 2824 . . . . . . . . . . . . . . . . . . 19 ((𝑧 +P 𝑤) = 𝑦 → (𝑦 ·P 𝑣) = ((𝑧 +P 𝑤) ·P 𝑣))
28 vex 3495 . . . . . . . . . . . . . . . . . . . . 21 𝑧 ∈ V
29 vex 3495 . . . . . . . . . . . . . . . . . . . . 21 𝑤 ∈ V
30 vex 3495 . . . . . . . . . . . . . . . . . . . . 21 𝑣 ∈ V
31 mulcompr 10433 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 ·P 𝑓) = (𝑓 ·P 𝑢)
32 distrpr 10438 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 ·P (𝑓 +P 𝑥)) = ((𝑢 ·P 𝑓) +P (𝑢 ·P 𝑥))
3328, 29, 30, 31, 32caovdir 7371 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 +P 𝑤) ·P 𝑣) = ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑣))
34 oveq2 7153 . . . . . . . . . . . . . . . . . . . 20 ((𝑤 ·P 𝑣) = 1P → ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑣)) = ((𝑧 ·P 𝑣) +P 1P))
3533, 34syl5eq 2865 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ·P 𝑣) = 1P → ((𝑧 +P 𝑤) ·P 𝑣) = ((𝑧 ·P 𝑣) +P 1P))
3627, 35sylan9eqr 2875 . . . . . . . . . . . . . . . . . 18 (((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦) → (𝑦 ·P 𝑣) = ((𝑧 ·P 𝑣) +P 1P))
3736oveq1d 7160 . . . . . . . . . . . . . . . . 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 7178 . . . . . . . . . . . . . . . . . 18 (𝑧 ·P 𝑣) ∈ V
3914elexi 3511 . . . . . . . . . . . . . . . . . 18 1P ∈ V
40 ovex 7178 . . . . . . . . . . . . . . . . . 18 ((𝑦 ·P 1P) +P (𝑧 ·P 1P)) ∈ V
41 addcompr 10431 . . . . . . . . . . . . . . . . . 18 (𝑢 +P 𝑓) = (𝑓 +P 𝑢)
42 addasspr 10432 . . . . . . . . . . . . . . . . . 18 ((𝑢 +P 𝑓) +P 𝑥) = (𝑢 +P (𝑓 +P 𝑥))
4338, 39, 40, 41, 42caov32 7364 . . . . . . . . . . . . . . . . 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 2869 . . . . . . . . . . . . . . . 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 7160 . . . . . . . . . . . . . . 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 10432 . . . . . . . . . . . . . . 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 2869 . . . . . . . . . . . . . 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 10438 . . . . . . . . . . . . . . . . 17 (𝑦 ·P (𝑣 +P 1P)) = ((𝑦 ·P 𝑣) +P (𝑦 ·P 1P))
4948oveq1i 7155 . . . . . . . . . . . . . . . 16 ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) = (((𝑦 ·P 𝑣) +P (𝑦 ·P 1P)) +P (𝑧 ·P 1P))
50 addasspr 10432 . . . . . . . . . . . . . . . 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 2841 . . . . . . . . . . . . . . 15 ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) = ((𝑦 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P)))
5251oveq1i 7155 . . . . . . . . . . . . . 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 10438 . . . . . . . . . . . . . . . . 17 (𝑧 ·P (𝑣 +P 1P)) = ((𝑧 ·P 𝑣) +P (𝑧 ·P 1P))
5453oveq2i 7156 . . . . . . . . . . . . . . . 16 ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) = ((𝑦 ·P 1P) +P ((𝑧 ·P 𝑣) +P (𝑧 ·P 1P)))
55 ovex 7178 . . . . . . . . . . . . . . . . 17 (𝑦 ·P 1P) ∈ V
56 ovex 7178 . . . . . . . . . . . . . . . . 17 (𝑧 ·P 1P) ∈ V
5755, 38, 56, 41, 42caov12 7365 . . . . . . . . . . . . . . . 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 2841 . . . . . . . . . . . . . . 15 ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) = ((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P)))
5958oveq1i 7155 . . . . . . . . . . . . . 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 2878 . . . . . . . . . . . . 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 10430 . . . . . . . . . . . . . . . . . 18 ((𝑦P ∧ (𝑣 +P 1P) ∈ P) → (𝑦 ·P (𝑣 +P 1P)) ∈ P)
6216, 61sylan2 592 . . . . . . . . . . . . . . . . 17 ((𝑦P𝑣P) → (𝑦 ·P (𝑣 +P 1P)) ∈ P)
63 mulclpr 10430 . . . . . . . . . . . . . . . . . 18 ((𝑧P ∧ 1PP) → (𝑧 ·P 1P) ∈ P)
6414, 63mpan2 687 . . . . . . . . . . . . . . . . 17 (𝑧P → (𝑧 ·P 1P) ∈ P)
65 addclpr 10428 . . . . . . . . . . . . . . . . 17 (((𝑦 ·P (𝑣 +P 1P)) ∈ P ∧ (𝑧 ·P 1P) ∈ P) → ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) ∈ P)
6662, 64, 65syl2an 595 . . . . . . . . . . . . . . . 16 (((𝑦P𝑣P) ∧ 𝑧P) → ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) ∈ P)
6766an32s 648 . . . . . . . . . . . . . . 15 (((𝑦P𝑧P) ∧ 𝑣P) → ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) ∈ P)
68 mulclpr 10430 . . . . . . . . . . . . . . . . . 18 ((𝑦P ∧ 1PP) → (𝑦 ·P 1P) ∈ P)
6914, 68mpan2 687 . . . . . . . . . . . . . . . . 17 (𝑦P → (𝑦 ·P 1P) ∈ P)
70 mulclpr 10430 . . . . . . . . . . . . . . . . . 18 ((𝑧P ∧ (𝑣 +P 1P) ∈ P) → (𝑧 ·P (𝑣 +P 1P)) ∈ P)
7116, 70sylan2 592 . . . . . . . . . . . . . . . . 17 ((𝑧P𝑣P) → (𝑧 ·P (𝑣 +P 1P)) ∈ P)
72 addclpr 10428 . . . . . . . . . . . . . . . . 17 (((𝑦 ·P 1P) ∈ P ∧ (𝑧 ·P (𝑣 +P 1P)) ∈ P) → ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) ∈ P)
7369, 71, 72syl2an 595 . . . . . . . . . . . . . . . 16 ((𝑦P ∧ (𝑧P𝑣P)) → ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) ∈ P)
7473anassrs 468 . . . . . . . . . . . . . . 15 (((𝑦P𝑧P) ∧ 𝑣P) → ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) ∈ P)
7567, 74jca 512 . . . . . . . . . . . . . 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 10428 . . . . . . . . . . . . . . . 16 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
7714, 14, 76mp2an 688 . . . . . . . . . . . . . . 15 (1P +P 1P) ∈ P
7877, 14pm3.2i 471 . . . . . . . . . . . . . 14 ((1P +P 1P) ∈ P ∧ 1PP)
79 enreceq 10476 . . . . . . . . . . . . . 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, 80syl5ibr 247 . . . . . . . . . . . 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 407 . . . . . . . . . . 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 2853 . . . . . . . . . 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 10471 . . . . . . . . . 10 1R = [⟨(1P +P 1P), 1P⟩] ~R
8583, 84syl6eqr 2871 . . . . . . . . 9 ((((𝑦P𝑧P) ∧ 𝑣P) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ) = 1R)
86 oveq2 7153 . . . . . . . . . . 11 (𝑥 = [⟨(𝑣 +P 1P), 1P⟩] ~R → ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ))
8786eqeq1d 2820 . . . . . . . . . 10 (𝑥 = [⟨(𝑣 +P 1P), 1P⟩] ~R → (([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R ↔ ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ) = 1R))
8887rspcev 3620 . . . . . . . . 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 437 . . . . . . 7 ((𝑦P𝑧P) → (𝑣P → ((𝑤 ·P 𝑣) = 1P → ((𝑧 +P 𝑤) = 𝑦 → ∃𝑥R ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R))))
9190rexlimdv 3280 . . . . . 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 3280 . . . 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 8376 . 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 207  wa 396   = wceq 1528  wcel 2105  wrex 3136  cop 4563   class class class wbr 5057  (class class class)co 7145  [cec 8276  Pcnp 10269  1Pc1p 10270   +P cpp 10271   ·P cmp 10272  <P cltp 10273   ~R cer 10274  Rcnr 10275  0Rc0r 10276  1Rc1r 10277   ·R cmr 10280   <R cltr 10281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-omul 8096  df-er 8278  df-ec 8280  df-qs 8284  df-ni 10282  df-pli 10283  df-mi 10284  df-lti 10285  df-plpq 10318  df-mpq 10319  df-ltpq 10320  df-enq 10321  df-nq 10322  df-erq 10323  df-plq 10324  df-mq 10325  df-1nq 10326  df-rq 10327  df-ltnq 10328  df-np 10391  df-1p 10392  df-plp 10393  df-mp 10394  df-ltp 10395  df-enr 10465  df-nr 10466  df-mr 10468  df-ltr 10469  df-0r 10470  df-1r 10471
This theorem is referenced by:  recexsr  10517
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