ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  recexgt0sr GIF version

Theorem recexgt0sr 7012
Description: The reciprocal of a positive signed real exists and is positive. (Contributed by Jim Kingdon, 6-Feb-2020.)
Assertion
Ref Expression
recexgt0sr (0R <R 𝐴 → ∃𝑥R (0R <R 𝑥 ∧ (𝐴 ·R 𝑥) = 1R))
Distinct variable group:   𝑥,𝐴

Proof of Theorem recexgt0sr
Dummy variables 𝑦 𝑧 𝑤 𝑣 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelsr 6977 . . . 4 <R ⊆ (R × R)
21brel 4418 . . 3 (0R <R 𝐴 → (0RR𝐴R))
32simprd 112 . 2 (0R <R 𝐴𝐴R)
4 df-nr 6966 . . 3 R = ((P × P) / ~R )
5 breq2 3797 . . . 4 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → (0R <R [⟨𝑦, 𝑧⟩] ~R ↔ 0R <R 𝐴))
6 oveq1 5550 . . . . . . 7 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = (𝐴 ·R 𝑥))
76eqeq1d 2090 . . . . . 6 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → (([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R ↔ (𝐴 ·R 𝑥) = 1R))
87anbi2d 452 . . . . 5 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → ((0R <R 𝑥 ∧ ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R) ↔ (0R <R 𝑥 ∧ (𝐴 ·R 𝑥) = 1R)))
98rexbidv 2370 . . . 4 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → (∃𝑥R (0R <R 𝑥 ∧ ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R) ↔ ∃𝑥R (0R <R 𝑥 ∧ (𝐴 ·R 𝑥) = 1R)))
105, 9imbi12d 232 . . 3 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → ((0R <R [⟨𝑦, 𝑧⟩] ~R → ∃𝑥R (0R <R 𝑥 ∧ ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R)) ↔ (0R <R 𝐴 → ∃𝑥R (0R <R 𝑥 ∧ (𝐴 ·R 𝑥) = 1R))))
11 gt0srpr 6987 . . . . 5 (0R <R [⟨𝑦, 𝑧⟩] ~R𝑧<P 𝑦)
12 ltexpri 6865 . . . . 5 (𝑧<P 𝑦 → ∃𝑤P (𝑧 +P 𝑤) = 𝑦)
1311, 12sylbi 119 . . . 4 (0R <R [⟨𝑦, 𝑧⟩] ~R → ∃𝑤P (𝑧 +P 𝑤) = 𝑦)
14 recexpr 6890 . . . . . . 7 (𝑤P → ∃𝑣P (𝑤 ·P 𝑣) = 1P)
1514adantl 271 . . . . . 6 (((𝑦P𝑧P) ∧ 𝑤P) → ∃𝑣P (𝑤 ·P 𝑣) = 1P)
16 1pr 6806 . . . . . . . . . . . . . 14 1PP
17 addclpr 6789 . . . . . . . . . . . . . 14 ((𝑣P ∧ 1PP) → (𝑣 +P 1P) ∈ P)
1816, 17mpan2 416 . . . . . . . . . . . . 13 (𝑣P → (𝑣 +P 1P) ∈ P)
19 enrex 6976 . . . . . . . . . . . . . 14 ~R ∈ V
2019, 4ecopqsi 6227 . . . . . . . . . . . . 13 (((𝑣 +P 1P) ∈ P ∧ 1PP) → [⟨(𝑣 +P 1P), 1P⟩] ~RR)
2118, 16, 20sylancl 404 . . . . . . . . . . . 12 (𝑣P → [⟨(𝑣 +P 1P), 1P⟩] ~RR)
2221adantl 271 . . . . . . . . . . 11 ((𝑤P𝑣P) → [⟨(𝑣 +P 1P), 1P⟩] ~RR)
2322ad2antlr 473 . . . . . . . . . 10 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → [⟨(𝑣 +P 1P), 1P⟩] ~RR)
24 simprr 499 . . . . . . . . . . . 12 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → 𝑣P)
2524adantr 270 . . . . . . . . . . 11 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → 𝑣P)
26 ltaddpr 6849 . . . . . . . . . . . . . 14 ((1PP𝑣P) → 1P<P (1P +P 𝑣))
2716, 26mpan 415 . . . . . . . . . . . . 13 (𝑣P → 1P<P (1P +P 𝑣))
28 addcomprg 6830 . . . . . . . . . . . . . 14 ((1PP𝑣P) → (1P +P 𝑣) = (𝑣 +P 1P))
2916, 28mpan 415 . . . . . . . . . . . . 13 (𝑣P → (1P +P 𝑣) = (𝑣 +P 1P))
3027, 29breqtrd 3817 . . . . . . . . . . . 12 (𝑣P → 1P<P (𝑣 +P 1P))
31 gt0srpr 6987 . . . . . . . . . . . 12 (0R <R [⟨(𝑣 +P 1P), 1P⟩] ~R ↔ 1P<P (𝑣 +P 1P))
3230, 31sylibr 132 . . . . . . . . . . 11 (𝑣P → 0R <R [⟨(𝑣 +P 1P), 1P⟩] ~R )
3325, 32syl 14 . . . . . . . . . 10 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → 0R <R [⟨(𝑣 +P 1P), 1P⟩] ~R )
3418, 16jctir 306 . . . . . . . . . . . . . . . 16 (𝑣P → ((𝑣 +P 1P) ∈ P ∧ 1PP))
3534anim2i 334 . . . . . . . . . . . . . . 15 (((𝑦P𝑧P) ∧ 𝑣P) → ((𝑦P𝑧P) ∧ ((𝑣 +P 1P) ∈ P ∧ 1PP)))
3635adantr 270 . . . . . . . . . . . . . 14 ((((𝑦P𝑧P) ∧ 𝑣P) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ((𝑦P𝑧P) ∧ ((𝑣 +P 1P) ∈ P ∧ 1PP)))
37 mulsrpr 6985 . . . . . . . . . . . . . 14 (((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 )
3836, 37syl 14 . . . . . . . . . . . . 13 ((((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 )
3938adantlrl 466 . . . . . . . . . . . 12 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((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 )
40 oveq1 5550 . . . . . . . . . . . . . . . . . . . . 21 ((𝑧 +P 𝑤) = 𝑦 → ((𝑧 +P 𝑤) ·P 𝑣) = (𝑦 ·P 𝑣))
4140eqcomd 2087 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 +P 𝑤) = 𝑦 → (𝑦 ·P 𝑣) = ((𝑧 +P 𝑤) ·P 𝑣))
4241ad2antll 475 . . . . . . . . . . . . . . . . . . 19 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → (𝑦 ·P 𝑣) = ((𝑧 +P 𝑤) ·P 𝑣))
43 mulcomprg 6832 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓PP) → (𝑓 ·P ) = ( ·P 𝑓))
44433adant2 958 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓P𝑔PP) → (𝑓 ·P ) = ( ·P 𝑓))
45 mulcomprg 6832 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑔PP) → (𝑔 ·P ) = ( ·P 𝑔))
46453adant1 957 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓P𝑔PP) → (𝑔 ·P ) = ( ·P 𝑔))
4744, 46oveq12d 5561 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓P𝑔PP) → ((𝑓 ·P ) +P (𝑔 ·P )) = (( ·P 𝑓) +P ( ·P 𝑔)))
48 distrprg 6840 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((P𝑓P𝑔P) → ( ·P (𝑓 +P 𝑔)) = (( ·P 𝑓) +P ( ·P 𝑔)))
49483coml 1146 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓P𝑔PP) → ( ·P (𝑓 +P 𝑔)) = (( ·P 𝑓) +P ( ·P 𝑔)))
50 simp3 941 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓P𝑔PP) → P)
51 addclpr 6789 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓P𝑔P) → (𝑓 +P 𝑔) ∈ P)
52513adant3 959 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓P𝑔PP) → (𝑓 +P 𝑔) ∈ P)
53 mulcomprg 6832 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((P ∧ (𝑓 +P 𝑔) ∈ P) → ( ·P (𝑓 +P 𝑔)) = ((𝑓 +P 𝑔) ·P ))
5450, 52, 53syl2anc 403 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓P𝑔PP) → ( ·P (𝑓 +P 𝑔)) = ((𝑓 +P 𝑔) ·P ))
5547, 49, 543eqtr2rd 2121 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓P𝑔PP) → ((𝑓 +P 𝑔) ·P ) = ((𝑓 ·P ) +P (𝑔 ·P )))
5655adantl 271 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ (𝑓P𝑔PP)) → ((𝑓 +P 𝑔) ·P ) = ((𝑓 ·P ) +P (𝑔 ·P )))
57 simplr 497 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → 𝑧P)
58 simprl 498 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → 𝑤P)
5956, 57, 58, 24caovdird 5710 . . . . . . . . . . . . . . . . . . . . 21 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑧 +P 𝑤) ·P 𝑣) = ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑣)))
60 oveq2 5551 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤 ·P 𝑣) = 1P → ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑣)) = ((𝑧 ·P 𝑣) +P 1P))
6159, 60sylan9eq 2134 . . . . . . . . . . . . . . . . . . . 20 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ (𝑤 ·P 𝑣) = 1P) → ((𝑧 +P 𝑤) ·P 𝑣) = ((𝑧 ·P 𝑣) +P 1P))
6261adantrr 463 . . . . . . . . . . . . . . . . . . 19 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ((𝑧 +P 𝑤) ·P 𝑣) = ((𝑧 ·P 𝑣) +P 1P))
6342, 62eqtrd 2114 . . . . . . . . . . . . . . . . . 18 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → (𝑦 ·P 𝑣) = ((𝑧 ·P 𝑣) +P 1P))
6463oveq1d 5558 . . . . . . . . . . . . . . . . 17 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((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))))
65 mulclpr 6824 . . . . . . . . . . . . . . . . . . . 20 ((𝑧P𝑣P) → (𝑧 ·P 𝑣) ∈ P)
6657, 24, 65syl2anc 403 . . . . . . . . . . . . . . . . . . 19 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑧 ·P 𝑣) ∈ P)
6716a1i 9 . . . . . . . . . . . . . . . . . . 19 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → 1PP)
68 simpll 496 . . . . . . . . . . . . . . . . . . . . 21 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → 𝑦P)
69 mulclpr 6824 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦P ∧ 1PP) → (𝑦 ·P 1P) ∈ P)
7068, 16, 69sylancl 404 . . . . . . . . . . . . . . . . . . . 20 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑦 ·P 1P) ∈ P)
71 mulclpr 6824 . . . . . . . . . . . . . . . . . . . . 21 ((𝑧P ∧ 1PP) → (𝑧 ·P 1P) ∈ P)
7257, 16, 71sylancl 404 . . . . . . . . . . . . . . . . . . . 20 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑧 ·P 1P) ∈ P)
73 addclpr 6789 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 ·P 1P) ∈ P ∧ (𝑧 ·P 1P) ∈ P) → ((𝑦 ·P 1P) +P (𝑧 ·P 1P)) ∈ P)
7470, 72, 73syl2anc 403 . . . . . . . . . . . . . . . . . . 19 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑦 ·P 1P) +P (𝑧 ·P 1P)) ∈ P)
75 addcomprg 6830 . . . . . . . . . . . . . . . . . . . 20 ((𝑓P𝑔P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
7675adantl 271 . . . . . . . . . . . . . . . . . . 19 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
77 addassprg 6831 . . . . . . . . . . . . . . . . . . . 20 ((𝑓P𝑔PP) → ((𝑓 +P 𝑔) +P ) = (𝑓 +P (𝑔 +P )))
7877adantl 271 . . . . . . . . . . . . . . . . . . 19 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ (𝑓P𝑔PP)) → ((𝑓 +P 𝑔) +P ) = (𝑓 +P (𝑔 +P )))
7966, 67, 74, 76, 78caov32d 5712 . . . . . . . . . . . . . . . . . 18 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (((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))
8079adantr 270 . . . . . . . . . . . . . . . . 17 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → (((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))
8164, 80eqtrd 2114 . . . . . . . . . . . . . . . 16 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((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))
8281oveq1d 5558 . . . . . . . . . . . . . . 15 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((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))
83 addclpr 6789 . . . . . . . . . . . . . . . . . 18 (((𝑧 ·P 𝑣) ∈ P ∧ ((𝑦 ·P 1P) +P (𝑧 ·P 1P)) ∈ P) → ((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) ∈ P)
8466, 74, 83syl2anc 403 . . . . . . . . . . . . . . . . 17 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) ∈ P)
8584adantr 270 . . . . . . . . . . . . . . . 16 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) ∈ P)
8616a1i 9 . . . . . . . . . . . . . . . 16 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → 1PP)
87 addassprg 6831 . . . . . . . . . . . . . . . 16 ((((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) ∈ P ∧ 1PP ∧ 1PP) → ((((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)))
8885, 86, 86, 87syl3anc 1170 . . . . . . . . . . . . . . 15 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ((((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)))
8982, 88eqtrd 2114 . . . . . . . . . . . . . 14 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((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)))
90 distrprg 6840 . . . . . . . . . . . . . . . . . . 19 ((𝑦P𝑣P ∧ 1PP) → (𝑦 ·P (𝑣 +P 1P)) = ((𝑦 ·P 𝑣) +P (𝑦 ·P 1P)))
9168, 24, 67, 90syl3anc 1170 . . . . . . . . . . . . . . . . . 18 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑦 ·P (𝑣 +P 1P)) = ((𝑦 ·P 𝑣) +P (𝑦 ·P 1P)))
9291oveq1d 5558 . . . . . . . . . . . . . . . . 17 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) = (((𝑦 ·P 𝑣) +P (𝑦 ·P 1P)) +P (𝑧 ·P 1P)))
93 mulclpr 6824 . . . . . . . . . . . . . . . . . . 19 ((𝑦P𝑣P) → (𝑦 ·P 𝑣) ∈ P)
9468, 24, 93syl2anc 403 . . . . . . . . . . . . . . . . . 18 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑦 ·P 𝑣) ∈ P)
95 addassprg 6831 . . . . . . . . . . . . . . . . . 18 (((𝑦 ·P 𝑣) ∈ P ∧ (𝑦 ·P 1P) ∈ P ∧ (𝑧 ·P 1P) ∈ P) → (((𝑦 ·P 𝑣) +P (𝑦 ·P 1P)) +P (𝑧 ·P 1P)) = ((𝑦 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))))
9694, 70, 72, 95syl3anc 1170 . . . . . . . . . . . . . . . . 17 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (((𝑦 ·P 𝑣) +P (𝑦 ·P 1P)) +P (𝑧 ·P 1P)) = ((𝑦 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))))
9792, 96eqtrd 2114 . . . . . . . . . . . . . . . 16 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) = ((𝑦 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))))
9897oveq1d 5558 . . . . . . . . . . . . . . 15 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) +P 1P) = (((𝑦 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) +P 1P))
9998adantr 270 . . . . . . . . . . . . . 14 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → (((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) +P 1P) = (((𝑦 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) +P 1P))
100 distrprg 6840 . . . . . . . . . . . . . . . . . . 19 ((𝑧P𝑣P ∧ 1PP) → (𝑧 ·P (𝑣 +P 1P)) = ((𝑧 ·P 𝑣) +P (𝑧 ·P 1P)))
10157, 24, 67, 100syl3anc 1170 . . . . . . . . . . . . . . . . . 18 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑧 ·P (𝑣 +P 1P)) = ((𝑧 ·P 𝑣) +P (𝑧 ·P 1P)))
102101oveq2d 5559 . . . . . . . . . . . . . . . . 17 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) = ((𝑦 ·P 1P) +P ((𝑧 ·P 𝑣) +P (𝑧 ·P 1P))))
10370, 66, 72, 76, 78caov12d 5713 . . . . . . . . . . . . . . . . 17 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑦 ·P 1P) +P ((𝑧 ·P 𝑣) +P (𝑧 ·P 1P))) = ((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))))
104102, 103eqtrd 2114 . . . . . . . . . . . . . . . 16 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) = ((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))))
105104oveq1d 5558 . . . . . . . . . . . . . . 15 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (((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)))
106105adantr 270 . . . . . . . . . . . . . 14 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → (((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)))
10789, 99, 1063eqtr4d 2124 . . . . . . . . . . . . 13 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((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)))
10824, 16, 17sylancl 404 . . . . . . . . . . . . . . . . 17 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑣 +P 1P) ∈ P)
109 mulclpr 6824 . . . . . . . . . . . . . . . . 17 ((𝑦P ∧ (𝑣 +P 1P) ∈ P) → (𝑦 ·P (𝑣 +P 1P)) ∈ P)
11068, 108, 109syl2anc 403 . . . . . . . . . . . . . . . 16 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑦 ·P (𝑣 +P 1P)) ∈ P)
111 addclpr 6789 . . . . . . . . . . . . . . . 16 (((𝑦 ·P (𝑣 +P 1P)) ∈ P ∧ (𝑧 ·P 1P) ∈ P) → ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) ∈ P)
112110, 72, 111syl2anc 403 . . . . . . . . . . . . . . 15 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) ∈ P)
113104, 84eqeltrd 2156 . . . . . . . . . . . . . . 15 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) ∈ P)
114 addclpr 6789 . . . . . . . . . . . . . . . . 17 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
11516, 16, 114mp2an 417 . . . . . . . . . . . . . . . 16 (1P +P 1P) ∈ P
116115a1i 9 . . . . . . . . . . . . . . 15 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (1P +P 1P) ∈ P)
117 enreceq 6975 . . . . . . . . . . . . . . 15 (((((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))))
118112, 113, 116, 67, 117syl22anc 1171 . . . . . . . . . . . . . 14 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ([⟨((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))))
119118adantr 270 . . . . . . . . . . . . 13 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((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 ↔ (((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) +P 1P) = (((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) +P (1P +P 1P))))
120107, 119mpbird 165 . . . . . . . . . . . 12 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((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 )
12139, 120eqtrd 2114 . . . . . . . . . . 11 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ) = [⟨(1P +P 1P), 1P⟩] ~R )
122 df-1r 6971 . . . . . . . . . . 11 1R = [⟨(1P +P 1P), 1P⟩] ~R
123121, 122syl6eqr 2132 . . . . . . . . . 10 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ) = 1R)
124 breq2 3797 . . . . . . . . . . . 12 (𝑥 = [⟨(𝑣 +P 1P), 1P⟩] ~R → (0R <R 𝑥 ↔ 0R <R [⟨(𝑣 +P 1P), 1P⟩] ~R ))
125 oveq2 5551 . . . . . . . . . . . . 13 (𝑥 = [⟨(𝑣 +P 1P), 1P⟩] ~R → ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ))
126125eqeq1d 2090 . . . . . . . . . . . 12 (𝑥 = [⟨(𝑣 +P 1P), 1P⟩] ~R → (([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R ↔ ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ) = 1R))
127124, 126anbi12d 457 . . . . . . . . . . 11 (𝑥 = [⟨(𝑣 +P 1P), 1P⟩] ~R → ((0R <R 𝑥 ∧ ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R) ↔ (0R <R [⟨(𝑣 +P 1P), 1P⟩] ~R ∧ ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ) = 1R)))
128127rspcev 2702 . . . . . . . . . 10 (([⟨(𝑣 +P 1P), 1P⟩] ~RR ∧ (0R <R [⟨(𝑣 +P 1P), 1P⟩] ~R ∧ ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ) = 1R)) → ∃𝑥R (0R <R 𝑥 ∧ ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R))
12923, 33, 123, 128syl12anc 1168 . . . . . . . . 9 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ∃𝑥R (0R <R 𝑥 ∧ ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R))
130129exp32 357 . . . . . . . 8 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑤 ·P 𝑣) = 1P → ((𝑧 +P 𝑤) = 𝑦 → ∃𝑥R (0R <R 𝑥 ∧ ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R))))
131130anassrs 392 . . . . . . 7 ((((𝑦P𝑧P) ∧ 𝑤P) ∧ 𝑣P) → ((𝑤 ·P 𝑣) = 1P → ((𝑧 +P 𝑤) = 𝑦 → ∃𝑥R (0R <R 𝑥 ∧ ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R))))
132131rexlimdva 2478 . . . . . 6 (((𝑦P𝑧P) ∧ 𝑤P) → (∃𝑣P (𝑤 ·P 𝑣) = 1P → ((𝑧 +P 𝑤) = 𝑦 → ∃𝑥R (0R <R 𝑥 ∧ ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R))))
13315, 132mpd 13 . . . . 5 (((𝑦P𝑧P) ∧ 𝑤P) → ((𝑧 +P 𝑤) = 𝑦 → ∃𝑥R (0R <R 𝑥 ∧ ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R)))
134133rexlimdva 2478 . . . 4 ((𝑦P𝑧P) → (∃𝑤P (𝑧 +P 𝑤) = 𝑦 → ∃𝑥R (0R <R 𝑥 ∧ ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R)))
13513, 134syl5 32 . . 3 ((𝑦P𝑧P) → (0R <R [⟨𝑦, 𝑧⟩] ~R → ∃𝑥R (0R <R 𝑥 ∧ ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R)))
1364, 10, 135ecoptocl 6259 . 2 (𝐴R → (0R <R 𝐴 → ∃𝑥R (0R <R 𝑥 ∧ (𝐴 ·R 𝑥) = 1R)))
1373, 136mpcom 36 1 (0R <R 𝐴 → ∃𝑥R (0R <R 𝑥 ∧ (𝐴 ·R 𝑥) = 1R))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 920   = wceq 1285  wcel 1434  wrex 2350  cop 3409   class class class wbr 3793  (class class class)co 5543  [cec 6170  Pcnp 6543  1Pc1p 6544   +P cpp 6545   ·P cmp 6546  <P cltp 6547   ~R cer 6548  Rcnr 6549  0Rc0r 6550  1Rc1r 6551   ·R cmr 6554   <R cltr 6555
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-2o 6066  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-enq0 6676  df-nq0 6677  df-0nq0 6678  df-plq0 6679  df-mq0 6680  df-inp 6718  df-i1p 6719  df-iplp 6720  df-imp 6721  df-iltp 6722  df-enr 6965  df-nr 6966  df-mr 6968  df-ltr 6969  df-0r 6970  df-1r 6971
This theorem is referenced by:  recexsrlem  7013  axprecex  7108
  Copyright terms: Public domain W3C validator