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

Theorem recexgt0sr 7581
 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 7546 . . . 4 <R ⊆ (R × R)
21brel 4591 . . 3 (0R <R 𝐴 → (0RR𝐴R))
32simprd 113 . 2 (0R <R 𝐴𝐴R)
4 df-nr 7535 . . 3 R = ((P × P) / ~R )
5 breq2 3933 . . . 4 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → (0R <R [⟨𝑦, 𝑧⟩] ~R ↔ 0R <R 𝐴))
6 oveq1 5781 . . . . . . 7 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = (𝐴 ·R 𝑥))
76eqeq1d 2148 . . . . . 6 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → (([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R ↔ (𝐴 ·R 𝑥) = 1R))
87anbi2d 459 . . . . 5 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → ((0R <R 𝑥 ∧ ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R) ↔ (0R <R 𝑥 ∧ (𝐴 ·R 𝑥) = 1R)))
98rexbidv 2438 . . . 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 233 . . 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 7556 . . . . 5 (0R <R [⟨𝑦, 𝑧⟩] ~R𝑧<P 𝑦)
12 ltexpri 7421 . . . . 5 (𝑧<P 𝑦 → ∃𝑤P (𝑧 +P 𝑤) = 𝑦)
1311, 12sylbi 120 . . . 4 (0R <R [⟨𝑦, 𝑧⟩] ~R → ∃𝑤P (𝑧 +P 𝑤) = 𝑦)
14 recexpr 7446 . . . . . . 7 (𝑤P → ∃𝑣P (𝑤 ·P 𝑣) = 1P)
1514adantl 275 . . . . . 6 (((𝑦P𝑧P) ∧ 𝑤P) → ∃𝑣P (𝑤 ·P 𝑣) = 1P)
16 1pr 7362 . . . . . . . . . . . . . 14 1PP
17 addclpr 7345 . . . . . . . . . . . . . 14 ((𝑣P ∧ 1PP) → (𝑣 +P 1P) ∈ P)
1816, 17mpan2 421 . . . . . . . . . . . . 13 (𝑣P → (𝑣 +P 1P) ∈ P)
19 enrex 7545 . . . . . . . . . . . . . 14 ~R ∈ V
2019, 4ecopqsi 6484 . . . . . . . . . . . . 13 (((𝑣 +P 1P) ∈ P ∧ 1PP) → [⟨(𝑣 +P 1P), 1P⟩] ~RR)
2118, 16, 20sylancl 409 . . . . . . . . . . . 12 (𝑣P → [⟨(𝑣 +P 1P), 1P⟩] ~RR)
2221adantl 275 . . . . . . . . . . 11 ((𝑤P𝑣P) → [⟨(𝑣 +P 1P), 1P⟩] ~RR)
2322ad2antlr 480 . . . . . . . . . 10 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → [⟨(𝑣 +P 1P), 1P⟩] ~RR)
24 simprr 521 . . . . . . . . . . . 12 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → 𝑣P)
2524adantr 274 . . . . . . . . . . 11 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → 𝑣P)
26 ltaddpr 7405 . . . . . . . . . . . . . 14 ((1PP𝑣P) → 1P<P (1P +P 𝑣))
2716, 26mpan 420 . . . . . . . . . . . . 13 (𝑣P → 1P<P (1P +P 𝑣))
28 addcomprg 7386 . . . . . . . . . . . . . 14 ((1PP𝑣P) → (1P +P 𝑣) = (𝑣 +P 1P))
2916, 28mpan 420 . . . . . . . . . . . . 13 (𝑣P → (1P +P 𝑣) = (𝑣 +P 1P))
3027, 29breqtrd 3954 . . . . . . . . . . . 12 (𝑣P → 1P<P (𝑣 +P 1P))
31 gt0srpr 7556 . . . . . . . . . . . 12 (0R <R [⟨(𝑣 +P 1P), 1P⟩] ~R ↔ 1P<P (𝑣 +P 1P))
3230, 31sylibr 133 . . . . . . . . . . 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 311 . . . . . . . . . . . . . . . 16 (𝑣P → ((𝑣 +P 1P) ∈ P ∧ 1PP))
3534anim2i 339 . . . . . . . . . . . . . . 15 (((𝑦P𝑧P) ∧ 𝑣P) → ((𝑦P𝑧P) ∧ ((𝑣 +P 1P) ∈ P ∧ 1PP)))
3635adantr 274 . . . . . . . . . . . . . 14 ((((𝑦P𝑧P) ∧ 𝑣P) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ((𝑦P𝑧P) ∧ ((𝑣 +P 1P) ∈ P ∧ 1PP)))
37 mulsrpr 7554 . . . . . . . . . . . . . 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 473 . . . . . . . . . . . 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 5781 . . . . . . . . . . . . . . . . . . . . 21 ((𝑧 +P 𝑤) = 𝑦 → ((𝑧 +P 𝑤) ·P 𝑣) = (𝑦 ·P 𝑣))
4140eqcomd 2145 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 +P 𝑤) = 𝑦 → (𝑦 ·P 𝑣) = ((𝑧 +P 𝑤) ·P 𝑣))
4241ad2antll 482 . . . . . . . . . . . . . . . . . . 19 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → (𝑦 ·P 𝑣) = ((𝑧 +P 𝑤) ·P 𝑣))
43 mulcomprg 7388 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓PP) → (𝑓 ·P ) = ( ·P 𝑓))
44433adant2 1000 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓P𝑔PP) → (𝑓 ·P ) = ( ·P 𝑓))
45 mulcomprg 7388 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑔PP) → (𝑔 ·P ) = ( ·P 𝑔))
46453adant1 999 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓P𝑔PP) → (𝑔 ·P ) = ( ·P 𝑔))
4744, 46oveq12d 5792 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓P𝑔PP) → ((𝑓 ·P ) +P (𝑔 ·P )) = (( ·P 𝑓) +P ( ·P 𝑔)))
48 distrprg 7396 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((P𝑓P𝑔P) → ( ·P (𝑓 +P 𝑔)) = (( ·P 𝑓) +P ( ·P 𝑔)))
49483coml 1188 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓P𝑔PP) → ( ·P (𝑓 +P 𝑔)) = (( ·P 𝑓) +P ( ·P 𝑔)))
50 simp3 983 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓P𝑔PP) → P)
51 addclpr 7345 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓P𝑔P) → (𝑓 +P 𝑔) ∈ P)
52513adant3 1001 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓P𝑔PP) → (𝑓 +P 𝑔) ∈ P)
53 mulcomprg 7388 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((P ∧ (𝑓 +P 𝑔) ∈ P) → ( ·P (𝑓 +P 𝑔)) = ((𝑓 +P 𝑔) ·P ))
5450, 52, 53syl2anc 408 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓P𝑔PP) → ( ·P (𝑓 +P 𝑔)) = ((𝑓 +P 𝑔) ·P ))
5547, 49, 543eqtr2rd 2179 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓P𝑔PP) → ((𝑓 +P 𝑔) ·P ) = ((𝑓 ·P ) +P (𝑔 ·P )))
5655adantl 275 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ (𝑓P𝑔PP)) → ((𝑓 +P 𝑔) ·P ) = ((𝑓 ·P ) +P (𝑔 ·P )))
57 simplr 519 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → 𝑧P)
58 simprl 520 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → 𝑤P)
5956, 57, 58, 24caovdird 5949 . . . . . . . . . . . . . . . . . . . . 21 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑧 +P 𝑤) ·P 𝑣) = ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑣)))
60 oveq2 5782 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤 ·P 𝑣) = 1P → ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑣)) = ((𝑧 ·P 𝑣) +P 1P))
6159, 60sylan9eq 2192 . . . . . . . . . . . . . . . . . . . 20 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ (𝑤 ·P 𝑣) = 1P) → ((𝑧 +P 𝑤) ·P 𝑣) = ((𝑧 ·P 𝑣) +P 1P))
6261adantrr 470 . . . . . . . . . . . . . . . . . . 19 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ((𝑧 +P 𝑤) ·P 𝑣) = ((𝑧 ·P 𝑣) +P 1P))
6342, 62eqtrd 2172 . . . . . . . . . . . . . . . . . 18 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → (𝑦 ·P 𝑣) = ((𝑧 ·P 𝑣) +P 1P))
6463oveq1d 5789 . . . . . . . . . . . . . . . . 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 7380 . . . . . . . . . . . . . . . . . . . 20 ((𝑧P𝑣P) → (𝑧 ·P 𝑣) ∈ P)
6657, 24, 65syl2anc 408 . . . . . . . . . . . . . . . . . . 19 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑧 ·P 𝑣) ∈ P)
6716a1i 9 . . . . . . . . . . . . . . . . . . 19 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → 1PP)
68 simpll 518 . . . . . . . . . . . . . . . . . . . . 21 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → 𝑦P)
69 mulclpr 7380 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦P ∧ 1PP) → (𝑦 ·P 1P) ∈ P)
7068, 16, 69sylancl 409 . . . . . . . . . . . . . . . . . . . 20 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑦 ·P 1P) ∈ P)
71 mulclpr 7380 . . . . . . . . . . . . . . . . . . . . 21 ((𝑧P ∧ 1PP) → (𝑧 ·P 1P) ∈ P)
7257, 16, 71sylancl 409 . . . . . . . . . . . . . . . . . . . 20 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑧 ·P 1P) ∈ P)
73 addclpr 7345 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 ·P 1P) ∈ P ∧ (𝑧 ·P 1P) ∈ P) → ((𝑦 ·P 1P) +P (𝑧 ·P 1P)) ∈ P)
7470, 72, 73syl2anc 408 . . . . . . . . . . . . . . . . . . 19 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑦 ·P 1P) +P (𝑧 ·P 1P)) ∈ P)
75 addcomprg 7386 . . . . . . . . . . . . . . . . . . . 20 ((𝑓P𝑔P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
7675adantl 275 . . . . . . . . . . . . . . . . . . 19 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
77 addassprg 7387 . . . . . . . . . . . . . . . . . . . 20 ((𝑓P𝑔PP) → ((𝑓 +P 𝑔) +P ) = (𝑓 +P (𝑔 +P )))
7877adantl 275 . . . . . . . . . . . . . . . . . . 19 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ (𝑓P𝑔PP)) → ((𝑓 +P 𝑔) +P ) = (𝑓 +P (𝑔 +P )))
7966, 67, 74, 76, 78caov32d 5951 . . . . . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . . . . 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 2172 . . . . . . . . . . . . . . . 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 5789 . . . . . . . . . . . . . . 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 7345 . . . . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . . . . 17 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) ∈ P)
8584adantr 274 . . . . . . . . . . . . . . . 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 7387 . . . . . . . . . . . . . . . 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 1216 . . . . . . . . . . . . . . 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 2172 . . . . . . . . . . . . . 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 7396 . . . . . . . . . . . . . . . . . . 19 ((𝑦P𝑣P ∧ 1PP) → (𝑦 ·P (𝑣 +P 1P)) = ((𝑦 ·P 𝑣) +P (𝑦 ·P 1P)))
9168, 24, 67, 90syl3anc 1216 . . . . . . . . . . . . . . . . . 18 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑦 ·P (𝑣 +P 1P)) = ((𝑦 ·P 𝑣) +P (𝑦 ·P 1P)))
9291oveq1d 5789 . . . . . . . . . . . . . . . . 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 7380 . . . . . . . . . . . . . . . . . . 19 ((𝑦P𝑣P) → (𝑦 ·P 𝑣) ∈ P)
9468, 24, 93syl2anc 408 . . . . . . . . . . . . . . . . . 18 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑦 ·P 𝑣) ∈ P)
95 addassprg 7387 . . . . . . . . . . . . . . . . . 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 1216 . . . . . . . . . . . . . . . . 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 2172 . . . . . . . . . . . . . . . 16 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) = ((𝑦 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))))
9897oveq1d 5789 . . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . 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 7396 . . . . . . . . . . . . . . . . . . 19 ((𝑧P𝑣P ∧ 1PP) → (𝑧 ·P (𝑣 +P 1P)) = ((𝑧 ·P 𝑣) +P (𝑧 ·P 1P)))
10157, 24, 67, 100syl3anc 1216 . . . . . . . . . . . . . . . . . 18 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑧 ·P (𝑣 +P 1P)) = ((𝑧 ·P 𝑣) +P (𝑧 ·P 1P)))
102101oveq2d 5790 . . . . . . . . . . . . . . . . 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 5952 . . . . . . . . . . . . . . . . 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 2172 . . . . . . . . . . . . . . . 16 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) = ((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))))
105104oveq1d 5789 . . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . 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 2182 . . . . . . . . . . . . 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 409 . . . . . . . . . . . . . . . . 17 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑣 +P 1P) ∈ P)
109 mulclpr 7380 . . . . . . . . . . . . . . . . 17 ((𝑦P ∧ (𝑣 +P 1P) ∈ P) → (𝑦 ·P (𝑣 +P 1P)) ∈ P)
11068, 108, 109syl2anc 408 . . . . . . . . . . . . . . . 16 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑦 ·P (𝑣 +P 1P)) ∈ P)
111 addclpr 7345 . . . . . . . . . . . . . . . 16 (((𝑦 ·P (𝑣 +P 1P)) ∈ P ∧ (𝑧 ·P 1P) ∈ P) → ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) ∈ P)
112110, 72, 111syl2anc 408 . . . . . . . . . . . . . . 15 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) ∈ P)
113104, 84eqeltrd 2216 . . . . . . . . . . . . . . 15 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) ∈ P)
114 addclpr 7345 . . . . . . . . . . . . . . . . 17 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
11516, 16, 114mp2an 422 . . . . . . . . . . . . . . . 16 (1P +P 1P) ∈ P
116115a1i 9 . . . . . . . . . . . . . . 15 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (1P +P 1P) ∈ P)
117 enreceq 7544 . . . . . . . . . . . . . . 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 1217 . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . 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 166 . . . . . . . . . . . 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 2172 . . . . . . . . . . 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 7540 . . . . . . . . . . 11 1R = [⟨(1P +P 1P), 1P⟩] ~R
123121, 122syl6eqr 2190 . . . . . . . . . 10 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ) = 1R)
124 breq2 3933 . . . . . . . . . . . 12 (𝑥 = [⟨(𝑣 +P 1P), 1P⟩] ~R → (0R <R 𝑥 ↔ 0R <R [⟨(𝑣 +P 1P), 1P⟩] ~R ))
125 oveq2 5782 . . . . . . . . . . . . 13 (𝑥 = [⟨(𝑣 +P 1P), 1P⟩] ~R → ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ))
126125eqeq1d 2148 . . . . . . . . . . . 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 464 . . . . . . . . . . 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 2789 . . . . . . . . . 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 1214 . . . . . . . . 9 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ∃𝑥R (0R <R 𝑥 ∧ ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R))
130129exp32 362 . . . . . . . 8 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑤 ·P 𝑣) = 1P → ((𝑧 +P 𝑤) = 𝑦 → ∃𝑥R (0R <R 𝑥 ∧ ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R))))
131130anassrs 397 . . . . . . 7 ((((𝑦P𝑧P) ∧ 𝑤P) ∧ 𝑣P) → ((𝑤 ·P 𝑣) = 1P → ((𝑧 +P 𝑤) = 𝑦 → ∃𝑥R (0R <R 𝑥 ∧ ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R))))
132131rexlimdva 2549 . . . . . 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 2549 . . . 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 6516 . 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 103   ↔ wb 104   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  ∃wrex 2417  ⟨cop 3530   class class class wbr 3929  (class class class)co 5774  [cec 6427  Pcnp 7099  1Pc1p 7100   +P cpp 7101   ·P cmp 7102
 Copyright terms: Public domain W3C validator