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Theorem recexgt0sr 6856
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 6821 . . . 4 <R ⊆ (R × R)
21brel 4392 . . 3 (0R <R 𝐴 → (0RR𝐴R))
32simprd 107 . 2 (0R <R 𝐴𝐴R)
4 df-nr 6810 . . 3 R = ((P × P) / ~R )
5 breq2 3768 . . . 4 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → (0R <R [⟨𝑦, 𝑧⟩] ~R ↔ 0R <R 𝐴))
6 oveq1 5519 . . . . . . 7 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = (𝐴 ·R 𝑥))
76eqeq1d 2048 . . . . . 6 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → (([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R ↔ (𝐴 ·R 𝑥) = 1R))
87anbi2d 437 . . . . 5 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → ((0R <R 𝑥 ∧ ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R) ↔ (0R <R 𝑥 ∧ (𝐴 ·R 𝑥) = 1R)))
98rexbidv 2327 . . . 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 223 . . 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 6831 . . . . 5 (0R <R [⟨𝑦, 𝑧⟩] ~R𝑧<P 𝑦)
12 ltexpri 6709 . . . . 5 (𝑧<P 𝑦 → ∃𝑤P (𝑧 +P 𝑤) = 𝑦)
1311, 12sylbi 114 . . . 4 (0R <R [⟨𝑦, 𝑧⟩] ~R → ∃𝑤P (𝑧 +P 𝑤) = 𝑦)
14 recexpr 6734 . . . . . . 7 (𝑤P → ∃𝑣P (𝑤 ·P 𝑣) = 1P)
1514adantl 262 . . . . . 6 (((𝑦P𝑧P) ∧ 𝑤P) → ∃𝑣P (𝑤 ·P 𝑣) = 1P)
16 1pr 6650 . . . . . . . . . . . . . 14 1PP
17 addclpr 6633 . . . . . . . . . . . . . 14 ((𝑣P ∧ 1PP) → (𝑣 +P 1P) ∈ P)
1816, 17mpan2 401 . . . . . . . . . . . . 13 (𝑣P → (𝑣 +P 1P) ∈ P)
19 enrex 6820 . . . . . . . . . . . . . 14 ~R ∈ V
2019, 4ecopqsi 6161 . . . . . . . . . . . . 13 (((𝑣 +P 1P) ∈ P ∧ 1PP) → [⟨(𝑣 +P 1P), 1P⟩] ~RR)
2118, 16, 20sylancl 392 . . . . . . . . . . . 12 (𝑣P → [⟨(𝑣 +P 1P), 1P⟩] ~RR)
2221adantl 262 . . . . . . . . . . 11 ((𝑤P𝑣P) → [⟨(𝑣 +P 1P), 1P⟩] ~RR)
2322ad2antlr 458 . . . . . . . . . 10 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → [⟨(𝑣 +P 1P), 1P⟩] ~RR)
24 simprr 484 . . . . . . . . . . . 12 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → 𝑣P)
2524adantr 261 . . . . . . . . . . 11 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → 𝑣P)
26 ltaddpr 6693 . . . . . . . . . . . . . 14 ((1PP𝑣P) → 1P<P (1P +P 𝑣))
2716, 26mpan 400 . . . . . . . . . . . . 13 (𝑣P → 1P<P (1P +P 𝑣))
28 addcomprg 6674 . . . . . . . . . . . . . 14 ((1PP𝑣P) → (1P +P 𝑣) = (𝑣 +P 1P))
2916, 28mpan 400 . . . . . . . . . . . . 13 (𝑣P → (1P +P 𝑣) = (𝑣 +P 1P))
3027, 29breqtrd 3788 . . . . . . . . . . . 12 (𝑣P → 1P<P (𝑣 +P 1P))
31 gt0srpr 6831 . . . . . . . . . . . 12 (0R <R [⟨(𝑣 +P 1P), 1P⟩] ~R ↔ 1P<P (𝑣 +P 1P))
3230, 31sylibr 137 . . . . . . . . . . 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 296 . . . . . . . . . . . . . . . 16 (𝑣P → ((𝑣 +P 1P) ∈ P ∧ 1PP))
3534anim2i 324 . . . . . . . . . . . . . . 15 (((𝑦P𝑧P) ∧ 𝑣P) → ((𝑦P𝑧P) ∧ ((𝑣 +P 1P) ∈ P ∧ 1PP)))
3635adantr 261 . . . . . . . . . . . . . 14 ((((𝑦P𝑧P) ∧ 𝑣P) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ((𝑦P𝑧P) ∧ ((𝑣 +P 1P) ∈ P ∧ 1PP)))
37 mulsrpr 6829 . . . . . . . . . . . . . 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 451 . . . . . . . . . . . 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 5519 . . . . . . . . . . . . . . . . . . . . 21 ((𝑧 +P 𝑤) = 𝑦 → ((𝑧 +P 𝑤) ·P 𝑣) = (𝑦 ·P 𝑣))
4140eqcomd 2045 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 +P 𝑤) = 𝑦 → (𝑦 ·P 𝑣) = ((𝑧 +P 𝑤) ·P 𝑣))
4241ad2antll 460 . . . . . . . . . . . . . . . . . . 19 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → (𝑦 ·P 𝑣) = ((𝑧 +P 𝑤) ·P 𝑣))
43 mulcomprg 6676 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓PP) → (𝑓 ·P ) = ( ·P 𝑓))
44433adant2 923 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓P𝑔PP) → (𝑓 ·P ) = ( ·P 𝑓))
45 mulcomprg 6676 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑔PP) → (𝑔 ·P ) = ( ·P 𝑔))
46453adant1 922 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓P𝑔PP) → (𝑔 ·P ) = ( ·P 𝑔))
4744, 46oveq12d 5530 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓P𝑔PP) → ((𝑓 ·P ) +P (𝑔 ·P )) = (( ·P 𝑓) +P ( ·P 𝑔)))
48 distrprg 6684 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((P𝑓P𝑔P) → ( ·P (𝑓 +P 𝑔)) = (( ·P 𝑓) +P ( ·P 𝑔)))
49483coml 1111 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓P𝑔PP) → ( ·P (𝑓 +P 𝑔)) = (( ·P 𝑓) +P ( ·P 𝑔)))
50 simp3 906 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓P𝑔PP) → P)
51 addclpr 6633 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓P𝑔P) → (𝑓 +P 𝑔) ∈ P)
52513adant3 924 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓P𝑔PP) → (𝑓 +P 𝑔) ∈ P)
53 mulcomprg 6676 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((P ∧ (𝑓 +P 𝑔) ∈ P) → ( ·P (𝑓 +P 𝑔)) = ((𝑓 +P 𝑔) ·P ))
5450, 52, 53syl2anc 391 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓P𝑔PP) → ( ·P (𝑓 +P 𝑔)) = ((𝑓 +P 𝑔) ·P ))
5547, 49, 543eqtr2rd 2079 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓P𝑔PP) → ((𝑓 +P 𝑔) ·P ) = ((𝑓 ·P ) +P (𝑔 ·P )))
5655adantl 262 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ (𝑓P𝑔PP)) → ((𝑓 +P 𝑔) ·P ) = ((𝑓 ·P ) +P (𝑔 ·P )))
57 simplr 482 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → 𝑧P)
58 simprl 483 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → 𝑤P)
5956, 57, 58, 24caovdird 5679 . . . . . . . . . . . . . . . . . . . . 21 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑧 +P 𝑤) ·P 𝑣) = ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑣)))
60 oveq2 5520 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤 ·P 𝑣) = 1P → ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑣)) = ((𝑧 ·P 𝑣) +P 1P))
6159, 60sylan9eq 2092 . . . . . . . . . . . . . . . . . . . 20 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ (𝑤 ·P 𝑣) = 1P) → ((𝑧 +P 𝑤) ·P 𝑣) = ((𝑧 ·P 𝑣) +P 1P))
6261adantrr 448 . . . . . . . . . . . . . . . . . . 19 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ((𝑧 +P 𝑤) ·P 𝑣) = ((𝑧 ·P 𝑣) +P 1P))
6342, 62eqtrd 2072 . . . . . . . . . . . . . . . . . 18 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → (𝑦 ·P 𝑣) = ((𝑧 ·P 𝑣) +P 1P))
6463oveq1d 5527 . . . . . . . . . . . . . . . . 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 6668 . . . . . . . . . . . . . . . . . . . 20 ((𝑧P𝑣P) → (𝑧 ·P 𝑣) ∈ P)
6657, 24, 65syl2anc 391 . . . . . . . . . . . . . . . . . . 19 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑧 ·P 𝑣) ∈ P)
6716a1i 9 . . . . . . . . . . . . . . . . . . 19 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → 1PP)
68 simpll 481 . . . . . . . . . . . . . . . . . . . . 21 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → 𝑦P)
69 mulclpr 6668 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦P ∧ 1PP) → (𝑦 ·P 1P) ∈ P)
7068, 16, 69sylancl 392 . . . . . . . . . . . . . . . . . . . 20 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑦 ·P 1P) ∈ P)
71 mulclpr 6668 . . . . . . . . . . . . . . . . . . . . 21 ((𝑧P ∧ 1PP) → (𝑧 ·P 1P) ∈ P)
7257, 16, 71sylancl 392 . . . . . . . . . . . . . . . . . . . 20 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑧 ·P 1P) ∈ P)
73 addclpr 6633 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 ·P 1P) ∈ P ∧ (𝑧 ·P 1P) ∈ P) → ((𝑦 ·P 1P) +P (𝑧 ·P 1P)) ∈ P)
7470, 72, 73syl2anc 391 . . . . . . . . . . . . . . . . . . 19 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑦 ·P 1P) +P (𝑧 ·P 1P)) ∈ P)
75 addcomprg 6674 . . . . . . . . . . . . . . . . . . . 20 ((𝑓P𝑔P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
7675adantl 262 . . . . . . . . . . . . . . . . . . 19 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
77 addassprg 6675 . . . . . . . . . . . . . . . . . . . 20 ((𝑓P𝑔PP) → ((𝑓 +P 𝑔) +P ) = (𝑓 +P (𝑔 +P )))
7877adantl 262 . . . . . . . . . . . . . . . . . . 19 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ (𝑓P𝑔PP)) → ((𝑓 +P 𝑔) +P ) = (𝑓 +P (𝑔 +P )))
7966, 67, 74, 76, 78caov32d 5681 . . . . . . . . . . . . . . . . . 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 261 . . . . . . . . . . . . . . . . 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 2072 . . . . . . . . . . . . . . . 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 5527 . . . . . . . . . . . . . . 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 6633 . . . . . . . . . . . . . . . . . 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 391 . . . . . . . . . . . . . . . . 17 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) ∈ P)
8584adantr 261 . . . . . . . . . . . . . . . 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 6675 . . . . . . . . . . . . . . . 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 1135 . . . . . . . . . . . . . . 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 2072 . . . . . . . . . . . . . 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 6684 . . . . . . . . . . . . . . . . . . 19 ((𝑦P𝑣P ∧ 1PP) → (𝑦 ·P (𝑣 +P 1P)) = ((𝑦 ·P 𝑣) +P (𝑦 ·P 1P)))
9168, 24, 67, 90syl3anc 1135 . . . . . . . . . . . . . . . . . 18 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑦 ·P (𝑣 +P 1P)) = ((𝑦 ·P 𝑣) +P (𝑦 ·P 1P)))
9291oveq1d 5527 . . . . . . . . . . . . . . . . 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 6668 . . . . . . . . . . . . . . . . . . 19 ((𝑦P𝑣P) → (𝑦 ·P 𝑣) ∈ P)
9468, 24, 93syl2anc 391 . . . . . . . . . . . . . . . . . 18 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑦 ·P 𝑣) ∈ P)
95 addassprg 6675 . . . . . . . . . . . . . . . . . 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 1135 . . . . . . . . . . . . . . . . 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 2072 . . . . . . . . . . . . . . . 16 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) = ((𝑦 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))))
9897oveq1d 5527 . . . . . . . . . . . . . . 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 261 . . . . . . . . . . . . . 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 6684 . . . . . . . . . . . . . . . . . . 19 ((𝑧P𝑣P ∧ 1PP) → (𝑧 ·P (𝑣 +P 1P)) = ((𝑧 ·P 𝑣) +P (𝑧 ·P 1P)))
10157, 24, 67, 100syl3anc 1135 . . . . . . . . . . . . . . . . . 18 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑧 ·P (𝑣 +P 1P)) = ((𝑧 ·P 𝑣) +P (𝑧 ·P 1P)))
102101oveq2d 5528 . . . . . . . . . . . . . . . . 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 5682 . . . . . . . . . . . . . . . . 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 2072 . . . . . . . . . . . . . . . 16 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) = ((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))))
105104oveq1d 5527 . . . . . . . . . . . . . . 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 261 . . . . . . . . . . . . . 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 2082 . . . . . . . . . . . . 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 392 . . . . . . . . . . . . . . . . 17 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑣 +P 1P) ∈ P)
109 mulclpr 6668 . . . . . . . . . . . . . . . . 17 ((𝑦P ∧ (𝑣 +P 1P) ∈ P) → (𝑦 ·P (𝑣 +P 1P)) ∈ P)
11068, 108, 109syl2anc 391 . . . . . . . . . . . . . . . 16 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑦 ·P (𝑣 +P 1P)) ∈ P)
111 addclpr 6633 . . . . . . . . . . . . . . . 16 (((𝑦 ·P (𝑣 +P 1P)) ∈ P ∧ (𝑧 ·P 1P) ∈ P) → ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) ∈ P)
112110, 72, 111syl2anc 391 . . . . . . . . . . . . . . 15 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) ∈ P)
113104, 84eqeltrd 2114 . . . . . . . . . . . . . . 15 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) ∈ P)
114 addclpr 6633 . . . . . . . . . . . . . . . . 17 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
11516, 16, 114mp2an 402 . . . . . . . . . . . . . . . 16 (1P +P 1P) ∈ P
116115a1i 9 . . . . . . . . . . . . . . 15 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (1P +P 1P) ∈ P)
117 enreceq 6819 . . . . . . . . . . . . . . 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 1136 . . . . . . . . . . . . . 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 261 . . . . . . . . . . . . 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 156 . . . . . . . . . . . 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 2072 . . . . . . . . . . 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 6815 . . . . . . . . . . 11 1R = [⟨(1P +P 1P), 1P⟩] ~R
123121, 122syl6eqr 2090 . . . . . . . . . 10 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ) = 1R)
124 breq2 3768 . . . . . . . . . . . 12 (𝑥 = [⟨(𝑣 +P 1P), 1P⟩] ~R → (0R <R 𝑥 ↔ 0R <R [⟨(𝑣 +P 1P), 1P⟩] ~R ))
125 oveq2 5520 . . . . . . . . . . . . 13 (𝑥 = [⟨(𝑣 +P 1P), 1P⟩] ~R → ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ))
126125eqeq1d 2048 . . . . . . . . . . . 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 442 . . . . . . . . . . 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 2656 . . . . . . . . . 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 1133 . . . . . . . . 9 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ∃𝑥R (0R <R 𝑥 ∧ ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R))
130129exp32 347 . . . . . . . 8 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑤 ·P 𝑣) = 1P → ((𝑧 +P 𝑤) = 𝑦 → ∃𝑥R (0R <R 𝑥 ∧ ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R))))
131130anassrs 380 . . . . . . 7 ((((𝑦P𝑧P) ∧ 𝑤P) ∧ 𝑣P) → ((𝑤 ·P 𝑣) = 1P → ((𝑧 +P 𝑤) = 𝑦 → ∃𝑥R (0R <R 𝑥 ∧ ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R))))
132131rexlimdva 2433 . . . . . 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 2433 . . . 4 ((𝑦P𝑧P) → (∃𝑤P (𝑧 +P 𝑤) = 𝑦 → ∃𝑥R (0R <R 𝑥 ∧ ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R)))
13513, 134syl5 28 . . 3 ((𝑦P𝑧P) → (0R <R [⟨𝑦, 𝑧⟩] ~R → ∃𝑥R (0R <R 𝑥 ∧ ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R)))
1364, 10, 135ecoptocl 6193 . 2 (𝐴R → (0R <R 𝐴 → ∃𝑥R (0R <R 𝑥 ∧ (𝐴 ·R 𝑥) = 1R)))
1373, 136mpcom 32 1 (0R <R 𝐴 → ∃𝑥R (0R <R 𝑥 ∧ (𝐴 ·R 𝑥) = 1R))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  w3a 885   = wceq 1243  wcel 1393  wrex 2307  cop 3378   class class class wbr 3764  (class class class)co 5512  [cec 6104  Pcnp 6387  1Pc1p 6388   +P cpp 6389   ·P cmp 6390  <P cltp 6391   ~R cer 6392  Rcnr 6393  0Rc0r 6394  1Rc1r 6395   ·R cmr 6398   <R cltr 6399
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6400  df-pli 6401  df-mi 6402  df-lti 6403  df-plpq 6440  df-mpq 6441  df-enq 6443  df-nqqs 6444  df-plqqs 6445  df-mqqs 6446  df-1nqqs 6447  df-rq 6448  df-ltnqqs 6449  df-enq0 6520  df-nq0 6521  df-0nq0 6522  df-plq0 6523  df-mq0 6524  df-inp 6562  df-i1p 6563  df-iplp 6564  df-imp 6565  df-iltp 6566  df-enr 6809  df-nr 6810  df-mr 6812  df-ltr 6813  df-0r 6814  df-1r 6815
This theorem is referenced by:  recexsrlem  6857  axprecex  6952
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