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Theorem recexgt0sr 7747
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 7712 . . . 4 <R ⊆ (R × R)
21brel 4672 . . 3 (0R <R 𝐴 → (0RR𝐴R))
32simprd 114 . 2 (0R <R 𝐴𝐴R)
4 df-nr 7701 . . 3 R = ((P × P) / ~R )
5 breq2 4002 . . . 4 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → (0R <R [⟨𝑦, 𝑧⟩] ~R ↔ 0R <R 𝐴))
6 oveq1 5872 . . . . . . 7 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = (𝐴 ·R 𝑥))
76eqeq1d 2184 . . . . . 6 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → (([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R ↔ (𝐴 ·R 𝑥) = 1R))
87anbi2d 464 . . . . 5 ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → ((0R <R 𝑥 ∧ ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R) ↔ (0R <R 𝑥 ∧ (𝐴 ·R 𝑥) = 1R)))
98rexbidv 2476 . . . 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 234 . . 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 7722 . . . . 5 (0R <R [⟨𝑦, 𝑧⟩] ~R𝑧<P 𝑦)
12 ltexpri 7587 . . . . 5 (𝑧<P 𝑦 → ∃𝑤P (𝑧 +P 𝑤) = 𝑦)
1311, 12sylbi 121 . . . 4 (0R <R [⟨𝑦, 𝑧⟩] ~R → ∃𝑤P (𝑧 +P 𝑤) = 𝑦)
14 recexpr 7612 . . . . . . 7 (𝑤P → ∃𝑣P (𝑤 ·P 𝑣) = 1P)
1514adantl 277 . . . . . 6 (((𝑦P𝑧P) ∧ 𝑤P) → ∃𝑣P (𝑤 ·P 𝑣) = 1P)
16 1pr 7528 . . . . . . . . . . . . . 14 1PP
17 addclpr 7511 . . . . . . . . . . . . . 14 ((𝑣P ∧ 1PP) → (𝑣 +P 1P) ∈ P)
1816, 17mpan2 425 . . . . . . . . . . . . 13 (𝑣P → (𝑣 +P 1P) ∈ P)
19 enrex 7711 . . . . . . . . . . . . . 14 ~R ∈ V
2019, 4ecopqsi 6580 . . . . . . . . . . . . 13 (((𝑣 +P 1P) ∈ P ∧ 1PP) → [⟨(𝑣 +P 1P), 1P⟩] ~RR)
2118, 16, 20sylancl 413 . . . . . . . . . . . 12 (𝑣P → [⟨(𝑣 +P 1P), 1P⟩] ~RR)
2221adantl 277 . . . . . . . . . . 11 ((𝑤P𝑣P) → [⟨(𝑣 +P 1P), 1P⟩] ~RR)
2322ad2antlr 489 . . . . . . . . . 10 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → [⟨(𝑣 +P 1P), 1P⟩] ~RR)
24 simprr 531 . . . . . . . . . . . 12 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → 𝑣P)
2524adantr 276 . . . . . . . . . . 11 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → 𝑣P)
26 ltaddpr 7571 . . . . . . . . . . . . . 14 ((1PP𝑣P) → 1P<P (1P +P 𝑣))
2716, 26mpan 424 . . . . . . . . . . . . 13 (𝑣P → 1P<P (1P +P 𝑣))
28 addcomprg 7552 . . . . . . . . . . . . . 14 ((1PP𝑣P) → (1P +P 𝑣) = (𝑣 +P 1P))
2916, 28mpan 424 . . . . . . . . . . . . 13 (𝑣P → (1P +P 𝑣) = (𝑣 +P 1P))
3027, 29breqtrd 4024 . . . . . . . . . . . 12 (𝑣P → 1P<P (𝑣 +P 1P))
31 gt0srpr 7722 . . . . . . . . . . . 12 (0R <R [⟨(𝑣 +P 1P), 1P⟩] ~R ↔ 1P<P (𝑣 +P 1P))
3230, 31sylibr 134 . . . . . . . . . . 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 313 . . . . . . . . . . . . . . . 16 (𝑣P → ((𝑣 +P 1P) ∈ P ∧ 1PP))
3534anim2i 342 . . . . . . . . . . . . . . 15 (((𝑦P𝑧P) ∧ 𝑣P) → ((𝑦P𝑧P) ∧ ((𝑣 +P 1P) ∈ P ∧ 1PP)))
3635adantr 276 . . . . . . . . . . . . . 14 ((((𝑦P𝑧P) ∧ 𝑣P) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ((𝑦P𝑧P) ∧ ((𝑣 +P 1P) ∈ P ∧ 1PP)))
37 mulsrpr 7720 . . . . . . . . . . . . . 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 482 . . . . . . . . . . . 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 5872 . . . . . . . . . . . . . . . . . . . . 21 ((𝑧 +P 𝑤) = 𝑦 → ((𝑧 +P 𝑤) ·P 𝑣) = (𝑦 ·P 𝑣))
4140eqcomd 2181 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 +P 𝑤) = 𝑦 → (𝑦 ·P 𝑣) = ((𝑧 +P 𝑤) ·P 𝑣))
4241ad2antll 491 . . . . . . . . . . . . . . . . . . 19 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → (𝑦 ·P 𝑣) = ((𝑧 +P 𝑤) ·P 𝑣))
43 mulcomprg 7554 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓PP) → (𝑓 ·P ) = ( ·P 𝑓))
44433adant2 1016 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓P𝑔PP) → (𝑓 ·P ) = ( ·P 𝑓))
45 mulcomprg 7554 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑔PP) → (𝑔 ·P ) = ( ·P 𝑔))
46453adant1 1015 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓P𝑔PP) → (𝑔 ·P ) = ( ·P 𝑔))
4744, 46oveq12d 5883 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓P𝑔PP) → ((𝑓 ·P ) +P (𝑔 ·P )) = (( ·P 𝑓) +P ( ·P 𝑔)))
48 distrprg 7562 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((P𝑓P𝑔P) → ( ·P (𝑓 +P 𝑔)) = (( ·P 𝑓) +P ( ·P 𝑔)))
49483coml 1210 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓P𝑔PP) → ( ·P (𝑓 +P 𝑔)) = (( ·P 𝑓) +P ( ·P 𝑔)))
50 simp3 999 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓P𝑔PP) → P)
51 addclpr 7511 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓P𝑔P) → (𝑓 +P 𝑔) ∈ P)
52513adant3 1017 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓P𝑔PP) → (𝑓 +P 𝑔) ∈ P)
53 mulcomprg 7554 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((P ∧ (𝑓 +P 𝑔) ∈ P) → ( ·P (𝑓 +P 𝑔)) = ((𝑓 +P 𝑔) ·P ))
5450, 52, 53syl2anc 411 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓P𝑔PP) → ( ·P (𝑓 +P 𝑔)) = ((𝑓 +P 𝑔) ·P ))
5547, 49, 543eqtr2rd 2215 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓P𝑔PP) → ((𝑓 +P 𝑔) ·P ) = ((𝑓 ·P ) +P (𝑔 ·P )))
5655adantl 277 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ (𝑓P𝑔PP)) → ((𝑓 +P 𝑔) ·P ) = ((𝑓 ·P ) +P (𝑔 ·P )))
57 simplr 528 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → 𝑧P)
58 simprl 529 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → 𝑤P)
5956, 57, 58, 24caovdird 6043 . . . . . . . . . . . . . . . . . . . . 21 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑧 +P 𝑤) ·P 𝑣) = ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑣)))
60 oveq2 5873 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤 ·P 𝑣) = 1P → ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑣)) = ((𝑧 ·P 𝑣) +P 1P))
6159, 60sylan9eq 2228 . . . . . . . . . . . . . . . . . . . 20 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ (𝑤 ·P 𝑣) = 1P) → ((𝑧 +P 𝑤) ·P 𝑣) = ((𝑧 ·P 𝑣) +P 1P))
6261adantrr 479 . . . . . . . . . . . . . . . . . . 19 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ((𝑧 +P 𝑤) ·P 𝑣) = ((𝑧 ·P 𝑣) +P 1P))
6342, 62eqtrd 2208 . . . . . . . . . . . . . . . . . 18 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → (𝑦 ·P 𝑣) = ((𝑧 ·P 𝑣) +P 1P))
6463oveq1d 5880 . . . . . . . . . . . . . . . . 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 7546 . . . . . . . . . . . . . . . . . . . 20 ((𝑧P𝑣P) → (𝑧 ·P 𝑣) ∈ P)
6657, 24, 65syl2anc 411 . . . . . . . . . . . . . . . . . . 19 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑧 ·P 𝑣) ∈ P)
6716a1i 9 . . . . . . . . . . . . . . . . . . 19 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → 1PP)
68 simpll 527 . . . . . . . . . . . . . . . . . . . . 21 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → 𝑦P)
69 mulclpr 7546 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦P ∧ 1PP) → (𝑦 ·P 1P) ∈ P)
7068, 16, 69sylancl 413 . . . . . . . . . . . . . . . . . . . 20 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑦 ·P 1P) ∈ P)
71 mulclpr 7546 . . . . . . . . . . . . . . . . . . . . 21 ((𝑧P ∧ 1PP) → (𝑧 ·P 1P) ∈ P)
7257, 16, 71sylancl 413 . . . . . . . . . . . . . . . . . . . 20 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑧 ·P 1P) ∈ P)
73 addclpr 7511 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 ·P 1P) ∈ P ∧ (𝑧 ·P 1P) ∈ P) → ((𝑦 ·P 1P) +P (𝑧 ·P 1P)) ∈ P)
7470, 72, 73syl2anc 411 . . . . . . . . . . . . . . . . . . 19 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑦 ·P 1P) +P (𝑧 ·P 1P)) ∈ P)
75 addcomprg 7552 . . . . . . . . . . . . . . . . . . . 20 ((𝑓P𝑔P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
7675adantl 277 . . . . . . . . . . . . . . . . . . 19 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
77 addassprg 7553 . . . . . . . . . . . . . . . . . . . 20 ((𝑓P𝑔PP) → ((𝑓 +P 𝑔) +P ) = (𝑓 +P (𝑔 +P )))
7877adantl 277 . . . . . . . . . . . . . . . . . . 19 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ (𝑓P𝑔PP)) → ((𝑓 +P 𝑔) +P ) = (𝑓 +P (𝑔 +P )))
7966, 67, 74, 76, 78caov32d 6045 . . . . . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . . . . 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 2208 . . . . . . . . . . . . . . . 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 5880 . . . . . . . . . . . . . . 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 7511 . . . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . . . 17 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) ∈ P)
8584adantr 276 . . . . . . . . . . . . . . . 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 7553 . . . . . . . . . . . . . . . 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 1238 . . . . . . . . . . . . . . 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 2208 . . . . . . . . . . . . . 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 7562 . . . . . . . . . . . . . . . . . . 19 ((𝑦P𝑣P ∧ 1PP) → (𝑦 ·P (𝑣 +P 1P)) = ((𝑦 ·P 𝑣) +P (𝑦 ·P 1P)))
9168, 24, 67, 90syl3anc 1238 . . . . . . . . . . . . . . . . . 18 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑦 ·P (𝑣 +P 1P)) = ((𝑦 ·P 𝑣) +P (𝑦 ·P 1P)))
9291oveq1d 5880 . . . . . . . . . . . . . . . . 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 7546 . . . . . . . . . . . . . . . . . . 19 ((𝑦P𝑣P) → (𝑦 ·P 𝑣) ∈ P)
9468, 24, 93syl2anc 411 . . . . . . . . . . . . . . . . . 18 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑦 ·P 𝑣) ∈ P)
95 addassprg 7553 . . . . . . . . . . . . . . . . . 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 1238 . . . . . . . . . . . . . . . . 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 2208 . . . . . . . . . . . . . . . 16 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) = ((𝑦 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))))
9897oveq1d 5880 . . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . 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 7562 . . . . . . . . . . . . . . . . . . 19 ((𝑧P𝑣P ∧ 1PP) → (𝑧 ·P (𝑣 +P 1P)) = ((𝑧 ·P 𝑣) +P (𝑧 ·P 1P)))
10157, 24, 67, 100syl3anc 1238 . . . . . . . . . . . . . . . . . 18 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑧 ·P (𝑣 +P 1P)) = ((𝑧 ·P 𝑣) +P (𝑧 ·P 1P)))
102101oveq2d 5881 . . . . . . . . . . . . . . . . 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 6046 . . . . . . . . . . . . . . . . 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 2208 . . . . . . . . . . . . . . . 16 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) = ((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))))
105104oveq1d 5880 . . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . 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 2218 . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . . 17 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑣 +P 1P) ∈ P)
109 mulclpr 7546 . . . . . . . . . . . . . . . . 17 ((𝑦P ∧ (𝑣 +P 1P) ∈ P) → (𝑦 ·P (𝑣 +P 1P)) ∈ P)
11068, 108, 109syl2anc 411 . . . . . . . . . . . . . . . 16 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (𝑦 ·P (𝑣 +P 1P)) ∈ P)
111 addclpr 7511 . . . . . . . . . . . . . . . 16 (((𝑦 ·P (𝑣 +P 1P)) ∈ P ∧ (𝑧 ·P 1P) ∈ P) → ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) ∈ P)
112110, 72, 111syl2anc 411 . . . . . . . . . . . . . . 15 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) ∈ P)
113104, 84eqeltrd 2252 . . . . . . . . . . . . . . 15 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) ∈ P)
114 addclpr 7511 . . . . . . . . . . . . . . . . 17 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
11516, 16, 114mp2an 426 . . . . . . . . . . . . . . . 16 (1P +P 1P) ∈ P
116115a1i 9 . . . . . . . . . . . . . . 15 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → (1P +P 1P) ∈ P)
117 enreceq 7710 . . . . . . . . . . . . . . 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 1239 . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . 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 167 . . . . . . . . . . . 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 2208 . . . . . . . . . . 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 7706 . . . . . . . . . . 11 1R = [⟨(1P +P 1P), 1P⟩] ~R
123121, 122eqtr4di 2226 . . . . . . . . . 10 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ) = 1R)
124 breq2 4002 . . . . . . . . . . . 12 (𝑥 = [⟨(𝑣 +P 1P), 1P⟩] ~R → (0R <R 𝑥 ↔ 0R <R [⟨(𝑣 +P 1P), 1P⟩] ~R ))
125 oveq2 5873 . . . . . . . . . . . . 13 (𝑥 = [⟨(𝑣 +P 1P), 1P⟩] ~R → ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ))
126125eqeq1d 2184 . . . . . . . . . . . 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 473 . . . . . . . . . . 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 2839 . . . . . . . . . 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 1236 . . . . . . . . 9 ((((𝑦P𝑧P) ∧ (𝑤P𝑣P)) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ∃𝑥R (0R <R 𝑥 ∧ ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R))
130129exp32 365 . . . . . . . 8 (((𝑦P𝑧P) ∧ (𝑤P𝑣P)) → ((𝑤 ·P 𝑣) = 1P → ((𝑧 +P 𝑤) = 𝑦 → ∃𝑥R (0R <R 𝑥 ∧ ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R))))
131130anassrs 400 . . . . . . 7 ((((𝑦P𝑧P) ∧ 𝑤P) ∧ 𝑣P) → ((𝑤 ·P 𝑣) = 1P → ((𝑧 +P 𝑤) = 𝑦 → ∃𝑥R (0R <R 𝑥 ∧ ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R))))
132131rexlimdva 2592 . . . . . 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 2592 . . . 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 6612 . 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 104  wb 105  w3a 978   = wceq 1353  wcel 2146  wrex 2454  cop 3592   class class class wbr 3998  (class class class)co 5865  [cec 6523  Pcnp 7265  1Pc1p 7266   +P cpp 7267   ·P cmp 7268  <P cltp 7269   ~R cer 7270  Rcnr 7271  0Rc0r 7272  1Rc1r 7273   ·R cmr 7276   <R cltr 7277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-eprel 4283  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-1o 6407  df-2o 6408  df-oadd 6411  df-omul 6412  df-er 6525  df-ec 6527  df-qs 6531  df-ni 7278  df-pli 7279  df-mi 7280  df-lti 7281  df-plpq 7318  df-mpq 7319  df-enq 7321  df-nqqs 7322  df-plqqs 7323  df-mqqs 7324  df-1nqqs 7325  df-rq 7326  df-ltnqqs 7327  df-enq0 7398  df-nq0 7399  df-0nq0 7400  df-plq0 7401  df-mq0 7402  df-inp 7440  df-i1p 7441  df-iplp 7442  df-imp 7443  df-iltp 7444  df-enr 7700  df-nr 7701  df-mr 7703  df-ltr 7704  df-0r 7705  df-1r 7706
This theorem is referenced by:  recexsrlem  7748  axprecex  7854
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