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

Theorem axprecex 6764
 Description: Existence of positive reciprocal of positive real number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-precex 6793. In treatments which assume excluded middle, the 0 <ℝ A condition is generally replaced by A ≠ 0, and it may not be necessary to state that the reciproacal is positive. (Contributed by Jim Kingdon, 6-Feb-2020.) (New usage is discouraged.)
Assertion
Ref Expression
axprecex ((A 0 < A) → x ℝ (0 < x (A · x) = 1))
Distinct variable group:   x,A

Proof of Theorem axprecex
Dummy variables y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elreal 6727 . . . 4 (A ℝ ↔ y Ry, 0R⟩ = A)
2 df-rex 2306 . . . 4 (y Ry, 0R⟩ = Ay(y R y, 0R⟩ = A))
31, 2bitri 173 . . 3 (A ℝ ↔ y(y R y, 0R⟩ = A))
4 breq2 3759 . . . 4 (⟨y, 0R⟩ = A → (0 <y, 0R⟩ ↔ 0 < A))
5 oveq1 5462 . . . . . . 7 (⟨y, 0R⟩ = A → (⟨y, 0R⟩ · x) = (A · x))
65eqeq1d 2045 . . . . . 6 (⟨y, 0R⟩ = A → ((⟨y, 0R⟩ · x) = 1 ↔ (A · x) = 1))
76anbi2d 437 . . . . 5 (⟨y, 0R⟩ = A → ((0 < x (⟨y, 0R⟩ · x) = 1) ↔ (0 < x (A · x) = 1)))
87rexbidv 2321 . . . 4 (⟨y, 0R⟩ = A → (x ℝ (0 < x (⟨y, 0R⟩ · x) = 1) ↔ x ℝ (0 < x (A · x) = 1)))
94, 8imbi12d 223 . . 3 (⟨y, 0R⟩ = A → ((0 <y, 0R⟩ → x ℝ (0 < x (⟨y, 0R⟩ · x) = 1)) ↔ (0 < Ax ℝ (0 < x (A · x) = 1))))
10 df-0 6718 . . . . . 6 0 = ⟨0R, 0R
1110breq1i 3762 . . . . 5 (0 <y, 0R⟩ ↔ ⟨0R, 0R⟩ <y, 0R⟩)
12 ltresr 6736 . . . . 5 (⟨0R, 0R⟩ <y, 0R⟩ ↔ 0R <R y)
1311, 12bitri 173 . . . 4 (0 <y, 0R⟩ ↔ 0R <R y)
14 recexgt0sr 6701 . . . . 5 (0R <R yz R (0R <R z (y ·R z) = 1R))
15 opelreal 6726 . . . . . . . . . 10 (⟨z, 0R ℝ ↔ z R)
1615anbi1i 431 . . . . . . . . 9 ((⟨z, 0R (0 <z, 0R (⟨y, 0R⟩ · ⟨z, 0R⟩) = 1)) ↔ (z R (0 <z, 0R (⟨y, 0R⟩ · ⟨z, 0R⟩) = 1)))
1710breq1i 3762 . . . . . . . . . . . . 13 (0 <z, 0R⟩ ↔ ⟨0R, 0R⟩ <z, 0R⟩)
18 ltresr 6736 . . . . . . . . . . . . 13 (⟨0R, 0R⟩ <z, 0R⟩ ↔ 0R <R z)
1917, 18bitri 173 . . . . . . . . . . . 12 (0 <z, 0R⟩ ↔ 0R <R z)
2019a1i 9 . . . . . . . . . . 11 ((y R z R) → (0 <z, 0R⟩ ↔ 0R <R z))
21 mulresr 6735 . . . . . . . . . . . . 13 ((y R z R) → (⟨y, 0R⟩ · ⟨z, 0R⟩) = ⟨(y ·R z), 0R⟩)
2221eqeq1d 2045 . . . . . . . . . . . 12 ((y R z R) → ((⟨y, 0R⟩ · ⟨z, 0R⟩) = 1 ↔ ⟨(y ·R z), 0R⟩ = 1))
23 df-1 6719 . . . . . . . . . . . . . 14 1 = ⟨1R, 0R
2423eqeq2i 2047 . . . . . . . . . . . . 13 (⟨(y ·R z), 0R⟩ = 1 ↔ ⟨(y ·R z), 0R⟩ = ⟨1R, 0R⟩)
25 eqid 2037 . . . . . . . . . . . . . 14 0R = 0R
26 1sr 6679 . . . . . . . . . . . . . . 15 1R R
27 0r 6678 . . . . . . . . . . . . . . 15 0R R
28 opthg2 3967 . . . . . . . . . . . . . . 15 ((1R R 0R R) → (⟨(y ·R z), 0R⟩ = ⟨1R, 0R⟩ ↔ ((y ·R z) = 1R 0R = 0R)))
2926, 27, 28mp2an 402 . . . . . . . . . . . . . 14 (⟨(y ·R z), 0R⟩ = ⟨1R, 0R⟩ ↔ ((y ·R z) = 1R 0R = 0R))
3025, 29mpbiran2 847 . . . . . . . . . . . . 13 (⟨(y ·R z), 0R⟩ = ⟨1R, 0R⟩ ↔ (y ·R z) = 1R)
3124, 30bitri 173 . . . . . . . . . . . 12 (⟨(y ·R z), 0R⟩ = 1 ↔ (y ·R z) = 1R)
3222, 31syl6bb 185 . . . . . . . . . . 11 ((y R z R) → ((⟨y, 0R⟩ · ⟨z, 0R⟩) = 1 ↔ (y ·R z) = 1R))
3320, 32anbi12d 442 . . . . . . . . . 10 ((y R z R) → ((0 <z, 0R (⟨y, 0R⟩ · ⟨z, 0R⟩) = 1) ↔ (0R <R z (y ·R z) = 1R)))
3433pm5.32da 425 . . . . . . . . 9 (y R → ((z R (0 <z, 0R (⟨y, 0R⟩ · ⟨z, 0R⟩) = 1)) ↔ (z R (0R <R z (y ·R z) = 1R))))
3516, 34syl5bb 181 . . . . . . . 8 (y R → ((⟨z, 0R (0 <z, 0R (⟨y, 0R⟩ · ⟨z, 0R⟩) = 1)) ↔ (z R (0R <R z (y ·R z) = 1R))))
36 breq2 3759 . . . . . . . . . 10 (x = ⟨z, 0R⟩ → (0 < x ↔ 0 <z, 0R⟩))
37 oveq2 5463 . . . . . . . . . . 11 (x = ⟨z, 0R⟩ → (⟨y, 0R⟩ · x) = (⟨y, 0R⟩ · ⟨z, 0R⟩))
3837eqeq1d 2045 . . . . . . . . . 10 (x = ⟨z, 0R⟩ → ((⟨y, 0R⟩ · x) = 1 ↔ (⟨y, 0R⟩ · ⟨z, 0R⟩) = 1))
3936, 38anbi12d 442 . . . . . . . . 9 (x = ⟨z, 0R⟩ → ((0 < x (⟨y, 0R⟩ · x) = 1) ↔ (0 <z, 0R (⟨y, 0R⟩ · ⟨z, 0R⟩) = 1)))
4039rspcev 2650 . . . . . . . 8 ((⟨z, 0R (0 <z, 0R (⟨y, 0R⟩ · ⟨z, 0R⟩) = 1)) → x ℝ (0 < x (⟨y, 0R⟩ · x) = 1))
4135, 40syl6bir 153 . . . . . . 7 (y R → ((z R (0R <R z (y ·R z) = 1R)) → x ℝ (0 < x (⟨y, 0R⟩ · x) = 1)))
4241expd 245 . . . . . 6 (y R → (z R → ((0R <R z (y ·R z) = 1R) → x ℝ (0 < x (⟨y, 0R⟩ · x) = 1))))
4342rexlimdv 2426 . . . . 5 (y R → (z R (0R <R z (y ·R z) = 1R) → x ℝ (0 < x (⟨y, 0R⟩ · x) = 1)))
4414, 43syl5 28 . . . 4 (y R → (0R <R yx ℝ (0 < x (⟨y, 0R⟩ · x) = 1)))
4513, 44syl5bi 141 . . 3 (y R → (0 <y, 0R⟩ → x ℝ (0 < x (⟨y, 0R⟩ · x) = 1)))
463, 9, 45gencl 2580 . 2 (A ℝ → (0 < Ax ℝ (0 < x (A · x) = 1)))
4746imp 115 1 ((A 0 < A) → x ℝ (0 < x (A · x) = 1))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃wrex 2301  ⟨cop 3370   class class class wbr 3755  (class class class)co 5455  Rcnr 6281  0Rc0r 6282  1Rc1r 6283   ·R cmr 6286
 Copyright terms: Public domain W3C validator