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Theorem axprecex 6716
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 6745.

In treatments which assume excluded middle, the 0 < 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 6679 . . . 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 3758 . . . 4 (⟨y, 0R⟩ = A → (0 <y, 0R⟩ ↔ 0 < A))
5 oveq1 5459 . . . . . . 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 6670 . . . . . 6 0 = ⟨0R, 0R
1110breq1i 3761 . . . . 5 (0 <y, 0R⟩ ↔ ⟨0R, 0R⟩ <y, 0R⟩)
12 ltresr 6688 . . . . 5 (⟨0R, 0R⟩ <y, 0R⟩ ↔ 0R <R y)
1311, 12bitri 173 . . . 4 (0 <y, 0R⟩ ↔ 0R <R y)
14 recexgt0sr 6653 . . . . 5 (0R <R yz R (0R <R z (y ·R z) = 1R))
15 opelreal 6678 . . . . . . . . . 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 3761 . . . . . . . . . . . . 13 (0 <z, 0R⟩ ↔ ⟨0R, 0R⟩ <z, 0R⟩)
18 ltresr 6688 . . . . . . . . . . . . 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 6687 . . . . . . . . . . . . 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 6671 . . . . . . . . . . . . . 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 6631 . . . . . . . . . . . . . . 15 1R R
27 0r 6630 . . . . . . . . . . . . . . 15 0R R
28 opthg2 3966 . . . . . . . . . . . . . . 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 3758 . . . . . . . . . 10 (x = ⟨z, 0R⟩ → (0 < x ↔ 0 <z, 0R⟩))
37 oveq2 5460 . . . . . . . . . . 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 3369   class class class wbr 3754  (class class class)co 5452  Rcnr 6274  0Rc0r 6275  1Rc1r 6276   ·R cmr 6279   <R cltr 6280  cr 6662  0cc0 6663  1c1 6664   < cltrr 6667   · cmul 6668
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3862  ax-sep 3865  ax-nul 3873  ax-pow 3917  ax-pr 3934  ax-un 4135  ax-setind 4219  ax-iinf 4253
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3352  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-int 3606  df-iun 3649  df-br 3755  df-opab 3809  df-mpt 3810  df-tr 3845  df-eprel 4016  df-id 4020  df-po 4023  df-iso 4024  df-iord 4068  df-on 4070  df-suc 4073  df-iom 4256  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-rn 4298  df-res 4299  df-ima 4300  df-iota 4809  df-fun 4846  df-fn 4847  df-f 4848  df-f1 4849  df-fo 4850  df-f1o 4851  df-fv 4852  df-ov 5455  df-oprab 5456  df-mpt2 5457  df-1st 5706  df-2nd 5707  df-recs 5858  df-irdg 5894  df-1o 5933  df-2o 5934  df-oadd 5937  df-omul 5938  df-er 6035  df-ec 6037  df-qs 6041  df-ni 6281  df-pli 6282  df-mi 6283  df-lti 6284  df-plpq 6321  df-mpq 6322  df-enq 6324  df-nqqs 6325  df-plqqs 6326  df-mqqs 6327  df-1nqqs 6328  df-rq 6329  df-ltnqqs 6330  df-enq0 6399  df-nq0 6400  df-0nq0 6401  df-plq0 6402  df-mq0 6403  df-inp 6441  df-i1p 6442  df-iplp 6443  df-imp 6444  df-iltp 6445  df-enr 6606  df-nr 6607  df-plr 6608  df-mr 6609  df-ltr 6610  df-0r 6611  df-1r 6612  df-m1r 6613  df-c 6669  df-0 6670  df-1 6671  df-r 6673  df-mul 6675  df-lt 6676
This theorem is referenced by: (None)
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