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

Theorem axpre-mulext 7862
Description: Strong extensionality of multiplication (expressed in terms of <). Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulext 7904.

(Contributed by Jim Kingdon, 18-Feb-2020.) (New usage is discouraged.)

Assertion
Ref Expression
axpre-mulext ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐶) < (𝐵 · 𝐶) → (𝐴 < 𝐵𝐵 < 𝐴)))

Proof of Theorem axpre-mulext
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elreal 7802 . 2 (𝐴 ∈ ℝ ↔ ∃𝑥R𝑥, 0R⟩ = 𝐴)
2 elreal 7802 . 2 (𝐵 ∈ ℝ ↔ ∃𝑦R𝑦, 0R⟩ = 𝐵)
3 elreal 7802 . 2 (𝐶 ∈ ℝ ↔ ∃𝑧R𝑧, 0R⟩ = 𝐶)
4 oveq1 5872 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) = (𝐴 · ⟨𝑧, 0R⟩))
54breq1d 4008 . . 3 (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) < (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) ↔ (𝐴 · ⟨𝑧, 0R⟩) < (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩)))
6 breq1 4001 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝐴 <𝑦, 0R⟩))
7 breq2 4002 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ <𝑥, 0R⟩ ↔ ⟨𝑦, 0R⟩ < 𝐴))
86, 7orbi12d 793 . . 3 (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩) ↔ (𝐴 <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴)))
95, 8imbi12d 234 . 2 (⟨𝑥, 0R⟩ = 𝐴 → (((⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) < (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩)) ↔ ((𝐴 · ⟨𝑧, 0R⟩) < (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) → (𝐴 <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴))))
10 oveq1 5872 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = (𝐵 · ⟨𝑧, 0R⟩))
1110breq2d 4010 . . 3 (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 · ⟨𝑧, 0R⟩) < (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) ↔ (𝐴 · ⟨𝑧, 0R⟩) < (𝐵 · ⟨𝑧, 0R⟩)))
12 breq2 4002 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <𝑦, 0R⟩ ↔ 𝐴 < 𝐵))
13 breq1 4001 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑦, 0R⟩ < 𝐴𝐵 < 𝐴))
1412, 13orbi12d 793 . . 3 (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴) ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
1511, 14imbi12d 234 . 2 (⟨𝑦, 0R⟩ = 𝐵 → (((𝐴 · ⟨𝑧, 0R⟩) < (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) → (𝐴 <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴)) ↔ ((𝐴 · ⟨𝑧, 0R⟩) < (𝐵 · ⟨𝑧, 0R⟩) → (𝐴 < 𝐵𝐵 < 𝐴))))
16 oveq2 5873 . . . 4 (⟨𝑧, 0R⟩ = 𝐶 → (𝐴 · ⟨𝑧, 0R⟩) = (𝐴 · 𝐶))
17 oveq2 5873 . . . 4 (⟨𝑧, 0R⟩ = 𝐶 → (𝐵 · ⟨𝑧, 0R⟩) = (𝐵 · 𝐶))
1816, 17breq12d 4011 . . 3 (⟨𝑧, 0R⟩ = 𝐶 → ((𝐴 · ⟨𝑧, 0R⟩) < (𝐵 · ⟨𝑧, 0R⟩) ↔ (𝐴 · 𝐶) < (𝐵 · 𝐶)))
1918imbi1d 231 . 2 (⟨𝑧, 0R⟩ = 𝐶 → (((𝐴 · ⟨𝑧, 0R⟩) < (𝐵 · ⟨𝑧, 0R⟩) → (𝐴 < 𝐵𝐵 < 𝐴)) ↔ ((𝐴 · 𝐶) < (𝐵 · 𝐶) → (𝐴 < 𝐵𝐵 < 𝐴))))
20 mulextsr1 7755 . . 3 ((𝑥R𝑦R𝑧R) → ((𝑥 ·R 𝑧) <R (𝑦 ·R 𝑧) → (𝑥 <R 𝑦𝑦 <R 𝑥)))
21 mulresr 7812 . . . . . 6 ((𝑥R𝑧R) → (⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑥 ·R 𝑧), 0R⟩)
22213adant2 1016 . . . . 5 ((𝑥R𝑦R𝑧R) → (⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑥 ·R 𝑧), 0R⟩)
23 mulresr 7812 . . . . . 6 ((𝑦R𝑧R) → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑦 ·R 𝑧), 0R⟩)
24233adant1 1015 . . . . 5 ((𝑥R𝑦R𝑧R) → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑦 ·R 𝑧), 0R⟩)
2522, 24breq12d 4011 . . . 4 ((𝑥R𝑦R𝑧R) → ((⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) < (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) ↔ ⟨(𝑥 ·R 𝑧), 0R⟩ < ⟨(𝑦 ·R 𝑧), 0R⟩))
26 ltresr 7813 . . . 4 (⟨(𝑥 ·R 𝑧), 0R⟩ < ⟨(𝑦 ·R 𝑧), 0R⟩ ↔ (𝑥 ·R 𝑧) <R (𝑦 ·R 𝑧))
2725, 26bitrdi 196 . . 3 ((𝑥R𝑦R𝑧R) → ((⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) < (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) ↔ (𝑥 ·R 𝑧) <R (𝑦 ·R 𝑧)))
28 ltresr 7813 . . . . 5 (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
29 ltresr 7813 . . . . 5 (⟨𝑦, 0R⟩ <𝑥, 0R⟩ ↔ 𝑦 <R 𝑥)
3028, 29orbi12i 764 . . . 4 ((⟨𝑥, 0R⟩ <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩) ↔ (𝑥 <R 𝑦𝑦 <R 𝑥))
3130a1i 9 . . 3 ((𝑥R𝑦R𝑧R) → ((⟨𝑥, 0R⟩ <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩) ↔ (𝑥 <R 𝑦𝑦 <R 𝑥)))
3220, 27, 313imtr4d 203 . 2 ((𝑥R𝑦R𝑧R) → ((⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) < (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩)))
331, 2, 3, 9, 15, 19, 323gencl 2769 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐶) < (𝐵 · 𝐶) → (𝐴 < 𝐵𝐵 < 𝐴)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wo 708  w3a 978   = wceq 1353  wcel 2146  cop 3592   class class class wbr 3998  (class class class)co 5865  Rcnr 7271  0Rc0r 7272   ·R cmr 7276   <R cltr 7277  cr 7785   < cltrr 7790   · cmul 7791
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-plr 7702  df-mr 7703  df-ltr 7704  df-0r 7705  df-m1r 7707  df-c 7792  df-r 7796  df-mul 7798  df-lt 7799
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator