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

Theorem axpre-mulext 7878
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 7920.

(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 7818 . 2 (𝐴 ∈ ℝ ↔ ∃𝑥R𝑥, 0R⟩ = 𝐴)
2 elreal 7818 . 2 (𝐵 ∈ ℝ ↔ ∃𝑦R𝑦, 0R⟩ = 𝐵)
3 elreal 7818 . 2 (𝐶 ∈ ℝ ↔ ∃𝑧R𝑧, 0R⟩ = 𝐶)
4 oveq1 5876 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) = (𝐴 · ⟨𝑧, 0R⟩))
54breq1d 4010 . . 3 (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) < (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) ↔ (𝐴 · ⟨𝑧, 0R⟩) < (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩)))
6 breq1 4003 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝐴 <𝑦, 0R⟩))
7 breq2 4004 . . . 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 5876 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = (𝐵 · ⟨𝑧, 0R⟩))
1110breq2d 4012 . . 3 (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 · ⟨𝑧, 0R⟩) < (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) ↔ (𝐴 · ⟨𝑧, 0R⟩) < (𝐵 · ⟨𝑧, 0R⟩)))
12 breq2 4004 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <𝑦, 0R⟩ ↔ 𝐴 < 𝐵))
13 breq1 4003 . . . 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 5877 . . . 4 (⟨𝑧, 0R⟩ = 𝐶 → (𝐴 · ⟨𝑧, 0R⟩) = (𝐴 · 𝐶))
17 oveq2 5877 . . . 4 (⟨𝑧, 0R⟩ = 𝐶 → (𝐵 · ⟨𝑧, 0R⟩) = (𝐵 · 𝐶))
1816, 17breq12d 4013 . . 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 7771 . . 3 ((𝑥R𝑦R𝑧R) → ((𝑥 ·R 𝑧) <R (𝑦 ·R 𝑧) → (𝑥 <R 𝑦𝑦 <R 𝑥)))
21 mulresr 7828 . . . . . 6 ((𝑥R𝑧R) → (⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑥 ·R 𝑧), 0R⟩)
22213adant2 1016 . . . . 5 ((𝑥R𝑦R𝑧R) → (⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑥 ·R 𝑧), 0R⟩)
23 mulresr 7828 . . . . . 6 ((𝑦R𝑧R) → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑦 ·R 𝑧), 0R⟩)
24233adant1 1015 . . . . 5 ((𝑥R𝑦R𝑧R) → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑦 ·R 𝑧), 0R⟩)
2522, 24breq12d 4013 . . . 4 ((𝑥R𝑦R𝑧R) → ((⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) < (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) ↔ ⟨(𝑥 ·R 𝑧), 0R⟩ < ⟨(𝑦 ·R 𝑧), 0R⟩))
26 ltresr 7829 . . . 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 7829 . . . . 5 (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
29 ltresr 7829 . . . . 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 2771 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 2148  cop 3594   class class class wbr 4000  (class class class)co 5869  Rcnr 7287  0Rc0r 7288   ·R cmr 7292   <R cltr 7293  cr 7801   < cltrr 7806   · cmul 7807
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
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 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-enq0 7414  df-nq0 7415  df-0nq0 7416  df-plq0 7417  df-mq0 7418  df-inp 7456  df-i1p 7457  df-iplp 7458  df-imp 7459  df-iltp 7460  df-enr 7716  df-nr 7717  df-plr 7718  df-mr 7719  df-ltr 7720  df-0r 7721  df-m1r 7723  df-c 7808  df-r 7812  df-mul 7814  df-lt 7815
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator