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Theorem axpre-mulext 6960
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 7000.

(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 6903 . 2 (𝐴 ∈ ℝ ↔ ∃𝑥R𝑥, 0R⟩ = 𝐴)
2 elreal 6903 . 2 (𝐵 ∈ ℝ ↔ ∃𝑦R𝑦, 0R⟩ = 𝐵)
3 elreal 6903 . 2 (𝐶 ∈ ℝ ↔ ∃𝑧R𝑧, 0R⟩ = 𝐶)
4 oveq1 5519 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) = (𝐴 · ⟨𝑧, 0R⟩))
54breq1d 3774 . . 3 (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) < (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) ↔ (𝐴 · ⟨𝑧, 0R⟩) < (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩)))
6 breq1 3767 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝐴 <𝑦, 0R⟩))
7 breq2 3768 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ <𝑥, 0R⟩ ↔ ⟨𝑦, 0R⟩ < 𝐴))
86, 7orbi12d 707 . . 3 (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩) ↔ (𝐴 <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴)))
95, 8imbi12d 223 . 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 5519 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = (𝐵 · ⟨𝑧, 0R⟩))
1110breq2d 3776 . . 3 (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 · ⟨𝑧, 0R⟩) < (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) ↔ (𝐴 · ⟨𝑧, 0R⟩) < (𝐵 · ⟨𝑧, 0R⟩)))
12 breq2 3768 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <𝑦, 0R⟩ ↔ 𝐴 < 𝐵))
13 breq1 3767 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑦, 0R⟩ < 𝐴𝐵 < 𝐴))
1412, 13orbi12d 707 . . 3 (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴) ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
1511, 14imbi12d 223 . 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 5520 . . . 4 (⟨𝑧, 0R⟩ = 𝐶 → (𝐴 · ⟨𝑧, 0R⟩) = (𝐴 · 𝐶))
17 oveq2 5520 . . . 4 (⟨𝑧, 0R⟩ = 𝐶 → (𝐵 · ⟨𝑧, 0R⟩) = (𝐵 · 𝐶))
1816, 17breq12d 3777 . . 3 (⟨𝑧, 0R⟩ = 𝐶 → ((𝐴 · ⟨𝑧, 0R⟩) < (𝐵 · ⟨𝑧, 0R⟩) ↔ (𝐴 · 𝐶) < (𝐵 · 𝐶)))
1918imbi1d 220 . 2 (⟨𝑧, 0R⟩ = 𝐶 → (((𝐴 · ⟨𝑧, 0R⟩) < (𝐵 · ⟨𝑧, 0R⟩) → (𝐴 < 𝐵𝐵 < 𝐴)) ↔ ((𝐴 · 𝐶) < (𝐵 · 𝐶) → (𝐴 < 𝐵𝐵 < 𝐴))))
20 mulextsr1 6863 . . 3 ((𝑥R𝑦R𝑧R) → ((𝑥 ·R 𝑧) <R (𝑦 ·R 𝑧) → (𝑥 <R 𝑦𝑦 <R 𝑥)))
21 mulresr 6912 . . . . . 6 ((𝑥R𝑧R) → (⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑥 ·R 𝑧), 0R⟩)
22213adant2 923 . . . . 5 ((𝑥R𝑦R𝑧R) → (⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑥 ·R 𝑧), 0R⟩)
23 mulresr 6912 . . . . . 6 ((𝑦R𝑧R) → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑦 ·R 𝑧), 0R⟩)
24233adant1 922 . . . . 5 ((𝑥R𝑦R𝑧R) → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑦 ·R 𝑧), 0R⟩)
2522, 24breq12d 3777 . . . 4 ((𝑥R𝑦R𝑧R) → ((⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) < (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) ↔ ⟨(𝑥 ·R 𝑧), 0R⟩ < ⟨(𝑦 ·R 𝑧), 0R⟩))
26 ltresr 6913 . . . 4 (⟨(𝑥 ·R 𝑧), 0R⟩ < ⟨(𝑦 ·R 𝑧), 0R⟩ ↔ (𝑥 ·R 𝑧) <R (𝑦 ·R 𝑧))
2725, 26syl6bb 185 . . 3 ((𝑥R𝑦R𝑧R) → ((⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) < (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) ↔ (𝑥 ·R 𝑧) <R (𝑦 ·R 𝑧)))
28 ltresr 6913 . . . . 5 (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
29 ltresr 6913 . . . . 5 (⟨𝑦, 0R⟩ <𝑥, 0R⟩ ↔ 𝑦 <R 𝑥)
3028, 29orbi12i 681 . . . 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 192 . 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 2588 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 98  wo 629  w3a 885   = wceq 1243  wcel 1393  cop 3378   class class class wbr 3764  (class class class)co 5512  Rcnr 6393  0Rc0r 6394   ·R cmr 6398   <R cltr 6399  cr 6886   < cltrr 6891   · cmul 6892
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6400  df-pli 6401  df-mi 6402  df-lti 6403  df-plpq 6440  df-mpq 6441  df-enq 6443  df-nqqs 6444  df-plqqs 6445  df-mqqs 6446  df-1nqqs 6447  df-rq 6448  df-ltnqqs 6449  df-enq0 6520  df-nq0 6521  df-0nq0 6522  df-plq0 6523  df-mq0 6524  df-inp 6562  df-i1p 6563  df-iplp 6564  df-imp 6565  df-iltp 6566  df-enr 6809  df-nr 6810  df-plr 6811  df-mr 6812  df-ltr 6813  df-0r 6814  df-m1r 6816  df-c 6893  df-r 6897  df-mul 6899  df-lt 6900
This theorem is referenced by: (None)
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