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Theorem axpre-ltadd 10026
Description: Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axltadd 10149. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 10050. (Contributed by NM, 11-May-1996.) (New usage is discouraged.)
Assertion
Ref Expression
axpre-ltadd ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐶 + 𝐴) < (𝐶 + 𝐵)))

Proof of Theorem axpre-ltadd
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elreal 9990 . . 3 (𝐴 ∈ ℝ ↔ ∃𝑥R𝑥, 0R⟩ = 𝐴)
2 elreal 9990 . . 3 (𝐵 ∈ ℝ ↔ ∃𝑦R𝑦, 0R⟩ = 𝐵)
3 elreal 9990 . . 3 (𝐶 ∈ ℝ ↔ ∃𝑧R𝑧, 0R⟩ = 𝐶)
4 breq1 4688 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝐴 <𝑦, 0R⟩))
5 oveq2 6698 . . . . 5 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) = (⟨𝑧, 0R⟩ + 𝐴))
65breq1d 4695 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ (⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)))
74, 6bibi12d 334 . . 3 (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)) ↔ (𝐴 <𝑦, 0R⟩ ↔ (⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩))))
8 breq2 4689 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <𝑦, 0R⟩ ↔ 𝐴 < 𝐵))
9 oveq2 6698 . . . . 5 (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) = (⟨𝑧, 0R⟩ + 𝐵))
109breq2d 4697 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → ((⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ (⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + 𝐵)))
118, 10bibi12d 334 . . 3 (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 <𝑦, 0R⟩ ↔ (⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)) ↔ (𝐴 < 𝐵 ↔ (⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + 𝐵))))
12 oveq1 6697 . . . . 5 (⟨𝑧, 0R⟩ = 𝐶 → (⟨𝑧, 0R⟩ + 𝐴) = (𝐶 + 𝐴))
13 oveq1 6697 . . . . 5 (⟨𝑧, 0R⟩ = 𝐶 → (⟨𝑧, 0R⟩ + 𝐵) = (𝐶 + 𝐵))
1412, 13breq12d 4698 . . . 4 (⟨𝑧, 0R⟩ = 𝐶 → ((⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + 𝐵) ↔ (𝐶 + 𝐴) < (𝐶 + 𝐵)))
1514bibi2d 331 . . 3 (⟨𝑧, 0R⟩ = 𝐶 → ((𝐴 < 𝐵 ↔ (⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + 𝐵)) ↔ (𝐴 < 𝐵 ↔ (𝐶 + 𝐴) < (𝐶 + 𝐵))))
16 ltasr 9959 . . . . . . 7 (𝑧R → (𝑥 <R 𝑦 ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦)))
1716adantr 480 . . . . . 6 ((𝑧R ∧ (𝑥R𝑦R)) → (𝑥 <R 𝑦 ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦)))
18 ltresr 9999 . . . . . . 7 (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
1918a1i 11 . . . . . 6 ((𝑧R ∧ (𝑥R𝑦R)) → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝑥 <R 𝑦))
20 addresr 9997 . . . . . . . . 9 ((𝑧R𝑥R) → (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) = ⟨(𝑧 +R 𝑥), 0R⟩)
21 addresr 9997 . . . . . . . . 9 ((𝑧R𝑦R) → (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) = ⟨(𝑧 +R 𝑦), 0R⟩)
2220, 21breqan12d 4701 . . . . . . . 8 (((𝑧R𝑥R) ∧ (𝑧R𝑦R)) → ((⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ ⟨(𝑧 +R 𝑥), 0R⟩ < ⟨(𝑧 +R 𝑦), 0R⟩))
2322anandis 890 . . . . . . 7 ((𝑧R ∧ (𝑥R𝑦R)) → ((⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ ⟨(𝑧 +R 𝑥), 0R⟩ < ⟨(𝑧 +R 𝑦), 0R⟩))
24 ltresr 9999 . . . . . . 7 (⟨(𝑧 +R 𝑥), 0R⟩ < ⟨(𝑧 +R 𝑦), 0R⟩ ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦))
2523, 24syl6bb 276 . . . . . 6 ((𝑧R ∧ (𝑥R𝑦R)) → ((⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦)))
2617, 19, 253bitr4d 300 . . . . 5 ((𝑧R ∧ (𝑥R𝑦R)) → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)))
2726ancoms 468 . . . 4 (((𝑥R𝑦R) ∧ 𝑧R) → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)))
28273impa 1278 . . 3 ((𝑥R𝑦R𝑧R) → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)))
291, 2, 3, 7, 11, 15, 283gencl 3268 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐶 + 𝐴) < (𝐶 + 𝐵)))
3029biimpd 219 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐶 + 𝐴) < (𝐶 + 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  cop 4216   class class class wbr 4685  (class class class)co 6690  Rcnr 9725  0Rc0r 9726   +R cplr 9729   <R cltr 9731  cr 9973   + caddc 9977   < cltrr 9978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-er 7787  df-ec 7789  df-qs 7793  df-ni 9732  df-pli 9733  df-mi 9734  df-lti 9735  df-plpq 9768  df-mpq 9769  df-ltpq 9770  df-enq 9771  df-nq 9772  df-erq 9773  df-plq 9774  df-mq 9775  df-1nq 9776  df-rq 9777  df-ltnq 9778  df-np 9841  df-1p 9842  df-plp 9843  df-ltp 9845  df-enr 9915  df-nr 9916  df-plr 9917  df-ltr 9919  df-0r 9920  df-c 9980  df-r 9984  df-add 9985  df-lt 9987
This theorem is referenced by: (None)
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