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Theorem axpre-ltadd 7915
Description: Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 7957. (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 7857 . . 3 (𝐴 ∈ ℝ ↔ ∃𝑥R𝑥, 0R⟩ = 𝐴)
2 elreal 7857 . . 3 (𝐵 ∈ ℝ ↔ ∃𝑦R𝑦, 0R⟩ = 𝐵)
3 elreal 7857 . . 3 (𝐶 ∈ ℝ ↔ ∃𝑧R𝑧, 0R⟩ = 𝐶)
4 breq1 4021 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝐴 <𝑦, 0R⟩))
5 oveq2 5904 . . . . 5 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) = (⟨𝑧, 0R⟩ + 𝐴))
65breq1d 4028 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ (⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)))
74, 6bibi12d 235 . . 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 4022 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <𝑦, 0R⟩ ↔ 𝐴 < 𝐵))
9 oveq2 5904 . . . . 5 (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) = (⟨𝑧, 0R⟩ + 𝐵))
109breq2d 4030 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → ((⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ (⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + 𝐵)))
118, 10bibi12d 235 . . 3 (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 <𝑦, 0R⟩ ↔ (⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)) ↔ (𝐴 < 𝐵 ↔ (⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + 𝐵))))
12 oveq1 5903 . . . . 5 (⟨𝑧, 0R⟩ = 𝐶 → (⟨𝑧, 0R⟩ + 𝐴) = (𝐶 + 𝐴))
13 oveq1 5903 . . . . 5 (⟨𝑧, 0R⟩ = 𝐶 → (⟨𝑧, 0R⟩ + 𝐵) = (𝐶 + 𝐵))
1412, 13breq12d 4031 . . . 4 (⟨𝑧, 0R⟩ = 𝐶 → ((⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + 𝐵) ↔ (𝐶 + 𝐴) < (𝐶 + 𝐵)))
1514bibi2d 232 . . 3 (⟨𝑧, 0R⟩ = 𝐶 → ((𝐴 < 𝐵 ↔ (⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + 𝐵)) ↔ (𝐴 < 𝐵 ↔ (𝐶 + 𝐴) < (𝐶 + 𝐵))))
16 ltasrg 7799 . . . 4 ((𝑥R𝑦R𝑧R) → (𝑥 <R 𝑦 ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦)))
17 ltresr 7868 . . . . 5 (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
1817a1i 9 . . . 4 ((𝑥R𝑦R𝑧R) → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝑥 <R 𝑦))
19 simp3 1001 . . . . . 6 ((𝑥R𝑦R𝑧R) → 𝑧R)
20 simp1 999 . . . . . 6 ((𝑥R𝑦R𝑧R) → 𝑥R)
21 simp2 1000 . . . . . 6 ((𝑥R𝑦R𝑧R) → 𝑦R)
22 addresr 7866 . . . . . . 7 ((𝑧R𝑥R) → (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) = ⟨(𝑧 +R 𝑥), 0R⟩)
23 addresr 7866 . . . . . . 7 ((𝑧R𝑦R) → (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) = ⟨(𝑧 +R 𝑦), 0R⟩)
2422, 23breqan12d 4034 . . . . . 6 (((𝑧R𝑥R) ∧ (𝑧R𝑦R)) → ((⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ ⟨(𝑧 +R 𝑥), 0R⟩ < ⟨(𝑧 +R 𝑦), 0R⟩))
2519, 20, 19, 21, 24syl22anc 1250 . . . . 5 ((𝑥R𝑦R𝑧R) → ((⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ ⟨(𝑧 +R 𝑥), 0R⟩ < ⟨(𝑧 +R 𝑦), 0R⟩))
26 ltresr 7868 . . . . 5 (⟨(𝑧 +R 𝑥), 0R⟩ < ⟨(𝑧 +R 𝑦), 0R⟩ ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦))
2725, 26bitrdi 196 . . . 4 ((𝑥R𝑦R𝑧R) → ((⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦)))
2816, 18, 273bitr4d 220 . . 3 ((𝑥R𝑦R𝑧R) → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)))
291, 2, 3, 7, 11, 15, 283gencl 2786 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐶 + 𝐴) < (𝐶 + 𝐵)))
3029biimpd 144 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐶 + 𝐴) < (𝐶 + 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 980   = wceq 1364  wcel 2160  cop 3610   class class class wbr 4018  (class class class)co 5896  Rcnr 7326  0Rc0r 7327   +R cplr 7330   <R cltr 7332  cr 7840   + caddc 7844   < cltrr 7845
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-irdg 6395  df-1o 6441  df-2o 6442  df-oadd 6445  df-omul 6446  df-er 6559  df-ec 6561  df-qs 6565  df-ni 7333  df-pli 7334  df-mi 7335  df-lti 7336  df-plpq 7373  df-mpq 7374  df-enq 7376  df-nqqs 7377  df-plqqs 7378  df-mqqs 7379  df-1nqqs 7380  df-rq 7381  df-ltnqqs 7382  df-enq0 7453  df-nq0 7454  df-0nq0 7455  df-plq0 7456  df-mq0 7457  df-inp 7495  df-i1p 7496  df-iplp 7497  df-iltp 7499  df-enr 7755  df-nr 7756  df-plr 7757  df-ltr 7759  df-0r 7760  df-c 7847  df-r 7851  df-add 7852  df-lt 7854
This theorem is referenced by: (None)
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