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Theorem axpre-ltadd 7706
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 7748. (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 7648 . . 3 (𝐴 ∈ ℝ ↔ ∃𝑥R𝑥, 0R⟩ = 𝐴)
2 elreal 7648 . . 3 (𝐵 ∈ ℝ ↔ ∃𝑦R𝑦, 0R⟩ = 𝐵)
3 elreal 7648 . . 3 (𝐶 ∈ ℝ ↔ ∃𝑧R𝑧, 0R⟩ = 𝐶)
4 breq1 3932 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝐴 <𝑦, 0R⟩))
5 oveq2 5782 . . . . 5 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) = (⟨𝑧, 0R⟩ + 𝐴))
65breq1d 3939 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ (⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)))
74, 6bibi12d 234 . . 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 3933 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <𝑦, 0R⟩ ↔ 𝐴 < 𝐵))
9 oveq2 5782 . . . . 5 (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) = (⟨𝑧, 0R⟩ + 𝐵))
109breq2d 3941 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → ((⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ (⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + 𝐵)))
118, 10bibi12d 234 . . 3 (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 <𝑦, 0R⟩ ↔ (⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)) ↔ (𝐴 < 𝐵 ↔ (⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + 𝐵))))
12 oveq1 5781 . . . . 5 (⟨𝑧, 0R⟩ = 𝐶 → (⟨𝑧, 0R⟩ + 𝐴) = (𝐶 + 𝐴))
13 oveq1 5781 . . . . 5 (⟨𝑧, 0R⟩ = 𝐶 → (⟨𝑧, 0R⟩ + 𝐵) = (𝐶 + 𝐵))
1412, 13breq12d 3942 . . . 4 (⟨𝑧, 0R⟩ = 𝐶 → ((⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + 𝐵) ↔ (𝐶 + 𝐴) < (𝐶 + 𝐵)))
1514bibi2d 231 . . 3 (⟨𝑧, 0R⟩ = 𝐶 → ((𝐴 < 𝐵 ↔ (⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + 𝐵)) ↔ (𝐴 < 𝐵 ↔ (𝐶 + 𝐴) < (𝐶 + 𝐵))))
16 ltasrg 7590 . . . 4 ((𝑥R𝑦R𝑧R) → (𝑥 <R 𝑦 ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦)))
17 ltresr 7659 . . . . 5 (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
1817a1i 9 . . . 4 ((𝑥R𝑦R𝑧R) → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝑥 <R 𝑦))
19 simp3 983 . . . . . 6 ((𝑥R𝑦R𝑧R) → 𝑧R)
20 simp1 981 . . . . . 6 ((𝑥R𝑦R𝑧R) → 𝑥R)
21 simp2 982 . . . . . 6 ((𝑥R𝑦R𝑧R) → 𝑦R)
22 addresr 7657 . . . . . . 7 ((𝑧R𝑥R) → (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) = ⟨(𝑧 +R 𝑥), 0R⟩)
23 addresr 7657 . . . . . . 7 ((𝑧R𝑦R) → (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) = ⟨(𝑧 +R 𝑦), 0R⟩)
2422, 23breqan12d 3945 . . . . . 6 (((𝑧R𝑥R) ∧ (𝑧R𝑦R)) → ((⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ ⟨(𝑧 +R 𝑥), 0R⟩ < ⟨(𝑧 +R 𝑦), 0R⟩))
2519, 20, 19, 21, 24syl22anc 1217 . . . . 5 ((𝑥R𝑦R𝑧R) → ((⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ ⟨(𝑧 +R 𝑥), 0R⟩ < ⟨(𝑧 +R 𝑦), 0R⟩))
26 ltresr 7659 . . . . 5 (⟨(𝑧 +R 𝑥), 0R⟩ < ⟨(𝑧 +R 𝑦), 0R⟩ ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦))
2725, 26syl6bb 195 . . . 4 ((𝑥R𝑦R𝑧R) → ((⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦)))
2816, 18, 273bitr4d 219 . . 3 ((𝑥R𝑦R𝑧R) → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)))
291, 2, 3, 7, 11, 15, 283gencl 2720 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐶 + 𝐴) < (𝐶 + 𝐵)))
3029biimpd 143 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐶 + 𝐴) < (𝐶 + 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 962   = wceq 1331  wcel 1480  cop 3530   class class class wbr 3929  (class class class)co 5774  Rcnr 7117  0Rc0r 7118   +R cplr 7121   <R cltr 7123  cr 7631   + caddc 7635   < cltrr 7636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7124  df-pli 7125  df-mi 7126  df-lti 7127  df-plpq 7164  df-mpq 7165  df-enq 7167  df-nqqs 7168  df-plqqs 7169  df-mqqs 7170  df-1nqqs 7171  df-rq 7172  df-ltnqqs 7173  df-enq0 7244  df-nq0 7245  df-0nq0 7246  df-plq0 7247  df-mq0 7248  df-inp 7286  df-i1p 7287  df-iplp 7288  df-iltp 7290  df-enr 7546  df-nr 7547  df-plr 7548  df-ltr 7550  df-0r 7551  df-c 7638  df-r 7642  df-add 7643  df-lt 7645
This theorem is referenced by: (None)
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