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Theorem axpre-ltadd 11192
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 11319. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 11216. (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 11156 . . 3 (𝐴 ∈ ℝ ↔ ∃𝑥R𝑥, 0R⟩ = 𝐴)
2 elreal 11156 . . 3 (𝐵 ∈ ℝ ↔ ∃𝑦R𝑦, 0R⟩ = 𝐵)
3 elreal 11156 . . 3 (𝐶 ∈ ℝ ↔ ∃𝑧R𝑧, 0R⟩ = 𝐶)
4 breq1 5152 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝐴 <𝑦, 0R⟩))
5 oveq2 7427 . . . . 5 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) = (⟨𝑧, 0R⟩ + 𝐴))
65breq1d 5159 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ (⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)))
74, 6bibi12d 344 . . 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 5153 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <𝑦, 0R⟩ ↔ 𝐴 < 𝐵))
9 oveq2 7427 . . . . 5 (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) = (⟨𝑧, 0R⟩ + 𝐵))
109breq2d 5161 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → ((⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ (⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + 𝐵)))
118, 10bibi12d 344 . . 3 (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 <𝑦, 0R⟩ ↔ (⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)) ↔ (𝐴 < 𝐵 ↔ (⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + 𝐵))))
12 oveq1 7426 . . . . 5 (⟨𝑧, 0R⟩ = 𝐶 → (⟨𝑧, 0R⟩ + 𝐴) = (𝐶 + 𝐴))
13 oveq1 7426 . . . . 5 (⟨𝑧, 0R⟩ = 𝐶 → (⟨𝑧, 0R⟩ + 𝐵) = (𝐶 + 𝐵))
1412, 13breq12d 5162 . . . 4 (⟨𝑧, 0R⟩ = 𝐶 → ((⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + 𝐵) ↔ (𝐶 + 𝐴) < (𝐶 + 𝐵)))
1514bibi2d 341 . . 3 (⟨𝑧, 0R⟩ = 𝐶 → ((𝐴 < 𝐵 ↔ (⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + 𝐵)) ↔ (𝐴 < 𝐵 ↔ (𝐶 + 𝐴) < (𝐶 + 𝐵))))
16 ltasr 11125 . . . . . . 7 (𝑧R → (𝑥 <R 𝑦 ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦)))
1716adantr 479 . . . . . 6 ((𝑧R ∧ (𝑥R𝑦R)) → (𝑥 <R 𝑦 ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦)))
18 ltresr 11165 . . . . . . 7 (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
1918a1i 11 . . . . . 6 ((𝑧R ∧ (𝑥R𝑦R)) → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝑥 <R 𝑦))
20 addresr 11163 . . . . . . . . 9 ((𝑧R𝑥R) → (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) = ⟨(𝑧 +R 𝑥), 0R⟩)
21 addresr 11163 . . . . . . . . 9 ((𝑧R𝑦R) → (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) = ⟨(𝑧 +R 𝑦), 0R⟩)
2220, 21breqan12d 5165 . . . . . . . 8 (((𝑧R𝑥R) ∧ (𝑧R𝑦R)) → ((⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ ⟨(𝑧 +R 𝑥), 0R⟩ < ⟨(𝑧 +R 𝑦), 0R⟩))
2322anandis 676 . . . . . . 7 ((𝑧R ∧ (𝑥R𝑦R)) → ((⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ ⟨(𝑧 +R 𝑥), 0R⟩ < ⟨(𝑧 +R 𝑦), 0R⟩))
24 ltresr 11165 . . . . . . 7 (⟨(𝑧 +R 𝑥), 0R⟩ < ⟨(𝑧 +R 𝑦), 0R⟩ ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦))
2523, 24bitrdi 286 . . . . . 6 ((𝑧R ∧ (𝑥R𝑦R)) → ((⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦)))
2617, 19, 253bitr4d 310 . . . . 5 ((𝑧R ∧ (𝑥R𝑦R)) → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)))
2726ancoms 457 . . . 4 (((𝑥R𝑦R) ∧ 𝑧R) → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)))
28273impa 1107 . . 3 ((𝑥R𝑦R𝑧R) → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)))
291, 2, 3, 7, 11, 15, 283gencl 3506 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐶 + 𝐴) < (𝐶 + 𝐵)))
3029biimpd 228 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐶 + 𝐴) < (𝐶 + 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  cop 4636   class class class wbr 5149  (class class class)co 7419  Rcnr 10890  0Rc0r 10891   +R cplr 10894   <R cltr 10896  cr 11139   + caddc 11143   < cltrr 11144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9666
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-oadd 8491  df-omul 8492  df-er 8725  df-ec 8727  df-qs 8731  df-ni 10897  df-pli 10898  df-mi 10899  df-lti 10900  df-plpq 10933  df-mpq 10934  df-ltpq 10935  df-enq 10936  df-nq 10937  df-erq 10938  df-plq 10939  df-mq 10940  df-1nq 10941  df-rq 10942  df-ltnq 10943  df-np 11006  df-1p 11007  df-plp 11008  df-ltp 11010  df-enr 11080  df-nr 11081  df-plr 11082  df-ltr 11084  df-0r 11085  df-c 11146  df-r 11150  df-add 11151  df-lt 11153
This theorem is referenced by: (None)
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