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Theorem axpre-ltadd 10566
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 10691. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 10590. (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 10530 . . 3 (𝐴 ∈ ℝ ↔ ∃𝑥R𝑥, 0R⟩ = 𝐴)
2 elreal 10530 . . 3 (𝐵 ∈ ℝ ↔ ∃𝑦R𝑦, 0R⟩ = 𝐵)
3 elreal 10530 . . 3 (𝐶 ∈ ℝ ↔ ∃𝑧R𝑧, 0R⟩ = 𝐶)
4 breq1 5042 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝐴 <𝑦, 0R⟩))
5 oveq2 7138 . . . . 5 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) = (⟨𝑧, 0R⟩ + 𝐴))
65breq1d 5049 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ (⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)))
74, 6bibi12d 349 . . 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 5043 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <𝑦, 0R⟩ ↔ 𝐴 < 𝐵))
9 oveq2 7138 . . . . 5 (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) = (⟨𝑧, 0R⟩ + 𝐵))
109breq2d 5051 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → ((⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ (⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + 𝐵)))
118, 10bibi12d 349 . . 3 (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 <𝑦, 0R⟩ ↔ (⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)) ↔ (𝐴 < 𝐵 ↔ (⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + 𝐵))))
12 oveq1 7137 . . . . 5 (⟨𝑧, 0R⟩ = 𝐶 → (⟨𝑧, 0R⟩ + 𝐴) = (𝐶 + 𝐴))
13 oveq1 7137 . . . . 5 (⟨𝑧, 0R⟩ = 𝐶 → (⟨𝑧, 0R⟩ + 𝐵) = (𝐶 + 𝐵))
1412, 13breq12d 5052 . . . 4 (⟨𝑧, 0R⟩ = 𝐶 → ((⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + 𝐵) ↔ (𝐶 + 𝐴) < (𝐶 + 𝐵)))
1514bibi2d 346 . . 3 (⟨𝑧, 0R⟩ = 𝐶 → ((𝐴 < 𝐵 ↔ (⟨𝑧, 0R⟩ + 𝐴) < (⟨𝑧, 0R⟩ + 𝐵)) ↔ (𝐴 < 𝐵 ↔ (𝐶 + 𝐴) < (𝐶 + 𝐵))))
16 ltasr 10499 . . . . . . 7 (𝑧R → (𝑥 <R 𝑦 ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦)))
1716adantr 484 . . . . . 6 ((𝑧R ∧ (𝑥R𝑦R)) → (𝑥 <R 𝑦 ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦)))
18 ltresr 10539 . . . . . . 7 (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
1918a1i 11 . . . . . 6 ((𝑧R ∧ (𝑥R𝑦R)) → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝑥 <R 𝑦))
20 addresr 10537 . . . . . . . . 9 ((𝑧R𝑥R) → (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) = ⟨(𝑧 +R 𝑥), 0R⟩)
21 addresr 10537 . . . . . . . . 9 ((𝑧R𝑦R) → (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) = ⟨(𝑧 +R 𝑦), 0R⟩)
2220, 21breqan12d 5055 . . . . . . . 8 (((𝑧R𝑥R) ∧ (𝑧R𝑦R)) → ((⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ ⟨(𝑧 +R 𝑥), 0R⟩ < ⟨(𝑧 +R 𝑦), 0R⟩))
2322anandis 677 . . . . . . 7 ((𝑧R ∧ (𝑥R𝑦R)) → ((⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ ⟨(𝑧 +R 𝑥), 0R⟩ < ⟨(𝑧 +R 𝑦), 0R⟩))
24 ltresr 10539 . . . . . . 7 (⟨(𝑧 +R 𝑥), 0R⟩ < ⟨(𝑧 +R 𝑦), 0R⟩ ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦))
2523, 24syl6bb 290 . . . . . 6 ((𝑧R ∧ (𝑥R𝑦R)) → ((⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦)))
2617, 19, 253bitr4d 314 . . . . 5 ((𝑧R ∧ (𝑥R𝑦R)) → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) < (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)))
2726ancoms 462 . . . 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 3513 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐶 + 𝐴) < (𝐶 + 𝐵)))
3029biimpd 232 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐶 + 𝐴) < (𝐶 + 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  cop 4546   class class class wbr 5039  (class class class)co 7130  Rcnr 10264  0Rc0r 10265   +R cplr 10268   <R cltr 10270  cr 10513   + caddc 10517   < cltrr 10518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-inf2 9080
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-oadd 8081  df-omul 8082  df-er 8264  df-ec 8266  df-qs 8270  df-ni 10271  df-pli 10272  df-mi 10273  df-lti 10274  df-plpq 10307  df-mpq 10308  df-ltpq 10309  df-enq 10310  df-nq 10311  df-erq 10312  df-plq 10313  df-mq 10314  df-1nq 10315  df-rq 10316  df-ltnq 10317  df-np 10380  df-1p 10381  df-plp 10382  df-ltp 10384  df-enr 10454  df-nr 10455  df-plr 10456  df-ltr 10458  df-0r 10459  df-c 10520  df-r 10524  df-add 10525  df-lt 10527
This theorem is referenced by: (None)
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