Theorem axpre-ltadd 10591

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 10716. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 10615. (Contributed by NM, 11-May-1996.) (New usage is discouraged.)

Assertion

Ref	Expression
axpre-ltadd	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵)))

Proof of Theorem axpre-ltadd

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elreal 10555	. . 3 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R ⟨𝑥, 0R⟩ = 𝐴)
2		elreal 10555	. . 3 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐵)
3		elreal 10555	. . 3 ⊢ (𝐶 ∈ ℝ ↔ ∃𝑧 ∈ R ⟨𝑧, 0R⟩ = 𝐶)
4		breq1 5071	. . . 4 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ ⟨𝑦, 0R⟩))
5		oveq2 7166	. . . . 5 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) = (⟨𝑧, 0R⟩ + 𝐴))
6	5	breq1d 5078	. . . 4 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) <ℝ (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ (⟨𝑧, 0R⟩ + 𝐴) <ℝ (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)))
7	4, 6	bibi12d 348	. . 3 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) <ℝ (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)) ↔ (𝐴 <ℝ ⟨𝑦, 0R⟩ ↔ (⟨𝑧, 0R⟩ + 𝐴) <ℝ (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩))))
8		breq2 5072	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ 𝐵))
9		oveq2 7166	. . . . 5 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) = (⟨𝑧, 0R⟩ + 𝐵))
10	9	breq2d 5080	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → ((⟨𝑧, 0R⟩ + 𝐴) <ℝ (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ (⟨𝑧, 0R⟩ + 𝐴) <ℝ (⟨𝑧, 0R⟩ + 𝐵)))
11	8, 10	bibi12d 348	. . 3 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 <ℝ ⟨𝑦, 0R⟩ ↔ (⟨𝑧, 0R⟩ + 𝐴) <ℝ (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)) ↔ (𝐴 <ℝ 𝐵 ↔ (⟨𝑧, 0R⟩ + 𝐴) <ℝ (⟨𝑧, 0R⟩ + 𝐵))))
12		oveq1 7165	. . . . 5 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (⟨𝑧, 0R⟩ + 𝐴) = (𝐶 + 𝐴))
13		oveq1 7165	. . . . 5 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (⟨𝑧, 0R⟩ + 𝐵) = (𝐶 + 𝐵))
14	12, 13	breq12d 5081	. . . 4 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → ((⟨𝑧, 0R⟩ + 𝐴) <ℝ (⟨𝑧, 0R⟩ + 𝐵) ↔ (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵)))
15	14	bibi2d 345	. . 3 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → ((𝐴 <ℝ 𝐵 ↔ (⟨𝑧, 0R⟩ + 𝐴) <ℝ (⟨𝑧, 0R⟩ + 𝐵)) ↔ (𝐴 <ℝ 𝐵 ↔ (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵))))
16		ltasr 10524	. . . . . . 7 ⊢ (𝑧 ∈ R → (𝑥 <R 𝑦 ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦)))
17	16	adantr 483	. . . . . 6 ⊢ ((𝑧 ∈ R ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) → (𝑥 <R 𝑦 ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦)))
18		ltresr 10564	. . . . . . 7 ⊢ (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
19	18	a1i 11	. . . . . 6 ⊢ ((𝑧 ∈ R ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) → (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝑥 <R 𝑦))
20		addresr 10562	. . . . . . . . 9 ⊢ ((𝑧 ∈ R ∧ 𝑥 ∈ R) → (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) = ⟨(𝑧 +R 𝑥), 0R⟩)
21		addresr 10562	. . . . . . . . 9 ⊢ ((𝑧 ∈ R ∧ 𝑦 ∈ R) → (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) = ⟨(𝑧 +R 𝑦), 0R⟩)
22	20, 21	breqan12d 5084	. . . . . . . 8 ⊢ (((𝑧 ∈ R ∧ 𝑥 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑦 ∈ R)) → ((⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) <ℝ (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ ⟨(𝑧 +R 𝑥), 0R⟩ <ℝ ⟨(𝑧 +R 𝑦), 0R⟩))
23	22	anandis 676	. . . . . . 7 ⊢ ((𝑧 ∈ R ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) → ((⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) <ℝ (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ ⟨(𝑧 +R 𝑥), 0R⟩ <ℝ ⟨(𝑧 +R 𝑦), 0R⟩))
24		ltresr 10564	. . . . . . 7 ⊢ (⟨(𝑧 +R 𝑥), 0R⟩ <ℝ ⟨(𝑧 +R 𝑦), 0R⟩ ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦))
25	23, 24	syl6bb 289	. . . . . 6 ⊢ ((𝑧 ∈ R ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) → ((⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) <ℝ (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦)))
26	17, 19, 25	3bitr4d 313	. . . . 5 ⊢ ((𝑧 ∈ R ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) → (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) <ℝ (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)))
27	26	ancoms 461	. . . 4 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ 𝑧 ∈ R) → (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) <ℝ (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)))
28	27	3impa 1106	. . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R) → (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) <ℝ (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)))
29	1, 2, 3, 7, 11, 15, 28	3gencl 3538	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 ↔ (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵)))
30	29	biimpd 231	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ⟨cop 4575   class class class wbr 5068  (class class class)co 7158  Rcnr 10289  0_Rc0r 10290   +_R cplr 10293   <_R cltr 10295  ℝcr 10538   + caddc 10542   <_ℝ cltrr 10543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-inf2 9106

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-oadd 8108  df-omul 8109  df-er 8291  df-ec 8293  df-qs 8297  df-ni 10296  df-pli 10297  df-mi 10298  df-lti 10299  df-plpq 10332  df-mpq 10333  df-ltpq 10334  df-enq 10335  df-nq 10336  df-erq 10337  df-plq 10338  df-mq 10339  df-1nq 10340  df-rq 10341  df-ltnq 10342  df-np 10405  df-1p 10406  df-plp 10407  df-ltp 10409  df-enr 10479  df-nr 10480  df-plr 10481  df-ltr 10483  df-0r 10484  df-c 10545  df-r 10549  df-add 10550  df-lt 10552

This theorem is referenced by: (None)