Theorem axpre-ltadd 11153

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 11284. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 11177. (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 11117	. . 3 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R ⟨𝑥, 0R⟩ = 𝐴)
2		elreal 11117	. . 3 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐵)
3		elreal 11117	. . 3 ⊢ (𝐶 ∈ ℝ ↔ ∃𝑧 ∈ R ⟨𝑧, 0R⟩ = 𝐶)
4		breq1 5113	. . . 4 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ ⟨𝑦, 0R⟩))
5		oveq2 7420	. . . . 5 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) = (⟨𝑧, 0R⟩ + 𝐴))
6	5	breq1d 5120	. . . 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 5114	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ 𝐵))
9		oveq2 7420	. . . . 5 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) = (⟨𝑧, 0R⟩ + 𝐵))
10	9	breq2d 5122	. . . 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 7419	. . . . 5 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (⟨𝑧, 0R⟩ + 𝐴) = (𝐶 + 𝐴))
13		oveq1 7419	. . . . 5 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (⟨𝑧, 0R⟩ + 𝐵) = (𝐶 + 𝐵))
14	12, 13	breq12d 5123	. . . 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 11086	. . . . . . 7 ⊢ (𝑧 ∈ R → (𝑥 <R 𝑦 ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦)))
17	16	adantr 485	. . . . . 6 ⊢ ((𝑧 ∈ R ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) → (𝑥 <R 𝑦 ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦)))
18		ltresr 11126	. . . . . . 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 11124	. . . . . . . . 9 ⊢ ((𝑧 ∈ R ∧ 𝑥 ∈ R) → (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) = ⟨(𝑧 +R 𝑥), 0R⟩)
21		addresr 11124	. . . . . . . . 9 ⊢ ((𝑧 ∈ R ∧ 𝑦 ∈ R) → (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) = ⟨(𝑧 +R 𝑦), 0R⟩)
22	20, 21	breqan12d 5126	. . . . . . . 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 690	. . . . . . 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 11126	. . . . . . 7 ⊢ (⟨(𝑧 +R 𝑥), 0R⟩ <ℝ ⟨(𝑧 +R 𝑦), 0R⟩ ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦))
25	23, 24	bitrdi 290	. . . . . 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 314	. . . . 5 ⊢ ((𝑧 ∈ R ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) → (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) <ℝ (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)))
27	26	ancoms 463	. . . 4 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ 𝑧 ∈ R) → (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) <ℝ (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)))
28	27	3impa 1127	. . 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 3498	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 ↔ (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵)))
30	29	biimpd 232	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ⟨cop 4596   class class class wbr 5110  (class class class)co 7412  Rcnr 10851  0_Rc0r 10852   +_R cplr 10855   <_R cltr 10857  ℝcr 11100   + caddc 11104   <_ℝ cltrr 11105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-inf2 9611

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-oadd 8458  df-omul 8459  df-er 8695  df-ec 8697  df-qs 8701  df-ni 10858  df-pli 10859  df-mi 10860  df-lti 10861  df-plpq 10894  df-mpq 10895  df-ltpq 10896  df-enq 10897  df-nq 10898  df-erq 10899  df-plq 10900  df-mq 10901  df-1nq 10902  df-rq 10903  df-ltnq 10904  df-np 10967  df-1p 10968  df-plp 10969  df-ltp 10971  df-enr 11041  df-nr 11042  df-plr 11043  df-ltr 11045  df-0r 11046  df-c 11107  df-r 11111  df-add 11112  df-lt 11114

This theorem is referenced by: (None)