Theorem axpre-ltadd 8201

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 8243. (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 8143	. . 3 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R ⟨𝑥, 0R⟩ = 𝐴)
2		elreal 8143	. . 3 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐵)
3		elreal 8143	. . 3 ⊢ (𝐶 ∈ ℝ ↔ ∃𝑧 ∈ R ⟨𝑧, 0R⟩ = 𝐶)
4		breq1 4112	. . . 4 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ ⟨𝑦, 0R⟩))
5		oveq2 6058	. . . . 5 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) = (⟨𝑧, 0R⟩ + 𝐴))
6	5	breq1d 4119	. . . 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 235	. . 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 4113	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ 𝐵))
9		oveq2 6058	. . . . 5 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) = (⟨𝑧, 0R⟩ + 𝐵))
10	9	breq2d 4121	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → ((⟨𝑧, 0R⟩ + 𝐴) <ℝ (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ (⟨𝑧, 0R⟩ + 𝐴) <ℝ (⟨𝑧, 0R⟩ + 𝐵)))
11	8, 10	bibi12d 235	. . 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 6057	. . . . 5 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (⟨𝑧, 0R⟩ + 𝐴) = (𝐶 + 𝐴))
13		oveq1 6057	. . . . 5 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (⟨𝑧, 0R⟩ + 𝐵) = (𝐶 + 𝐵))
14	12, 13	breq12d 4122	. . . 4 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → ((⟨𝑧, 0R⟩ + 𝐴) <ℝ (⟨𝑧, 0R⟩ + 𝐵) ↔ (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵)))
15	14	bibi2d 232	. . 3 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → ((𝐴 <ℝ 𝐵 ↔ (⟨𝑧, 0R⟩ + 𝐴) <ℝ (⟨𝑧, 0R⟩ + 𝐵)) ↔ (𝐴 <ℝ 𝐵 ↔ (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵))))
16		ltasrg 8085	. . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R) → (𝑥 <R 𝑦 ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦)))
17		ltresr 8154	. . . . 5 ⊢ (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
18	17	a1i 9	. . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R) → (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝑥 <R 𝑦))
19		simp3 1026	. . . . . 6 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R) → 𝑧 ∈ R)
20		simp1 1024	. . . . . 6 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R) → 𝑥 ∈ R)
21		simp2 1025	. . . . . 6 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R) → 𝑦 ∈ R)
22		addresr 8152	. . . . . . 7 ⊢ ((𝑧 ∈ R ∧ 𝑥 ∈ R) → (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) = ⟨(𝑧 +R 𝑥), 0R⟩)
23		addresr 8152	. . . . . . 7 ⊢ ((𝑧 ∈ R ∧ 𝑦 ∈ R) → (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) = ⟨(𝑧 +R 𝑦), 0R⟩)
24	22, 23	breqan12d 4125	. . . . . 6 ⊢ (((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⟩))
25	19, 20, 19, 21, 24	syl22anc 1275	. . . . 5 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R) → ((⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) <ℝ (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ ⟨(𝑧 +R 𝑥), 0R⟩ <ℝ ⟨(𝑧 +R 𝑦), 0R⟩))
26		ltresr 8154	. . . . 5 ⊢ (⟨(𝑧 +R 𝑥), 0R⟩ <ℝ ⟨(𝑧 +R 𝑦), 0R⟩ ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦))
27	25, 26	bitrdi 196	. . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R) → ((⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) <ℝ (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦)))
28	16, 18, 27	3bitr4d 220	. . 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 2848	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 ↔ (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵)))
30	29	biimpd 144	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203  ⟨cop 3692   class class class wbr 4109  (class class class)co 6050  Rcnr 7612  0_Rc0r 7613   +_R cplr 7616   <_R cltr 7618  ℝcr 8126   + caddc 8130   <_ℝ cltrr 8131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-eprel 4410  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-1o 6647  df-2o 6648  df-oadd 6651  df-omul 6652  df-er 6767  df-ec 6769  df-qs 6773  df-ni 7619  df-pli 7620  df-mi 7621  df-lti 7622  df-plpq 7659  df-mpq 7660  df-enq 7662  df-nqqs 7663  df-plqqs 7664  df-mqqs 7665  df-1nqqs 7666  df-rq 7667  df-ltnqqs 7668  df-enq0 7739  df-nq0 7740  df-0nq0 7741  df-plq0 7742  df-mq0 7743  df-inp 7781  df-i1p 7782  df-iplp 7783  df-iltp 7785  df-enr 8041  df-nr 8042  df-plr 8043  df-ltr 8045  df-0r 8046  df-c 8133  df-r 8137  df-add 8138  df-lt 8140

This theorem is referenced by: (None)