Theorem axpre-ltadd 11081

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 11210. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 11105. (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 11045	. . 3 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R ⟨𝑥, 0R⟩ = 𝐴)
2		elreal 11045	. . 3 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐵)
3		elreal 11045	. . 3 ⊢ (𝐶 ∈ ℝ ↔ ∃𝑧 ∈ R ⟨𝑧, 0R⟩ = 𝐶)
4		breq1 5075	. . . 4 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ ⟨𝑦, 0R⟩))
5		oveq2 7364	. . . . 5 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) = (⟨𝑧, 0R⟩ + 𝐴))
6	5	breq1d 5082	. . . 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 346	. . 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 5076	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ 𝐵))
9		oveq2 7364	. . . . 5 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) = (⟨𝑧, 0R⟩ + 𝐵))
10	9	breq2d 5084	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → ((⟨𝑧, 0R⟩ + 𝐴) <ℝ (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ (⟨𝑧, 0R⟩ + 𝐴) <ℝ (⟨𝑧, 0R⟩ + 𝐵)))
11	8, 10	bibi12d 346	. . 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 7363	. . . . 5 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (⟨𝑧, 0R⟩ + 𝐴) = (𝐶 + 𝐴))
13		oveq1 7363	. . . . 5 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (⟨𝑧, 0R⟩ + 𝐵) = (𝐶 + 𝐵))
14	12, 13	breq12d 5085	. . . 4 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → ((⟨𝑧, 0R⟩ + 𝐴) <ℝ (⟨𝑧, 0R⟩ + 𝐵) ↔ (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵)))
15	14	bibi2d 343	. . 3 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → ((𝐴 <ℝ 𝐵 ↔ (⟨𝑧, 0R⟩ + 𝐴) <ℝ (⟨𝑧, 0R⟩ + 𝐵)) ↔ (𝐴 <ℝ 𝐵 ↔ (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵))))
16		ltasr 11014	. . . . . . 7 ⊢ (𝑧 ∈ R → (𝑥 <R 𝑦 ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦)))
17	16	adantr 481	. . . . . 6 ⊢ ((𝑧 ∈ R ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) → (𝑥 <R 𝑦 ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦)))
18		ltresr 11054	. . . . . . 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 11052	. . . . . . . . 9 ⊢ ((𝑧 ∈ R ∧ 𝑥 ∈ R) → (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) = ⟨(𝑧 +R 𝑥), 0R⟩)
21		addresr 11052	. . . . . . . . 9 ⊢ ((𝑧 ∈ R ∧ 𝑦 ∈ R) → (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) = ⟨(𝑧 +R 𝑦), 0R⟩)
22	20, 21	breqan12d 5088	. . . . . . . 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 684	. . . . . . 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 11054	. . . . . . 7 ⊢ (⟨(𝑧 +R 𝑥), 0R⟩ <ℝ ⟨(𝑧 +R 𝑦), 0R⟩ ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦))
25	23, 24	bitrdi 288	. . . . . 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 312	. . . . 5 ⊢ ((𝑧 ∈ R ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) → (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) <ℝ (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)))
27	26	ancoms 459	. . . 4 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ 𝑧 ∈ R) → (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) <ℝ (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)))
28	27	3impa 1115	. . 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 3474	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 ↔ (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵)))
30	29	biimpd 230	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ⟨cop 4561   class class class wbr 5072  (class class class)co 7356  Rcnr 10779  0_Rc0r 10780   +_R cplr 10783   <_R cltr 10785  ℝcr 11028   + caddc 11032   <_ℝ cltrr 11033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-inf2 9553

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-oadd 8399  df-omul 8400  df-er 8633  df-ec 8635  df-qs 8639  df-ni 10786  df-pli 10787  df-mi 10788  df-lti 10789  df-plpq 10822  df-mpq 10823  df-ltpq 10824  df-enq 10825  df-nq 10826  df-erq 10827  df-plq 10828  df-mq 10829  df-1nq 10830  df-rq 10831  df-ltnq 10832  df-np 10895  df-1p 10896  df-plp 10897  df-ltp 10899  df-enr 10969  df-nr 10970  df-plr 10971  df-ltr 10973  df-0r 10974  df-c 11035  df-r 11039  df-add 11040  df-lt 11042

This theorem is referenced by: (None)