Theorem axpre-ltadd 11090

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 11219. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 11114. (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 11054	. . 3 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R ⟨𝑥, 0R⟩ = 𝐴)
2		elreal 11054	. . 3 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐵)
3		elreal 11054	. . 3 ⊢ (𝐶 ∈ ℝ ↔ ∃𝑧 ∈ R ⟨𝑧, 0R⟩ = 𝐶)
4		breq1 5088	. . . 4 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ ⟨𝑦, 0R⟩))
5		oveq2 7375	. . . . 5 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) = (⟨𝑧, 0R⟩ + 𝐴))
6	5	breq1d 5095	. . . 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 345	. . 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 5089	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ 𝐵))
9		oveq2 7375	. . . . 5 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) = (⟨𝑧, 0R⟩ + 𝐵))
10	9	breq2d 5097	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → ((⟨𝑧, 0R⟩ + 𝐴) <ℝ (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) ↔ (⟨𝑧, 0R⟩ + 𝐴) <ℝ (⟨𝑧, 0R⟩ + 𝐵)))
11	8, 10	bibi12d 345	. . 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 7374	. . . . 5 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (⟨𝑧, 0R⟩ + 𝐴) = (𝐶 + 𝐴))
13		oveq1 7374	. . . . 5 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (⟨𝑧, 0R⟩ + 𝐵) = (𝐶 + 𝐵))
14	12, 13	breq12d 5098	. . . 4 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → ((⟨𝑧, 0R⟩ + 𝐴) <ℝ (⟨𝑧, 0R⟩ + 𝐵) ↔ (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵)))
15	14	bibi2d 342	. . 3 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → ((𝐴 <ℝ 𝐵 ↔ (⟨𝑧, 0R⟩ + 𝐴) <ℝ (⟨𝑧, 0R⟩ + 𝐵)) ↔ (𝐴 <ℝ 𝐵 ↔ (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵))))
16		ltasr 11023	. . . . . . 7 ⊢ (𝑧 ∈ R → (𝑥 <R 𝑦 ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦)))
17	16	adantr 480	. . . . . 6 ⊢ ((𝑧 ∈ R ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) → (𝑥 <R 𝑦 ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦)))
18		ltresr 11063	. . . . . . 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 11061	. . . . . . . . 9 ⊢ ((𝑧 ∈ R ∧ 𝑥 ∈ R) → (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) = ⟨(𝑧 +R 𝑥), 0R⟩)
21		addresr 11061	. . . . . . . . 9 ⊢ ((𝑧 ∈ R ∧ 𝑦 ∈ R) → (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩) = ⟨(𝑧 +R 𝑦), 0R⟩)
22	20, 21	breqan12d 5101	. . . . . . . 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 679	. . . . . . 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 11063	. . . . . . 7 ⊢ (⟨(𝑧 +R 𝑥), 0R⟩ <ℝ ⟨(𝑧 +R 𝑦), 0R⟩ ↔ (𝑧 +R 𝑥) <R (𝑧 +R 𝑦))
25	23, 24	bitrdi 287	. . . . . 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 311	. . . . 5 ⊢ ((𝑧 ∈ R ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) → (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) <ℝ (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)))
27	26	ancoms 458	. . . 4 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ 𝑧 ∈ R) → (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ (⟨𝑧, 0R⟩ + ⟨𝑥, 0R⟩) <ℝ (⟨𝑧, 0R⟩ + ⟨𝑦, 0R⟩)))
28	27	3impa 1110	. . 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 3473	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 ↔ (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵)))
30	29	biimpd 229	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ⟨cop 4573   class class class wbr 5085  (class class class)co 7367  Rcnr 10788  0_Rc0r 10789   +_R cplr 10792   <_R cltr 10794  ℝcr 11037   + caddc 11041   <_ℝ cltrr 11042

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-oadd 8409  df-omul 8410  df-er 8643  df-ec 8645  df-qs 8649  df-ni 10795  df-pli 10796  df-mi 10797  df-lti 10798  df-plpq 10831  df-mpq 10832  df-ltpq 10833  df-enq 10834  df-nq 10835  df-erq 10836  df-plq 10837  df-mq 10838  df-1nq 10839  df-rq 10840  df-ltnq 10841  df-np 10904  df-1p 10905  df-plp 10906  df-ltp 10908  df-enr 10978  df-nr 10979  df-plr 10980  df-ltr 10982  df-0r 10983  df-c 11044  df-r 11048  df-add 11049  df-lt 11051

This theorem is referenced by: (None)