Theorem nnaordi 6104

Description: Ordering property of addition. Proposition 8.4 of [TakeutiZaring] p. 58, limited to natural numbers. (Contributed by NM, 3-Feb-1996.) (Revised by Mario Carneiro, 15-Nov-2014.)

Assertion

Ref	Expression
nnaordi	⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ∈ 𝐵 → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))

Proof of Theorem nnaordi

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

Step	Hyp	Ref	Expression
1		oveq2 5540	. . . . . . . . 9 ⊢ (𝑥 = 𝐶 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝐶))
2		oveq2 5540	. . . . . . . . 9 ⊢ (𝑥 = 𝐶 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝐶))
3	1, 2	eleq12d 2149	. . . . . . . 8 ⊢ (𝑥 = 𝐶 → ((𝐴 +𝑜 𝑥) ∈ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 𝐶) ∈ (𝐵 +𝑜 𝐶)))
4	3	imbi2d 228	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐴 +𝑜 𝑥) ∈ (𝐵 +𝑜 𝑥)) ↔ ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐴 +𝑜 𝐶) ∈ (𝐵 +𝑜 𝐶))))
5		oveq2 5540	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 ∅))
6		oveq2 5540	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 ∅))
7	5, 6	eleq12d 2149	. . . . . . . 8 ⊢ (𝑥 = ∅ → ((𝐴 +𝑜 𝑥) ∈ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 ∅) ∈ (𝐵 +𝑜 ∅)))
8		oveq2 5540	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑦))
9		oveq2 5540	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝑦))
10	8, 9	eleq12d 2149	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝐴 +𝑜 𝑥) ∈ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑦)))
11		oveq2 5540	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 suc 𝑦))
12		oveq2 5540	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 suc 𝑦))
13	11, 12	eleq12d 2149	. . . . . . . 8 ⊢ (𝑥 = suc 𝑦 → ((𝐴 +𝑜 𝑥) ∈ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 suc 𝑦) ∈ (𝐵 +𝑜 suc 𝑦)))
14		simpr 108	. . . . . . . . 9 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵)
15		elnn 4346	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω)
16	15	ancoms 264	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ ω)
17		nna0 6076	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴)
18	16, 17	syl 14	. . . . . . . . 9 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐴 +𝑜 ∅) = 𝐴)
19		nna0 6076	. . . . . . . . . 10 ⊢ (𝐵 ∈ ω → (𝐵 +𝑜 ∅) = 𝐵)
20	19	adantr 270	. . . . . . . . 9 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐵 +𝑜 ∅) = 𝐵)
21	14, 18, 20	3eltr4d 2162	. . . . . . . 8 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐴 +𝑜 ∅) ∈ (𝐵 +𝑜 ∅))
22		simprl 497	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝐵 ∈ ω)
23		simpl 107	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝑦 ∈ ω)
24		nnacl 6082	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 𝑦) ∈ ω)
25	22, 23, 24	syl2anc 403	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → (𝐵 +𝑜 𝑦) ∈ ω)
26		nnsucelsuc 6093	. . . . . . . . . . . 12 ⊢ ((𝐵 +𝑜 𝑦) ∈ ω → ((𝐴 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑦) ↔ suc (𝐴 +𝑜 𝑦) ∈ suc (𝐵 +𝑜 𝑦)))
27	25, 26	syl 14	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → ((𝐴 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑦) ↔ suc (𝐴 +𝑜 𝑦) ∈ suc (𝐵 +𝑜 𝑦)))
28	16	adantl 271	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝐴 ∈ ω)
29		nnon 4350	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
30	28, 29	syl 14	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝐴 ∈ On)
31		nnon 4350	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On)
32	31	adantr 270	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝑦 ∈ On)
33		oasuc 6067	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
34	30, 32, 33	syl2anc 403	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
35		nnon 4350	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
36	35	ad2antrl 473	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝐵 ∈ On)
37		oasuc 6067	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
38	36, 32, 37	syl2anc 403	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
39	34, 38	eleq12d 2149	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → ((𝐴 +𝑜 suc 𝑦) ∈ (𝐵 +𝑜 suc 𝑦) ↔ suc (𝐴 +𝑜 𝑦) ∈ suc (𝐵 +𝑜 𝑦)))
40	27, 39	bitr4d 189	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → ((𝐴 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑦) ↔ (𝐴 +𝑜 suc 𝑦) ∈ (𝐵 +𝑜 suc 𝑦)))
41	40	biimpd 142	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → ((𝐴 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑦) → (𝐴 +𝑜 suc 𝑦) ∈ (𝐵 +𝑜 suc 𝑦)))
42	41	ex 113	. . . . . . . 8 ⊢ (𝑦 ∈ ω → ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → ((𝐴 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑦) → (𝐴 +𝑜 suc 𝑦) ∈ (𝐵 +𝑜 suc 𝑦))))
43	7, 10, 13, 21, 42	finds2 4342	. . . . . . 7 ⊢ (𝑥 ∈ ω → ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐴 +𝑜 𝑥) ∈ (𝐵 +𝑜 𝑥)))
44	4, 43	vtoclga 2664	. . . . . 6 ⊢ (𝐶 ∈ ω → ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐴 +𝑜 𝐶) ∈ (𝐵 +𝑜 𝐶)))
45	44	imp 122	. . . . 5 ⊢ ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → (𝐴 +𝑜 𝐶) ∈ (𝐵 +𝑜 𝐶))
46	16	adantl 271	. . . . . 6 ⊢ ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝐴 ∈ ω)
47		simpl 107	. . . . . 6 ⊢ ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝐶 ∈ ω)
48		nnacom 6086	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 +𝑜 𝐶) = (𝐶 +𝑜 𝐴))
49	46, 47, 48	syl2anc 403	. . . . 5 ⊢ ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → (𝐴 +𝑜 𝐶) = (𝐶 +𝑜 𝐴))
50		nnacom 6086	. . . . . . 7 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐵 +𝑜 𝐶) = (𝐶 +𝑜 𝐵))
51	50	ancoms 264	. . . . . 6 ⊢ ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 +𝑜 𝐶) = (𝐶 +𝑜 𝐵))
52	51	adantrr 462	. . . . 5 ⊢ ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → (𝐵 +𝑜 𝐶) = (𝐶 +𝑜 𝐵))
53	45, 49, 52	3eltr3d 2161	. . . 4 ⊢ ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵))
54	53	3impb 1134	. . 3 ⊢ ((𝐶 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵))
55	54	3com12 1142	. 2 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵))
56	55	3expia 1140	1 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ∈ 𝐵 → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284   ∈ wcel 1433  ∅c0 3251  Oncon0 4118  suc csuc 4120  ωcom 4331  (class class class)co 5532   +_𝑜 coa 6021

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-oadd 6028

This theorem is referenced by:  nnaord  6105  nnmordi  6112  addclpi  6517  addnidpig  6526  archnqq  6607  prarloclemarch2  6609  prarloclemlt  6683