Theorem nnaordi 6081

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 5520	. . . . . . . . 9 ⊢ (𝑥 = 𝐶 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝐶))
2		oveq2 5520	. . . . . . . . 9 ⊢ (𝑥 = 𝐶 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝐶))
3	1, 2	eleq12d 2108	. . . . . . . 8 ⊢ (𝑥 = 𝐶 → ((𝐴 +𝑜 𝑥) ∈ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 𝐶) ∈ (𝐵 +𝑜 𝐶)))
4	3	imbi2d 219	. . . . . . 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 5520	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 ∅))
6		oveq2 5520	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 ∅))
7	5, 6	eleq12d 2108	. . . . . . . 8 ⊢ (𝑥 = ∅ → ((𝐴 +𝑜 𝑥) ∈ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 ∅) ∈ (𝐵 +𝑜 ∅)))
8		oveq2 5520	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑦))
9		oveq2 5520	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝑦))
10	8, 9	eleq12d 2108	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝐴 +𝑜 𝑥) ∈ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑦)))
11		oveq2 5520	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 suc 𝑦))
12		oveq2 5520	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 suc 𝑦))
13	11, 12	eleq12d 2108	. . . . . . . 8 ⊢ (𝑥 = suc 𝑦 → ((𝐴 +𝑜 𝑥) ∈ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 suc 𝑦) ∈ (𝐵 +𝑜 suc 𝑦)))
14		simpr 103	. . . . . . . . 9 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵)
15		elnn 4328	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω)
16	15	ancoms 255	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ ω)
17		nna0 6053	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴)
18	16, 17	syl 14	. . . . . . . . 9 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐴 +𝑜 ∅) = 𝐴)
19		nna0 6053	. . . . . . . . . 10 ⊢ (𝐵 ∈ ω → (𝐵 +𝑜 ∅) = 𝐵)
20	19	adantr 261	. . . . . . . . 9 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐵 +𝑜 ∅) = 𝐵)
21	14, 18, 20	3eltr4d 2121	. . . . . . . 8 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐴 +𝑜 ∅) ∈ (𝐵 +𝑜 ∅))
22		simprl 483	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝐵 ∈ ω)
23		simpl 102	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝑦 ∈ ω)
24		nnacl 6059	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 𝑦) ∈ ω)
25	22, 23, 24	syl2anc 391	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → (𝐵 +𝑜 𝑦) ∈ ω)
26		nnsucelsuc 6070	. . . . . . . . . . . 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 262	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝐴 ∈ ω)
29		nnon 4332	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
30	28, 29	syl 14	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝐴 ∈ On)
31		nnon 4332	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On)
32	31	adantr 261	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝑦 ∈ On)
33		oasuc 6044	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
34	30, 32, 33	syl2anc 391	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
35		nnon 4332	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
36	35	ad2antrl 459	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝐵 ∈ On)
37		oasuc 6044	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
38	36, 32, 37	syl2anc 391	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
39	34, 38	eleq12d 2108	. . . . . . . . . . 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 180	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → ((𝐴 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑦) ↔ (𝐴 +𝑜 suc 𝑦) ∈ (𝐵 +𝑜 suc 𝑦)))
41	40	biimpd 132	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → ((𝐴 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑦) → (𝐴 +𝑜 suc 𝑦) ∈ (𝐵 +𝑜 suc 𝑦)))
42	41	ex 108	. . . . . . . 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 4324	. . . . . . 7 ⊢ (𝑥 ∈ ω → ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐴 +𝑜 𝑥) ∈ (𝐵 +𝑜 𝑥)))
44	4, 43	vtoclga 2619	. . . . . 6 ⊢ (𝐶 ∈ ω → ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐴 +𝑜 𝐶) ∈ (𝐵 +𝑜 𝐶)))
45	44	imp 115	. . . . 5 ⊢ ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → (𝐴 +𝑜 𝐶) ∈ (𝐵 +𝑜 𝐶))
46	16	adantl 262	. . . . . 6 ⊢ ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝐴 ∈ ω)
47		simpl 102	. . . . . 6 ⊢ ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝐶 ∈ ω)
48		nnacom 6063	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 +𝑜 𝐶) = (𝐶 +𝑜 𝐴))
49	46, 47, 48	syl2anc 391	. . . . 5 ⊢ ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → (𝐴 +𝑜 𝐶) = (𝐶 +𝑜 𝐴))
50		nnacom 6063	. . . . . . 7 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐵 +𝑜 𝐶) = (𝐶 +𝑜 𝐵))
51	50	ancoms 255	. . . . . 6 ⊢ ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 +𝑜 𝐶) = (𝐶 +𝑜 𝐵))
52	51	adantrr 448	. . . . 5 ⊢ ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → (𝐵 +𝑜 𝐶) = (𝐶 +𝑜 𝐵))
53	45, 49, 52	3eltr3d 2120	. . . 4 ⊢ ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵))
54	53	3impb 1100	. . 3 ⊢ ((𝐶 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵))
55	54	3com12 1108	. 2 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵))
56	55	3expia 1106	1 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ∈ 𝐵 → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  ∅c0 3224  Oncon0 4100  suc csuc 4102  ωcom 4313  (class class class)co 5512   +_𝑜 coa 5998

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005

This theorem is referenced by:  nnaord  6082  nnmordi  6089  addclpi  6425  addnidpig  6434  archnqq  6515  prarloclemarch2  6517  prarloclemlt  6591