Theorem nnaordi 6532

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 5903	. . . . . . . . 9 ⊢ (𝑥 = 𝐶 → (𝐴 +o 𝑥) = (𝐴 +o 𝐶))
2		oveq2 5903	. . . . . . . . 9 ⊢ (𝑥 = 𝐶 → (𝐵 +o 𝑥) = (𝐵 +o 𝐶))
3	1, 2	eleq12d 2260	. . . . . . . 8 ⊢ (𝑥 = 𝐶 → ((𝐴 +o 𝑥) ∈ (𝐵 +o 𝑥) ↔ (𝐴 +o 𝐶) ∈ (𝐵 +o 𝐶)))
4	3	imbi2d 230	. . . . . . 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 5903	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝐴 +o 𝑥) = (𝐴 +o ∅))
6		oveq2 5903	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
7	5, 6	eleq12d 2260	. . . . . . . 8 ⊢ (𝑥 = ∅ → ((𝐴 +o 𝑥) ∈ (𝐵 +o 𝑥) ↔ (𝐴 +o ∅) ∈ (𝐵 +o ∅)))
8		oveq2 5903	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
9		oveq2 5903	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
10	8, 9	eleq12d 2260	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝐴 +o 𝑥) ∈ (𝐵 +o 𝑥) ↔ (𝐴 +o 𝑦) ∈ (𝐵 +o 𝑦)))
11		oveq2 5903	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o suc 𝑦))
12		oveq2 5903	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
13	11, 12	eleq12d 2260	. . . . . . . 8 ⊢ (𝑥 = suc 𝑦 → ((𝐴 +o 𝑥) ∈ (𝐵 +o 𝑥) ↔ (𝐴 +o suc 𝑦) ∈ (𝐵 +o suc 𝑦)))
14		simpr 110	. . . . . . . . 9 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵)
15		elnn 4623	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω)
16	15	ancoms 268	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ ω)
17		nna0 6498	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴)
18	16, 17	syl 14	. . . . . . . . 9 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐴 +o ∅) = 𝐴)
19		nna0 6498	. . . . . . . . . 10 ⊢ (𝐵 ∈ ω → (𝐵 +o ∅) = 𝐵)
20	19	adantr 276	. . . . . . . . 9 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐵 +o ∅) = 𝐵)
21	14, 18, 20	3eltr4d 2273	. . . . . . . 8 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐴 +o ∅) ∈ (𝐵 +o ∅))
22		simprl 529	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝐵 ∈ ω)
23		simpl 109	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝑦 ∈ ω)
24		nnacl 6504	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o 𝑦) ∈ ω)
25	22, 23, 24	syl2anc 411	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → (𝐵 +o 𝑦) ∈ ω)
26		nnsucelsuc 6515	. . . . . . . . . . . 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 277	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝐴 ∈ ω)
29		nnon 4627	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
30	28, 29	syl 14	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝐴 ∈ On)
31		nnon 4627	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On)
32	31	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝑦 ∈ On)
33		oasuc 6488	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
34	30, 32, 33	syl2anc 411	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
35		nnon 4627	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
36	35	ad2antrl 490	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝐵 ∈ On)
37		oasuc 6488	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
38	36, 32, 37	syl2anc 411	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
39	34, 38	eleq12d 2260	. . . . . . . . . . 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 191	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → ((𝐴 +o 𝑦) ∈ (𝐵 +o 𝑦) ↔ (𝐴 +o suc 𝑦) ∈ (𝐵 +o suc 𝑦)))
41	40	biimpd 144	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → ((𝐴 +o 𝑦) ∈ (𝐵 +o 𝑦) → (𝐴 +o suc 𝑦) ∈ (𝐵 +o suc 𝑦)))
42	41	ex 115	. . . . . . . 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 4618	. . . . . . 7 ⊢ (𝑥 ∈ ω → ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐴 +o 𝑥) ∈ (𝐵 +o 𝑥)))
44	4, 43	vtoclga 2818	. . . . . 6 ⊢ (𝐶 ∈ ω → ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐴 +o 𝐶) ∈ (𝐵 +o 𝐶)))
45	44	imp 124	. . . . 5 ⊢ ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → (𝐴 +o 𝐶) ∈ (𝐵 +o 𝐶))
46	16	adantl 277	. . . . . 6 ⊢ ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝐴 ∈ ω)
47		simpl 109	. . . . . 6 ⊢ ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝐶 ∈ ω)
48		nnacom 6508	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 +o 𝐶) = (𝐶 +o 𝐴))
49	46, 47, 48	syl2anc 411	. . . . 5 ⊢ ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → (𝐴 +o 𝐶) = (𝐶 +o 𝐴))
50		nnacom 6508	. . . . . . 7 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐵 +o 𝐶) = (𝐶 +o 𝐵))
51	50	ancoms 268	. . . . . 6 ⊢ ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 +o 𝐶) = (𝐶 +o 𝐵))
52	51	adantrr 479	. . . . 5 ⊢ ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → (𝐵 +o 𝐶) = (𝐶 +o 𝐵))
53	45, 49, 52	3eltr3d 2272	. . . 4 ⊢ ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
54	53	3impb 1201	. . 3 ⊢ ((𝐶 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
55	54	3com12 1209	. 2 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
56	55	3expia 1207	1 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ∈ 𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2160  ∅c0 3437  Oncon0 4381  suc csuc 4383  ωcom 4607  (class class class)co 5895   +_o coa 6437

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-recs 6329  df-irdg 6394  df-oadd 6444

This theorem is referenced by:  nnaord  6533  nnmordi  6540  addclpi  7355  addnidpig  7364  archnqq  7445  prarloclemarch2  7447  prarloclemlt  7521