Theorem nnaordi 6566

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 5930	. . . . . . . . 9 ⊢ (𝑥 = 𝐶 → (𝐴 +o 𝑥) = (𝐴 +o 𝐶))
2		oveq2 5930	. . . . . . . . 9 ⊢ (𝑥 = 𝐶 → (𝐵 +o 𝑥) = (𝐵 +o 𝐶))
3	1, 2	eleq12d 2267	. . . . . . . 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 5930	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝐴 +o 𝑥) = (𝐴 +o ∅))
6		oveq2 5930	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
7	5, 6	eleq12d 2267	. . . . . . . 8 ⊢ (𝑥 = ∅ → ((𝐴 +o 𝑥) ∈ (𝐵 +o 𝑥) ↔ (𝐴 +o ∅) ∈ (𝐵 +o ∅)))
8		oveq2 5930	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
9		oveq2 5930	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
10	8, 9	eleq12d 2267	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝐴 +o 𝑥) ∈ (𝐵 +o 𝑥) ↔ (𝐴 +o 𝑦) ∈ (𝐵 +o 𝑦)))
11		oveq2 5930	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o suc 𝑦))
12		oveq2 5930	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
13	11, 12	eleq12d 2267	. . . . . . . 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 4642	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω)
16	15	ancoms 268	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ ω)
17		nna0 6532	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴)
18	16, 17	syl 14	. . . . . . . . 9 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐴 +o ∅) = 𝐴)
19		nna0 6532	. . . . . . . . . 10 ⊢ (𝐵 ∈ ω → (𝐵 +o ∅) = 𝐵)
20	19	adantr 276	. . . . . . . . 9 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐵 +o ∅) = 𝐵)
21	14, 18, 20	3eltr4d 2280	. . . . . . . 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 6538	. . . . . . . . . . . . 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 6549	. . . . . . . . . . . 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 4646	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
30	28, 29	syl 14	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝐴 ∈ On)
31		nnon 4646	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On)
32	31	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝑦 ∈ On)
33		oasuc 6522	. . . . . . . . . . . . 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 4646	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
36	35	ad2antrl 490	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝐵 ∈ On)
37		oasuc 6522	. . . . . . . . . . . . 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 2267	. . . . . . . . . . 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 4637	. . . . . . 7 ⊢ (𝑥 ∈ ω → ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐴 +o 𝑥) ∈ (𝐵 +o 𝑥)))
44	4, 43	vtoclga 2830	. . . . . 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 6542	. . . . . 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 6542	. . . . . . 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 2279	. . . 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 2167  ∅c0 3450  Oncon0 4398  suc csuc 4400  ωcom 4626  (class class class)co 5922   +_o coa 6471

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-oadd 6478

This theorem is referenced by:  nnaord  6567  nnmordi  6574  addclpi  7394  addnidpig  7403  archnqq  7484  prarloclemarch2  7486  prarloclemlt  7560