Theorem nnaordi 6771

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 6083	. . . . . . . . 9 ⊢ (𝑥 = 𝐶 → (𝐴 +o 𝑥) = (𝐴 +o 𝐶))
2		oveq2 6083	. . . . . . . . 9 ⊢ (𝑥 = 𝐶 → (𝐵 +o 𝑥) = (𝐵 +o 𝐶))
3	1, 2	eleq12d 2309	. . . . . . . 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 6083	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝐴 +o 𝑥) = (𝐴 +o ∅))
6		oveq2 6083	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
7	5, 6	eleq12d 2309	. . . . . . . 8 ⊢ (𝑥 = ∅ → ((𝐴 +o 𝑥) ∈ (𝐵 +o 𝑥) ↔ (𝐴 +o ∅) ∈ (𝐵 +o ∅)))
8		oveq2 6083	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
9		oveq2 6083	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
10	8, 9	eleq12d 2309	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝐴 +o 𝑥) ∈ (𝐵 +o 𝑥) ↔ (𝐴 +o 𝑦) ∈ (𝐵 +o 𝑦)))
11		oveq2 6083	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o suc 𝑦))
12		oveq2 6083	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
13	11, 12	eleq12d 2309	. . . . . . . 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 4748	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω)
16	15	ancoms 268	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ ω)
17		nna0 6737	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴)
18	16, 17	syl 14	. . . . . . . . 9 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐴 +o ∅) = 𝐴)
19		nna0 6737	. . . . . . . . . 10 ⊢ (𝐵 ∈ ω → (𝐵 +o ∅) = 𝐵)
20	19	adantr 276	. . . . . . . . 9 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐵 +o ∅) = 𝐵)
21	14, 18, 20	3eltr4d 2322	. . . . . . . 8 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐴 +o ∅) ∈ (𝐵 +o ∅))
22		simprl 535	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝐵 ∈ ω)
23		simpl 109	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝑦 ∈ ω)
24		nnacl 6743	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o 𝑦) ∈ ω)
25	22, 23, 24	syl2anc 415	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → (𝐵 +o 𝑦) ∈ ω)
26		nnsucelsuc 6754	. . . . . . . . . . . 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 4752	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
30	28, 29	syl 14	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝐴 ∈ On)
31		nnon 4752	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On)
32	31	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝑦 ∈ On)
33		oasuc 6727	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
34	30, 32, 33	syl2anc 415	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
35		nnon 4752	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
36	35	ad2antrl 494	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → 𝐵 ∈ On)
37		oasuc 6727	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
38	36, 32, 37	syl2anc 415	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
39	34, 38	eleq12d 2309	. . . . . . . . . . 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 4743	. . . . . . 7 ⊢ (𝑥 ∈ ω → ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐴 +o 𝑥) ∈ (𝐵 +o 𝑥)))
44	4, 43	vtoclga 2889	. . . . . 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 6747	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 +o 𝐶) = (𝐶 +o 𝐴))
49	46, 47, 48	syl2anc 415	. . . . 5 ⊢ ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → (𝐴 +o 𝐶) = (𝐶 +o 𝐴))
50		nnacom 6747	. . . . . . 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 483	. . . . 5 ⊢ ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → (𝐵 +o 𝐶) = (𝐶 +o 𝐵))
53	45, 49, 52	3eltr3d 2321	. . . 4 ⊢ ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵)) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
54	53	3impb 1230	. . 3 ⊢ ((𝐶 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
55	54	3com12 1238	. 2 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω ∧ 𝐴 ∈ 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
56	55	3expia 1236	1 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ∈ 𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∅c0 3520  Oncon0 4503  suc csuc 4505  ωcom 4732  (class class class)co 6075   +_o coa 6674

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-oadd 6681

This theorem is referenced by:  nnaord  6772  nnmordi  6779  addclpi  7684  addnidpig  7693  archnqq  7774  prarloclemarch2  7776  prarloclemlt  7850