Theorem nnaordi 8449

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		elnn 7723	. . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω)
2	1	ancoms 459	. . . . 5 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ ω)
3	2	adantll 711	. . . 4 ⊢ (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ ω)
4		nnord 7720	. . . . . . . . 9 ⊢ (𝐵 ∈ ω → Ord 𝐵)
5		ordsucss 7665	. . . . . . . . 9 ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
6	4, 5	syl 17	. . . . . . . 8 ⊢ (𝐵 ∈ ω → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
7	6	ad2antlr 724	. . . . . . 7 ⊢ (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ ω) → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
8		peano2b 7729	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
9		oveq2 7283	. . . . . . . . . . . . . 14 ⊢ (𝑥 = suc 𝐴 → (𝐶 +o 𝑥) = (𝐶 +o suc 𝐴))
10	9	sseq2d 3953	. . . . . . . . . . . . 13 ⊢ (𝑥 = suc 𝐴 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝐴)))
11	10	imbi2d 341	. . . . . . . . . . . 12 ⊢ (𝑥 = suc 𝐴 → ((𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥)) ↔ (𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝐴))))
12		oveq2 7283	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝐶 +o 𝑥) = (𝐶 +o 𝑦))
13	12	sseq2d 3953	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦)))
14	13	imbi2d 341	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥)) ↔ (𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦))))
15		oveq2 7283	. . . . . . . . . . . . . 14 ⊢ (𝑥 = suc 𝑦 → (𝐶 +o 𝑥) = (𝐶 +o suc 𝑦))
16	15	sseq2d 3953	. . . . . . . . . . . . 13 ⊢ (𝑥 = suc 𝑦 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦)))
17	16	imbi2d 341	. . . . . . . . . . . 12 ⊢ (𝑥 = suc 𝑦 → ((𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥)) ↔ (𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦))))
18		oveq2 7283	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐵 → (𝐶 +o 𝑥) = (𝐶 +o 𝐵))
19	18	sseq2d 3953	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))
20	19	imbi2d 341	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → ((𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥)) ↔ (𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵))))
21		ssid 3943	. . . . . . . . . . . . 13 ⊢ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝐴)
22	21	2a1i 12	. . . . . . . . . . . 12 ⊢ (suc 𝐴 ∈ ω → (𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝐴)))
23		sssucid 6343	. . . . . . . . . . . . . . . . 17 ⊢ (𝐶 +o 𝑦) ⊆ suc (𝐶 +o 𝑦)
24		sstr2 3928	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → ((𝐶 +o 𝑦) ⊆ suc (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ suc (𝐶 +o 𝑦)))
25	23, 24	mpi 20	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ suc (𝐶 +o 𝑦))
26		nnasuc 8437	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → (𝐶 +o suc 𝑦) = suc (𝐶 +o 𝑦))
27	26	ancoms 459	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 +o suc 𝑦) = suc (𝐶 +o 𝑦))
28	27	sseq2d 3953	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦) ↔ (𝐶 +o suc 𝐴) ⊆ suc (𝐶 +o 𝑦)))
29	25, 28	syl5ibr 245	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦)))
30	29	ex 413	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ω → (𝐶 ∈ ω → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦))))
31	30	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴 ⊆ 𝑦) → (𝐶 ∈ ω → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦))))
32	31	a2d 29	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴 ⊆ 𝑦) → ((𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦)) → (𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦))))
33	11, 14, 17, 20, 22, 32	findsg 7746	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴 ⊆ 𝐵) → (𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))
34	33	exp31 420	. . . . . . . . . 10 ⊢ (𝐵 ∈ ω → (suc 𝐴 ∈ ω → (suc 𝐴 ⊆ 𝐵 → (𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))))
35	8, 34	syl5bi 241	. . . . . . . . 9 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ω → (suc 𝐴 ⊆ 𝐵 → (𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))))
36	35	com4r 94	. . . . . . . 8 ⊢ (𝐶 ∈ ω → (𝐵 ∈ ω → (𝐴 ∈ ω → (suc 𝐴 ⊆ 𝐵 → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))))
37	36	imp31 418	. . . . . . 7 ⊢ (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ ω) → (suc 𝐴 ⊆ 𝐵 → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))
38		nnasuc 8437	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → (𝐶 +o suc 𝐴) = suc (𝐶 +o 𝐴))
39	38	sseq1d 3952	. . . . . . . . 9 ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵) ↔ suc (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)))
40		ovex 7308	. . . . . . . . . 10 ⊢ (𝐶 +o 𝐴) ∈ V
41		sucssel 6358	. . . . . . . . . 10 ⊢ ((𝐶 +o 𝐴) ∈ V → (suc (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
42	40, 41	ax-mp 5	. . . . . . . . 9 ⊢ (suc (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
43	39, 42	syl6bi 252	. . . . . . . 8 ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
44	43	adantlr 712	. . . . . . 7 ⊢ (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ ω) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
45	7, 37, 44	3syld 60	. . . . . 6 ⊢ (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ ω) → (𝐴 ∈ 𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
46	45	imp 407	. . . . 5 ⊢ ((((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ ω) ∧ 𝐴 ∈ 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
47	46	an32s 649	. . . 4 ⊢ ((((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ 𝐵) ∧ 𝐴 ∈ ω) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
48	3, 47	mpdan 684	. . 3 ⊢ (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
49	48	ex 413	. 2 ⊢ ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
50	49	ancoms 459	1 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ∈ 𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ⊆ wss 3887  Ord word 6265  suc csuc 6268  (class class class)co 7275  ωcom 7712   +_o coa 8294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-oadd 8301

This theorem is referenced by:  nnaord  8450  nnmordi  8462  addclpi  10648  addnidpi  10657