Theorem nnaordi 7984

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 7355	. . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω)
2	1	ancoms 452	. . . . 5 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ ω)
3	2	adantll 704	. . . 4 ⊢ (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ ω)
4		nnord 7353	. . . . . . . . 9 ⊢ (𝐵 ∈ ω → Ord 𝐵)
5		ordsucss 7298	. . . . . . . . 9 ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
6	4, 5	syl 17	. . . . . . . 8 ⊢ (𝐵 ∈ ω → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
7	6	ad2antlr 717	. . . . . . 7 ⊢ (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ ω) → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
8		peano2b 7361	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
9		oveq2 6932	. . . . . . . . . . . . . 14 ⊢ (𝑥 = suc 𝐴 → (𝐶 +o 𝑥) = (𝐶 +o suc 𝐴))
10	9	sseq2d 3852	. . . . . . . . . . . . 13 ⊢ (𝑥 = suc 𝐴 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝐴)))
11	10	imbi2d 332	. . . . . . . . . . . 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 6932	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝐶 +o 𝑥) = (𝐶 +o 𝑦))
13	12	sseq2d 3852	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦)))
14	13	imbi2d 332	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥)) ↔ (𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦))))
15		oveq2 6932	. . . . . . . . . . . . . 14 ⊢ (𝑥 = suc 𝑦 → (𝐶 +o 𝑥) = (𝐶 +o suc 𝑦))
16	15	sseq2d 3852	. . . . . . . . . . . . 13 ⊢ (𝑥 = suc 𝑦 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦)))
17	16	imbi2d 332	. . . . . . . . . . . 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 6932	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐵 → (𝐶 +o 𝑥) = (𝐶 +o 𝐵))
19	18	sseq2d 3852	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))
20	19	imbi2d 332	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → ((𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥)) ↔ (𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵))))
21		ssid 3842	. . . . . . . . . . . . 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 6055	. . . . . . . . . . . . . . . . 17 ⊢ (𝐶 +o 𝑦) ⊆ suc (𝐶 +o 𝑦)
24		sstr2 3828	. . . . . . . . . . . . . . . . 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 7972	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → (𝐶 +o suc 𝑦) = suc (𝐶 +o 𝑦))
27	26	ancoms 452	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 +o suc 𝑦) = suc (𝐶 +o 𝑦))
28	27	sseq2d 3852	. . . . . . . . . . . . . . . 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 238	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦)))
30	29	ex 403	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ω → (𝐶 ∈ ω → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦))))
31	30	ad2antrr 716	. . . . . . . . . . . . 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 7373	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴 ⊆ 𝐵) → (𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))
34	33	exp31 412	. . . . . . . . . 10 ⊢ (𝐵 ∈ ω → (suc 𝐴 ∈ ω → (suc 𝐴 ⊆ 𝐵 → (𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))))
35	8, 34	syl5bi 234	. . . . . . . . 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 410	. . . . . . 7 ⊢ (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ ω) → (suc 𝐴 ⊆ 𝐵 → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))
38		nnasuc 7972	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → (𝐶 +o suc 𝐴) = suc (𝐶 +o 𝐴))
39	38	sseq1d 3851	. . . . . . . . 9 ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵) ↔ suc (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)))
40		ovex 6956	. . . . . . . . . 10 ⊢ (𝐶 +o 𝐴) ∈ V
41		sucssel 6070	. . . . . . . . . 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 245	. . . . . . . 8 ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
44	43	adantlr 705	. . . . . . 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 397	. . . . 5 ⊢ ((((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ ω) ∧ 𝐴 ∈ 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
47	46	an32s 642	. . . 4 ⊢ ((((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ 𝐵) ∧ 𝐴 ∈ ω) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
48	3, 47	mpdan 677	. . 3 ⊢ (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
49	48	ex 403	. 2 ⊢ ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
50	49	ancoms 452	1 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ∈ 𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107  Vcvv 3398   ⊆ wss 3792  Ord word 5977  suc csuc 5980  (class class class)co 6924  ωcom 7345   +_o coa 7842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-oadd 7849

This theorem is referenced by:  nnaord  7985  nnmordi  7997  addclpi  10051  addnidpi  10060