Theorem nnaordi 8615

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 7863	. . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω)
2	1	ancoms 460	. . . . 5 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ ω)
3	2	adantll 713	. . . 4 ⊢ (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ ω)
4		nnord 7860	. . . . . . . . 9 ⊢ (𝐵 ∈ ω → Ord 𝐵)
5		ordsucss 7803	. . . . . . . . 9 ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
6	4, 5	syl 17	. . . . . . . 8 ⊢ (𝐵 ∈ ω → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
7	6	ad2antlr 726	. . . . . . 7 ⊢ (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ ω) → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
8		peano2b 7869	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
9		oveq2 7414	. . . . . . . . . . . . . 14 ⊢ (𝑥 = suc 𝐴 → (𝐶 +o 𝑥) = (𝐶 +o suc 𝐴))
10	9	sseq2d 4014	. . . . . . . . . . . . 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 7414	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝐶 +o 𝑥) = (𝐶 +o 𝑦))
13	12	sseq2d 4014	. . . . . . . . . . . . 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 7414	. . . . . . . . . . . . . 14 ⊢ (𝑥 = suc 𝑦 → (𝐶 +o 𝑥) = (𝐶 +o suc 𝑦))
16	15	sseq2d 4014	. . . . . . . . . . . . 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 7414	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐵 → (𝐶 +o 𝑥) = (𝐶 +o 𝐵))
19	18	sseq2d 4014	. . . . . . . . . . . . 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 4004	. . . . . . . . . . . . 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 6442	. . . . . . . . . . . . . . . . 17 ⊢ (𝐶 +o 𝑦) ⊆ suc (𝐶 +o 𝑦)
24		sstr2 3989	. . . . . . . . . . . . . . . . 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 8603	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ ω ∧ 𝑦 ∈ ω) → (𝐶 +o suc 𝑦) = suc (𝐶 +o 𝑦))
27	26	ancoms 460	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 +o suc 𝑦) = suc (𝐶 +o 𝑦))
28	27	sseq2d 4014	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦) ↔ (𝐶 +o suc 𝐴) ⊆ suc (𝐶 +o 𝑦)))
29	25, 28	imbitrrid 245	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦)))
30	29	ex 414	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ω → (𝐶 ∈ ω → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦))))
31	30	ad2antrr 725	. . . . . . . . . . . . 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 7887	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ ω ∧ suc 𝐴 ∈ ω) ∧ suc 𝐴 ⊆ 𝐵) → (𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))
34	33	exp31 421	. . . . . . . . . 10 ⊢ (𝐵 ∈ ω → (suc 𝐴 ∈ ω → (suc 𝐴 ⊆ 𝐵 → (𝐶 ∈ ω → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))))
35	8, 34	biimtrid 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 419	. . . . . . 7 ⊢ (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ ω) → (suc 𝐴 ⊆ 𝐵 → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))
38		nnasuc 8603	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → (𝐶 +o suc 𝐴) = suc (𝐶 +o 𝐴))
39	38	sseq1d 4013	. . . . . . . . 9 ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵) ↔ suc (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)))
40		ovex 7439	. . . . . . . . . 10 ⊢ (𝐶 +o 𝐴) ∈ V
41		sucssel 6457	. . . . . . . . . 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 253	. . . . . . . 8 ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
44	43	adantlr 714	. . . . . . 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 408	. . . . 5 ⊢ ((((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ ω) ∧ 𝐴 ∈ 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
47	46	an32s 651	. . . 4 ⊢ ((((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ 𝐵) ∧ 𝐴 ∈ ω) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
48	3, 47	mpdan 686	. . 3 ⊢ (((𝐶 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ∈ 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
49	48	ex 414	. 2 ⊢ ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
50	49	ancoms 460	1 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ∈ 𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475   ⊆ wss 3948  Ord word 6361  suc csuc 6364  (class class class)co 7406  ωcom 7852   +_o coa 8460

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-oadd 8467

This theorem is referenced by:  nnaord  8616  nnmordi  8628  addclpi  10884  addnidpi  10893