Theorem nnaword 6743

Description: Weak ordering property of addition. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Assertion

Ref	Expression
nnaword	⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)))

Proof of Theorem nnaword

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 6056	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑥 +o 𝐴) = (𝐶 +o 𝐴))
2		oveq1 6056	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑥 +o 𝐵) = (𝐶 +o 𝐵))
3	1, 2	sseq12d 3268	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵) ↔ (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)))
4	3	bibi2d 232	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐴 ⊆ 𝐵 ↔ (𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵)) ↔ (𝐴 ⊆ 𝐵 ↔ (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵))))
5	4	imbi2d 230	. . . 4 ⊢ (𝑥 = 𝐶 → (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵))) ↔ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)))))
6		oveq1 6056	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 +o 𝐴) = (∅ +o 𝐴))
7		oveq1 6056	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 +o 𝐵) = (∅ +o 𝐵))
8	6, 7	sseq12d 3268	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵) ↔ (∅ +o 𝐴) ⊆ (∅ +o 𝐵)))
9	8	bibi2d 232	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐴 ⊆ 𝐵 ↔ (𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵)) ↔ (𝐴 ⊆ 𝐵 ↔ (∅ +o 𝐴) ⊆ (∅ +o 𝐵))))
10		oveq1 6056	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 +o 𝐴) = (𝑦 +o 𝐴))
11		oveq1 6056	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 +o 𝐵) = (𝑦 +o 𝐵))
12	10, 11	sseq12d 3268	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵) ↔ (𝑦 +o 𝐴) ⊆ (𝑦 +o 𝐵)))
13	12	bibi2d 232	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ⊆ 𝐵 ↔ (𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵)) ↔ (𝐴 ⊆ 𝐵 ↔ (𝑦 +o 𝐴) ⊆ (𝑦 +o 𝐵))))
14		oveq1 6056	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝑥 +o 𝐴) = (suc 𝑦 +o 𝐴))
15		oveq1 6056	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝑥 +o 𝐵) = (suc 𝑦 +o 𝐵))
16	14, 15	sseq12d 3268	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵) ↔ (suc 𝑦 +o 𝐴) ⊆ (suc 𝑦 +o 𝐵)))
17	16	bibi2d 232	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ⊆ 𝐵 ↔ (𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵)) ↔ (𝐴 ⊆ 𝐵 ↔ (suc 𝑦 +o 𝐴) ⊆ (suc 𝑦 +o 𝐵))))
18		nna0r 6710	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (∅ +o 𝐴) = 𝐴)
19	18	eqcomd 2238	. . . . . . 7 ⊢ (𝐴 ∈ ω → 𝐴 = (∅ +o 𝐴))
20	19	adantr 276	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 = (∅ +o 𝐴))
21		nna0r 6710	. . . . . . . 8 ⊢ (𝐵 ∈ ω → (∅ +o 𝐵) = 𝐵)
22	21	eqcomd 2238	. . . . . . 7 ⊢ (𝐵 ∈ ω → 𝐵 = (∅ +o 𝐵))
23	22	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐵 = (∅ +o 𝐵))
24	20, 23	sseq12d 3268	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (∅ +o 𝐴) ⊆ (∅ +o 𝐵)))
25		nnacl 6712	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝑦 +o 𝐴) ∈ ω)
26	25	3adant3 1044	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑦 +o 𝐴) ∈ ω)
27		nnacl 6712	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝑦 +o 𝐵) ∈ ω)
28	27	3adant2 1043	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑦 +o 𝐵) ∈ ω)
29		nnsucsssuc 6724	. . . . . . . . . 10 ⊢ (((𝑦 +o 𝐴) ∈ ω ∧ (𝑦 +o 𝐵) ∈ ω) → ((𝑦 +o 𝐴) ⊆ (𝑦 +o 𝐵) ↔ suc (𝑦 +o 𝐴) ⊆ suc (𝑦 +o 𝐵)))
30	26, 28, 29	syl2anc 411	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑦 +o 𝐴) ⊆ (𝑦 +o 𝐵) ↔ suc (𝑦 +o 𝐴) ⊆ suc (𝑦 +o 𝐵)))
31		nnasuc 6708	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
32		peano2 4716	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
33		nnacom 6716	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ω ∧ suc 𝑦 ∈ ω) → (𝐴 +o suc 𝑦) = (suc 𝑦 +o 𝐴))
34	32, 33	sylan2 286	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o suc 𝑦) = (suc 𝑦 +o 𝐴))
35		nnacom 6716	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o 𝑦) = (𝑦 +o 𝐴))
36		suceq 4522	. . . . . . . . . . . . . 14 ⊢ ((𝐴 +o 𝑦) = (𝑦 +o 𝐴) → suc (𝐴 +o 𝑦) = suc (𝑦 +o 𝐴))
37	35, 36	syl 14	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → suc (𝐴 +o 𝑦) = suc (𝑦 +o 𝐴))
38	31, 34, 37	3eqtr3rd 2274	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → suc (𝑦 +o 𝐴) = (suc 𝑦 +o 𝐴))
39	38	ancoms 268	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → suc (𝑦 +o 𝐴) = (suc 𝑦 +o 𝐴))
40	39	3adant3 1044	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → suc (𝑦 +o 𝐴) = (suc 𝑦 +o 𝐴))
41		nnasuc 6708	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
42		nnacom 6716	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ω ∧ suc 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = (suc 𝑦 +o 𝐵))
43	32, 42	sylan2 286	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = (suc 𝑦 +o 𝐵))
44		nnacom 6716	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o 𝑦) = (𝑦 +o 𝐵))
45		suceq 4522	. . . . . . . . . . . . . 14 ⊢ ((𝐵 +o 𝑦) = (𝑦 +o 𝐵) → suc (𝐵 +o 𝑦) = suc (𝑦 +o 𝐵))
46	44, 45	syl 14	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → suc (𝐵 +o 𝑦) = suc (𝑦 +o 𝐵))
47	41, 43, 46	3eqtr3rd 2274	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → suc (𝑦 +o 𝐵) = (suc 𝑦 +o 𝐵))
48	47	ancoms 268	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → suc (𝑦 +o 𝐵) = (suc 𝑦 +o 𝐵))
49	48	3adant2 1043	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → suc (𝑦 +o 𝐵) = (suc 𝑦 +o 𝐵))
50	40, 49	sseq12d 3268	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc (𝑦 +o 𝐴) ⊆ suc (𝑦 +o 𝐵) ↔ (suc 𝑦 +o 𝐴) ⊆ (suc 𝑦 +o 𝐵)))
51	30, 50	bitrd 188	. . . . . . . 8 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑦 +o 𝐴) ⊆ (𝑦 +o 𝐵) ↔ (suc 𝑦 +o 𝐴) ⊆ (suc 𝑦 +o 𝐵)))
52	51	bibi2d 232	. . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ⊆ 𝐵 ↔ (𝑦 +o 𝐴) ⊆ (𝑦 +o 𝐵)) ↔ (𝐴 ⊆ 𝐵 ↔ (suc 𝑦 +o 𝐴) ⊆ (suc 𝑦 +o 𝐵))))
53	52	biimpd 144	. . . . . 6 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ⊆ 𝐵 ↔ (𝑦 +o 𝐴) ⊆ (𝑦 +o 𝐵)) → (𝐴 ⊆ 𝐵 ↔ (suc 𝑦 +o 𝐴) ⊆ (suc 𝑦 +o 𝐵))))
54	53	3expib 1233	. . . . 5 ⊢ (𝑦 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ⊆ 𝐵 ↔ (𝑦 +o 𝐴) ⊆ (𝑦 +o 𝐵)) → (𝐴 ⊆ 𝐵 ↔ (suc 𝑦 +o 𝐴) ⊆ (suc 𝑦 +o 𝐵)))))
55	9, 13, 17, 24, 54	finds2 4722	. . . 4 ⊢ (𝑥 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵))))
56	5, 55	vtoclga 2880	. . 3 ⊢ (𝐶 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵))))
57	56	impcom 125	. 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐶 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)))
58	57	3impa 1221	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203   ⊆ wss 3210  ∅c0 3507  suc csuc 4485  ωcom 4711  (class class class)co 6049   +_o coa 6643

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-oadd 6650

This theorem is referenced by:  nnacan  6744  nnawordi  6747