Theorem nnaword 6540

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 5907	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑥 +o 𝐴) = (𝐶 +o 𝐴))
2		oveq1 5907	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑥 +o 𝐵) = (𝐶 +o 𝐵))
3	1, 2	sseq12d 3201	. . . . . 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 5907	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 +o 𝐴) = (∅ +o 𝐴))
7		oveq1 5907	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 +o 𝐵) = (∅ +o 𝐵))
8	6, 7	sseq12d 3201	. . . . . 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 5907	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 +o 𝐴) = (𝑦 +o 𝐴))
11		oveq1 5907	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 +o 𝐵) = (𝑦 +o 𝐵))
12	10, 11	sseq12d 3201	. . . . . 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 5907	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝑥 +o 𝐴) = (suc 𝑦 +o 𝐴))
15		oveq1 5907	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝑥 +o 𝐵) = (suc 𝑦 +o 𝐵))
16	14, 15	sseq12d 3201	. . . . . 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 6507	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (∅ +o 𝐴) = 𝐴)
19	18	eqcomd 2195	. . . . . . 7 ⊢ (𝐴 ∈ ω → 𝐴 = (∅ +o 𝐴))
20	19	adantr 276	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 = (∅ +o 𝐴))
21		nna0r 6507	. . . . . . . 8 ⊢ (𝐵 ∈ ω → (∅ +o 𝐵) = 𝐵)
22	21	eqcomd 2195	. . . . . . 7 ⊢ (𝐵 ∈ ω → 𝐵 = (∅ +o 𝐵))
23	22	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐵 = (∅ +o 𝐵))
24	20, 23	sseq12d 3201	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (∅ +o 𝐴) ⊆ (∅ +o 𝐵)))
25		nnacl 6509	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝑦 +o 𝐴) ∈ ω)
26	25	3adant3 1019	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑦 +o 𝐴) ∈ ω)
27		nnacl 6509	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝑦 +o 𝐵) ∈ ω)
28	27	3adant2 1018	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑦 +o 𝐵) ∈ ω)
29		nnsucsssuc 6521	. . . . . . . . . 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 6505	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
32		peano2 4615	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
33		nnacom 6513	. . . . . . . . . . . . . 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 6513	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o 𝑦) = (𝑦 +o 𝐴))
36		suceq 4423	. . . . . . . . . . . . . 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 2231	. . . . . . . . . . . 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 1019	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → suc (𝑦 +o 𝐴) = (suc 𝑦 +o 𝐴))
41		nnasuc 6505	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
42		nnacom 6513	. . . . . . . . . . . . . 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 6513	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o 𝑦) = (𝑦 +o 𝐵))
45		suceq 4423	. . . . . . . . . . . . . 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 2231	. . . . . . . . . . . 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 1018	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → suc (𝑦 +o 𝐵) = (suc 𝑦 +o 𝐵))
50	40, 49	sseq12d 3201	. . . . . . . . 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 1208	. . . . 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 4621	. . . 4 ⊢ (𝑥 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵))))
56	5, 55	vtoclga 2818	. . 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 1196	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2160   ⊆ wss 3144  ∅c0 3437  suc csuc 4386  ωcom 4610  (class class class)co 5900   +_o coa 6442

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4136  ax-sep 4139  ax-nul 4147  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-iinf 4608

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-tr 4120  df-id 4314  df-iord 4387  df-on 4389  df-suc 4392  df-iom 4611  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-ov 5903  df-oprab 5904  df-mpo 5905  df-1st 6169  df-2nd 6170  df-recs 6334  df-irdg 6399  df-oadd 6449

This theorem is referenced by:  nnacan  6541  nnawordi  6544