Theorem nnawordi 8603

Description: Adding to both sides of an inequality in ω. (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 12-May-2012.)

Assertion

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

Proof of Theorem nnawordi

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

Step	Hyp	Ref	Expression
1		oveq2 7418	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐴 +o 𝑥) = (𝐴 +o ∅))
2		oveq2 7418	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
3	1, 2	sseq12d 3970	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥) ↔ (𝐴 +o ∅) ⊆ (𝐵 +o ∅)))
4	3	imbi2d 343	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐴 ⊆ 𝐵 → (𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥)) ↔ (𝐴 ⊆ 𝐵 → (𝐴 +o ∅) ⊆ (𝐵 +o ∅))))
5	4	imbi2d 343	. . . 4 ⊢ (𝑥 = ∅ → (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥))) ↔ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +o ∅) ⊆ (𝐵 +o ∅)))))
6		oveq2 7418	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
7		oveq2 7418	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
8	6, 7	sseq12d 3970	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥) ↔ (𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦)))
9	8	imbi2d 343	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ⊆ 𝐵 → (𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥)) ↔ (𝐴 ⊆ 𝐵 → (𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦))))
10	9	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝑦 → (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥))) ↔ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦)))))
11		oveq2 7418	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o suc 𝑦))
12		oveq2 7418	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
13	11, 12	sseq12d 3970	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥) ↔ (𝐴 +o suc 𝑦) ⊆ (𝐵 +o suc 𝑦)))
14	13	imbi2d 343	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ⊆ 𝐵 → (𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥)) ↔ (𝐴 ⊆ 𝐵 → (𝐴 +o suc 𝑦) ⊆ (𝐵 +o suc 𝑦))))
15	14	imbi2d 343	. . . 4 ⊢ (𝑥 = suc 𝑦 → (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥))) ↔ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +o suc 𝑦) ⊆ (𝐵 +o suc 𝑦)))))
16		oveq2 7418	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐴 +o 𝑥) = (𝐴 +o 𝐶))
17		oveq2 7418	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐵 +o 𝑥) = (𝐵 +o 𝐶))
18	16, 17	sseq12d 3970	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥) ↔ (𝐴 +o 𝐶) ⊆ (𝐵 +o 𝐶)))
19	18	imbi2d 343	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐴 ⊆ 𝐵 → (𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥)) ↔ (𝐴 ⊆ 𝐵 → (𝐴 +o 𝐶) ⊆ (𝐵 +o 𝐶))))
20	19	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝐶 → (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥))) ↔ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +o 𝐶) ⊆ (𝐵 +o 𝐶)))))
21		nnon 7864	. . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
22		nnon 7864	. . . . 5 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
23		oa0 8497	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
24	23	adantr 485	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o ∅) = 𝐴)
25		oa0 8497	. . . . . . . 8 ⊢ (𝐵 ∈ On → (𝐵 +o ∅) = 𝐵)
26	25	adantl 486	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o ∅) = 𝐵)
27	24, 26	sseq12d 3970	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o ∅) ⊆ (𝐵 +o ∅) ↔ 𝐴 ⊆ 𝐵))
28	27	biimprd 251	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 +o ∅) ⊆ (𝐵 +o ∅)))
29	21, 22, 28	syl2an 607	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +o ∅) ⊆ (𝐵 +o ∅)))
30		nnacl 8593	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o 𝑦) ∈ ω)
31	30	ancoms 463	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝐴 +o 𝑦) ∈ ω)
32	31	adantrr 729	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐴 +o 𝑦) ∈ ω)
33		nnon 7864	. . . . . . . . . . . 12 ⊢ ((𝐴 +o 𝑦) ∈ ω → (𝐴 +o 𝑦) ∈ On)
34		eloni 6370	. . . . . . . . . . . 12 ⊢ ((𝐴 +o 𝑦) ∈ On → Ord (𝐴 +o 𝑦))
35	32, 33, 34	3syl 19	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → Ord (𝐴 +o 𝑦))
36		nnacl 8593	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o 𝑦) ∈ ω)
37	36	ancoms 463	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 +o 𝑦) ∈ ω)
38	37	adantrl 728	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐵 +o 𝑦) ∈ ω)
39		nnon 7864	. . . . . . . . . . . 12 ⊢ ((𝐵 +o 𝑦) ∈ ω → (𝐵 +o 𝑦) ∈ On)
40		eloni 6370	. . . . . . . . . . . 12 ⊢ ((𝐵 +o 𝑦) ∈ On → Ord (𝐵 +o 𝑦))
41	38, 39, 40	3syl 19	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → Ord (𝐵 +o 𝑦))
42		ordsucsssuc 7815	. . . . . . . . . . 11 ⊢ ((Ord (𝐴 +o 𝑦) ∧ Ord (𝐵 +o 𝑦)) → ((𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦) ↔ suc (𝐴 +o 𝑦) ⊆ suc (𝐵 +o 𝑦)))
43	35, 41, 42	syl2anc 595	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦) ↔ suc (𝐴 +o 𝑦) ⊆ suc (𝐵 +o 𝑦)))
44	43	biimpa 481	. . . . . . . . 9 ⊢ (((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) ∧ (𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦)) → suc (𝐴 +o 𝑦) ⊆ suc (𝐵 +o 𝑦))
45		nnasuc 8588	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
46	45	ancoms 463	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
47	46	adantrr 729	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
48		nnasuc 8588	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
49	48	ancoms 463	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
50	49	adantrl 728	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
51	47, 50	sseq12d 3970	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝐴 +o suc 𝑦) ⊆ (𝐵 +o suc 𝑦) ↔ suc (𝐴 +o 𝑦) ⊆ suc (𝐵 +o 𝑦)))
52	51	adantr 485	. . . . . . . . 9 ⊢ (((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) ∧ (𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦)) → ((𝐴 +o suc 𝑦) ⊆ (𝐵 +o suc 𝑦) ↔ suc (𝐴 +o 𝑦) ⊆ suc (𝐵 +o 𝑦)))
53	44, 52	mpbird 260	. . . . . . . 8 ⊢ (((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) ∧ (𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦)) → (𝐴 +o suc 𝑦) ⊆ (𝐵 +o suc 𝑦))
54	53	ex 417	. . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦) → (𝐴 +o suc 𝑦) ⊆ (𝐵 +o suc 𝑦)))
55	54	imim2d 58	. . . . . 6 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝐴 ⊆ 𝐵 → (𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦)) → (𝐴 ⊆ 𝐵 → (𝐴 +o suc 𝑦) ⊆ (𝐵 +o suc 𝑦))))
56	55	ex 417	. . . . 5 ⊢ (𝑦 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ⊆ 𝐵 → (𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦)) → (𝐴 ⊆ 𝐵 → (𝐴 +o suc 𝑦) ⊆ (𝐵 +o suc 𝑦)))))
57	56	a2d 30	. . . 4 ⊢ (𝑦 ∈ ω → (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦))) → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +o suc 𝑦) ⊆ (𝐵 +o suc 𝑦)))))
58	5, 10, 15, 20, 29, 57	finds 7889	. . 3 ⊢ (𝐶 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +o 𝐶) ⊆ (𝐵 +o 𝐶))))
59	58	com12 33	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 ∈ ω → (𝐴 ⊆ 𝐵 → (𝐴 +o 𝐶) ⊆ (𝐵 +o 𝐶))))
60	59	3impia 1135	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +o 𝐶) ⊆ (𝐵 +o 𝐶)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ⊆ wss 3905  ∅c0 4286  Ord word 6359  Oncon0 6360  suc csuc 6362  (class class class)co 7410  ωcom 7858   +_o coa 8446

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-oadd 8453

This theorem is used by:  omopthlem2  8642