Theorem omopthlem1 7622

Description: Lemma for omopthi 7624. (Contributed by Scott Fenton, 18-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)

Hypotheses

Ref	Expression
omopthlem1.1	⊢ 𝐴 ∈ ω
omopthlem1.2	⊢ 𝐶 ∈ ω

Assertion

Ref	Expression
omopthlem1	⊢ (𝐴 ∈ 𝐶 → ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) ∈ (𝐶 ·𝑜 𝐶))

Proof of Theorem omopthlem1

Step	Hyp	Ref	Expression
1		omopthlem1.1	. . . . 5 ⊢ 𝐴 ∈ ω
2		peano2 6978	. . . . 5 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
3	1, 2	ax-mp 5	. . . 4 ⊢ suc 𝐴 ∈ ω
4		omopthlem1.2	. . . 4 ⊢ 𝐶 ∈ ω
5		nnmwordi 7602	. . . 4 ⊢ ((suc 𝐴 ∈ ω ∧ 𝐶 ∈ ω ∧ suc 𝐴 ∈ ω) → (suc 𝐴 ⊆ 𝐶 → (suc 𝐴 ·𝑜 suc 𝐴) ⊆ (suc 𝐴 ·𝑜 𝐶)))
6	3, 4, 3, 5	mp3an 1416	. . 3 ⊢ (suc 𝐴 ⊆ 𝐶 → (suc 𝐴 ·𝑜 suc 𝐴) ⊆ (suc 𝐴 ·𝑜 𝐶))
7		nnmwordri 7603	. . . 4 ⊢ ((suc 𝐴 ∈ ω ∧ 𝐶 ∈ ω ∧ 𝐶 ∈ ω) → (suc 𝐴 ⊆ 𝐶 → (suc 𝐴 ·𝑜 𝐶) ⊆ (𝐶 ·𝑜 𝐶)))
8	3, 4, 4, 7	mp3an 1416	. . 3 ⊢ (suc 𝐴 ⊆ 𝐶 → (suc 𝐴 ·𝑜 𝐶) ⊆ (𝐶 ·𝑜 𝐶))
9	6, 8	sstrd 3578	. 2 ⊢ (suc 𝐴 ⊆ 𝐶 → (suc 𝐴 ·𝑜 suc 𝐴) ⊆ (𝐶 ·𝑜 𝐶))
10	1	nnoni 6964	. . 3 ⊢ 𝐴 ∈ On
11	4	nnoni 6964	. . 3 ⊢ 𝐶 ∈ On
12	10, 11	onsucssi 6933	. 2 ⊢ (𝐴 ∈ 𝐶 ↔ suc 𝐴 ⊆ 𝐶)
13	1, 1	nnmcli 7582	. . . . . 6 ⊢ (𝐴 ·𝑜 𝐴) ∈ ω
14		2onn 7607	. . . . . . 7 ⊢ 2𝑜 ∈ ω
15	1, 14	nnmcli 7582	. . . . . 6 ⊢ (𝐴 ·𝑜 2𝑜) ∈ ω
16	13, 15	nnacli 7581	. . . . 5 ⊢ ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) ∈ ω
17	16	nnoni 6964	. . . 4 ⊢ ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) ∈ On
18	4, 4	nnmcli 7582	. . . . 5 ⊢ (𝐶 ·𝑜 𝐶) ∈ ω
19	18	nnoni 6964	. . . 4 ⊢ (𝐶 ·𝑜 𝐶) ∈ On
20	17, 19	onsucssi 6933	. . 3 ⊢ (((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) ∈ (𝐶 ·𝑜 𝐶) ↔ suc ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) ⊆ (𝐶 ·𝑜 𝐶))
21	3, 1	nnmcli 7582	. . . . . 6 ⊢ (suc 𝐴 ·𝑜 𝐴) ∈ ω
22		nnasuc 7573	. . . . . 6 ⊢ (((suc 𝐴 ·𝑜 𝐴) ∈ ω ∧ 𝐴 ∈ ω) → ((suc 𝐴 ·𝑜 𝐴) +𝑜 suc 𝐴) = suc ((suc 𝐴 ·𝑜 𝐴) +𝑜 𝐴))
23	21, 1, 22	mp2an 704	. . . . 5 ⊢ ((suc 𝐴 ·𝑜 𝐴) +𝑜 suc 𝐴) = suc ((suc 𝐴 ·𝑜 𝐴) +𝑜 𝐴)
24		nnmsuc 7574	. . . . . 6 ⊢ ((suc 𝐴 ∈ ω ∧ 𝐴 ∈ ω) → (suc 𝐴 ·𝑜 suc 𝐴) = ((suc 𝐴 ·𝑜 𝐴) +𝑜 suc 𝐴))
25	3, 1, 24	mp2an 704	. . . . 5 ⊢ (suc 𝐴 ·𝑜 suc 𝐴) = ((suc 𝐴 ·𝑜 𝐴) +𝑜 suc 𝐴)
26		nnaass 7589	. . . . . . . 8 ⊢ (((𝐴 ·𝑜 𝐴) ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐴 ∈ ω) → (((𝐴 ·𝑜 𝐴) +𝑜 𝐴) +𝑜 𝐴) = ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 +𝑜 𝐴)))
27	13, 1, 1, 26	mp3an 1416	. . . . . . 7 ⊢ (((𝐴 ·𝑜 𝐴) +𝑜 𝐴) +𝑜 𝐴) = ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 +𝑜 𝐴))
28		nnmcom 7593	. . . . . . . . . 10 ⊢ ((suc 𝐴 ∈ ω ∧ 𝐴 ∈ ω) → (suc 𝐴 ·𝑜 𝐴) = (𝐴 ·𝑜 suc 𝐴))
29	3, 1, 28	mp2an 704	. . . . . . . . 9 ⊢ (suc 𝐴 ·𝑜 𝐴) = (𝐴 ·𝑜 suc 𝐴)
30		nnmsuc 7574	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ∈ ω) → (𝐴 ·𝑜 suc 𝐴) = ((𝐴 ·𝑜 𝐴) +𝑜 𝐴))
31	1, 1, 30	mp2an 704	. . . . . . . . 9 ⊢ (𝐴 ·𝑜 suc 𝐴) = ((𝐴 ·𝑜 𝐴) +𝑜 𝐴)
32	29, 31	eqtri 2632	. . . . . . . 8 ⊢ (suc 𝐴 ·𝑜 𝐴) = ((𝐴 ·𝑜 𝐴) +𝑜 𝐴)
33	32	oveq1i 6559	. . . . . . 7 ⊢ ((suc 𝐴 ·𝑜 𝐴) +𝑜 𝐴) = (((𝐴 ·𝑜 𝐴) +𝑜 𝐴) +𝑜 𝐴)
34		nnm2 7616	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (𝐴 ·𝑜 2𝑜) = (𝐴 +𝑜 𝐴))
35	1, 34	ax-mp 5	. . . . . . . 8 ⊢ (𝐴 ·𝑜 2𝑜) = (𝐴 +𝑜 𝐴)
36	35	oveq2i 6560	. . . . . . 7 ⊢ ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) = ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 +𝑜 𝐴))
37	27, 33, 36	3eqtr4ri 2643	. . . . . 6 ⊢ ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) = ((suc 𝐴 ·𝑜 𝐴) +𝑜 𝐴)
38		suceq 5707	. . . . . 6 ⊢ (((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) = ((suc 𝐴 ·𝑜 𝐴) +𝑜 𝐴) → suc ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) = suc ((suc 𝐴 ·𝑜 𝐴) +𝑜 𝐴))
39	37, 38	ax-mp 5	. . . . 5 ⊢ suc ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) = suc ((suc 𝐴 ·𝑜 𝐴) +𝑜 𝐴)
40	23, 25, 39	3eqtr4ri 2643	. . . 4 ⊢ suc ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) = (suc 𝐴 ·𝑜 suc 𝐴)
41	40	sseq1i 3592	. . 3 ⊢ (suc ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) ⊆ (𝐶 ·𝑜 𝐶) ↔ (suc 𝐴 ·𝑜 suc 𝐴) ⊆ (𝐶 ·𝑜 𝐶))
42	20, 41	bitri 263	. 2 ⊢ (((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) ∈ (𝐶 ·𝑜 𝐶) ↔ (suc 𝐴 ·𝑜 suc 𝐴) ⊆ (𝐶 ·𝑜 𝐶))
43	9, 12, 42	3imtr4i 280	1 ⊢ (𝐴 ∈ 𝐶 → ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) ∈ (𝐶 ·𝑜 𝐶))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  suc csuc 5642  (class class class)co 6549  ωcom 6957  2_𝑜c2o 7441   +_𝑜 coa 7444   ·_𝑜 comu 7445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452

This theorem is referenced by:  omopthlem2  7623