Theorem omopthlem1 7735

Description: Lemma for omopthi 7737. (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 7086	. . . . 5 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
3	1, 2	ax-mp 5	. . . 4 ⊢ suc 𝐴 ∈ ω
4		omopthlem1.2	. . . 4 ⊢ 𝐶 ∈ ω
5		nnmwordi 7715	. . . 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 1424	. . 3 ⊢ (suc 𝐴 ⊆ 𝐶 → (suc 𝐴 ·𝑜 suc 𝐴) ⊆ (suc 𝐴 ·𝑜 𝐶))
7		nnmwordri 7716	. . . 4 ⊢ ((suc 𝐴 ∈ ω ∧ 𝐶 ∈ ω ∧ 𝐶 ∈ ω) → (suc 𝐴 ⊆ 𝐶 → (suc 𝐴 ·𝑜 𝐶) ⊆ (𝐶 ·𝑜 𝐶)))
8	3, 4, 4, 7	mp3an 1424	. . 3 ⊢ (suc 𝐴 ⊆ 𝐶 → (suc 𝐴 ·𝑜 𝐶) ⊆ (𝐶 ·𝑜 𝐶))
9	6, 8	sstrd 3613	. 2 ⊢ (suc 𝐴 ⊆ 𝐶 → (suc 𝐴 ·𝑜 suc 𝐴) ⊆ (𝐶 ·𝑜 𝐶))
10	1	nnoni 7072	. . 3 ⊢ 𝐴 ∈ On
11	4	nnoni 7072	. . 3 ⊢ 𝐶 ∈ On
12	10, 11	onsucssi 7041	. 2 ⊢ (𝐴 ∈ 𝐶 ↔ suc 𝐴 ⊆ 𝐶)
13	1, 1	nnmcli 7695	. . . . . 6 ⊢ (𝐴 ·𝑜 𝐴) ∈ ω
14		2onn 7720	. . . . . . 7 ⊢ 2𝑜 ∈ ω
15	1, 14	nnmcli 7695	. . . . . 6 ⊢ (𝐴 ·𝑜 2𝑜) ∈ ω
16	13, 15	nnacli 7694	. . . . 5 ⊢ ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) ∈ ω
17	16	nnoni 7072	. . . 4 ⊢ ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) ∈ On
18	4, 4	nnmcli 7695	. . . . 5 ⊢ (𝐶 ·𝑜 𝐶) ∈ ω
19	18	nnoni 7072	. . . 4 ⊢ (𝐶 ·𝑜 𝐶) ∈ On
20	17, 19	onsucssi 7041	. . 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 7695	. . . . . 6 ⊢ (suc 𝐴 ·𝑜 𝐴) ∈ ω
22		nnasuc 7686	. . . . . 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 708	. . . . 5 ⊢ ((suc 𝐴 ·𝑜 𝐴) +𝑜 suc 𝐴) = suc ((suc 𝐴 ·𝑜 𝐴) +𝑜 𝐴)
24		nnmsuc 7687	. . . . . 6 ⊢ ((suc 𝐴 ∈ ω ∧ 𝐴 ∈ ω) → (suc 𝐴 ·𝑜 suc 𝐴) = ((suc 𝐴 ·𝑜 𝐴) +𝑜 suc 𝐴))
25	3, 1, 24	mp2an 708	. . . . 5 ⊢ (suc 𝐴 ·𝑜 suc 𝐴) = ((suc 𝐴 ·𝑜 𝐴) +𝑜 suc 𝐴)
26		nnaass 7702	. . . . . . . 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 1424	. . . . . . 7 ⊢ (((𝐴 ·𝑜 𝐴) +𝑜 𝐴) +𝑜 𝐴) = ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 +𝑜 𝐴))
28		nnmcom 7706	. . . . . . . . . 10 ⊢ ((suc 𝐴 ∈ ω ∧ 𝐴 ∈ ω) → (suc 𝐴 ·𝑜 𝐴) = (𝐴 ·𝑜 suc 𝐴))
29	3, 1, 28	mp2an 708	. . . . . . . . 9 ⊢ (suc 𝐴 ·𝑜 𝐴) = (𝐴 ·𝑜 suc 𝐴)
30		nnmsuc 7687	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ∈ ω) → (𝐴 ·𝑜 suc 𝐴) = ((𝐴 ·𝑜 𝐴) +𝑜 𝐴))
31	1, 1, 30	mp2an 708	. . . . . . . . 9 ⊢ (𝐴 ·𝑜 suc 𝐴) = ((𝐴 ·𝑜 𝐴) +𝑜 𝐴)
32	29, 31	eqtri 2644	. . . . . . . 8 ⊢ (suc 𝐴 ·𝑜 𝐴) = ((𝐴 ·𝑜 𝐴) +𝑜 𝐴)
33	32	oveq1i 6660	. . . . . . 7 ⊢ ((suc 𝐴 ·𝑜 𝐴) +𝑜 𝐴) = (((𝐴 ·𝑜 𝐴) +𝑜 𝐴) +𝑜 𝐴)
34		nnm2 7729	. . . . . . . . 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 6661	. . . . . . 7 ⊢ ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) = ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 +𝑜 𝐴))
37	27, 33, 36	3eqtr4ri 2655	. . . . . 6 ⊢ ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) = ((suc 𝐴 ·𝑜 𝐴) +𝑜 𝐴)
38		suceq 5790	. . . . . 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 2655	. . . 4 ⊢ suc ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) = (suc 𝐴 ·𝑜 suc 𝐴)
41	40	sseq1i 3629	. . 3 ⊢ (suc ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) ⊆ (𝐶 ·𝑜 𝐶) ↔ (suc 𝐴 ·𝑜 suc 𝐴) ⊆ (𝐶 ·𝑜 𝐶))
42	20, 41	bitri 264	. 2 ⊢ (((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) ∈ (𝐶 ·𝑜 𝐶) ↔ (suc 𝐴 ·𝑜 suc 𝐴) ⊆ (𝐶 ·𝑜 𝐶))
43	9, 12, 42	3imtr4i 281	1 ⊢ (𝐴 ∈ 𝐶 → ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) ∈ (𝐶 ·𝑜 𝐶))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990   ⊆ wss 3574  suc csuc 5725  (class class class)co 6650  ωcom 7065  2_𝑜c2o 7554   +_𝑜 coa 7557   ·_𝑜 comu 7558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565

This theorem is referenced by:  omopthlem2  7736