Theorem omopthlem2 7736

Description: Lemma for omopthi 7737. (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)

Hypotheses

Ref	Expression
omopthlem2.1	⊢ 𝐴 ∈ ω
omopthlem2.2	⊢ 𝐵 ∈ ω
omopthlem2.3	⊢ 𝐶 ∈ ω
omopthlem2.4	⊢ 𝐷 ∈ ω

Assertion

Ref	Expression
omopthlem2	⊢ ((𝐴 +𝑜 𝐵) ∈ 𝐶 → ¬ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷) = (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵))

Proof of Theorem omopthlem2

Step	Hyp	Ref	Expression
1		omopthlem2.3	. . . . . . 7 ⊢ 𝐶 ∈ ω
2	1, 1	nnmcli 7695	. . . . . 6 ⊢ (𝐶 ·𝑜 𝐶) ∈ ω
3		omopthlem2.4	. . . . . 6 ⊢ 𝐷 ∈ ω
4	2, 3	nnacli 7694	. . . . 5 ⊢ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷) ∈ ω
5	4	nnoni 7072	. . . 4 ⊢ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷) ∈ On
6	5	onirri 5834	. . 3 ⊢ ¬ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷) ∈ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷)
7		eleq1 2689	. . 3 ⊢ (((𝐶 ·𝑜 𝐶) +𝑜 𝐷) = (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) → (((𝐶 ·𝑜 𝐶) +𝑜 𝐷) ∈ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷) ↔ (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ∈ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷)))
8	6, 7	mtbii 316	. 2 ⊢ (((𝐶 ·𝑜 𝐶) +𝑜 𝐷) = (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) → ¬ (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ∈ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷))
9		nnaword1 7709	. . . 4 ⊢ (((𝐶 ·𝑜 𝐶) ∈ ω ∧ 𝐷 ∈ ω) → (𝐶 ·𝑜 𝐶) ⊆ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷))
10	2, 3, 9	mp2an 708	. . 3 ⊢ (𝐶 ·𝑜 𝐶) ⊆ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷)
11		omopthlem2.2	. . . . . . . . 9 ⊢ 𝐵 ∈ ω
12		omopthlem2.1	. . . . . . . . . . 11 ⊢ 𝐴 ∈ ω
13	12, 11	nnacli 7694	. . . . . . . . . 10 ⊢ (𝐴 +𝑜 𝐵) ∈ ω
14	13, 12	nnacli 7694	. . . . . . . . 9 ⊢ ((𝐴 +𝑜 𝐵) +𝑜 𝐴) ∈ ω
15		nnaword1 7709	. . . . . . . . 9 ⊢ ((𝐵 ∈ ω ∧ ((𝐴 +𝑜 𝐵) +𝑜 𝐴) ∈ ω) → 𝐵 ⊆ (𝐵 +𝑜 ((𝐴 +𝑜 𝐵) +𝑜 𝐴)))
16	11, 14, 15	mp2an 708	. . . . . . . 8 ⊢ 𝐵 ⊆ (𝐵 +𝑜 ((𝐴 +𝑜 𝐵) +𝑜 𝐴))
17		nnacom 7697	. . . . . . . . 9 ⊢ ((𝐵 ∈ ω ∧ ((𝐴 +𝑜 𝐵) +𝑜 𝐴) ∈ ω) → (𝐵 +𝑜 ((𝐴 +𝑜 𝐵) +𝑜 𝐴)) = (((𝐴 +𝑜 𝐵) +𝑜 𝐴) +𝑜 𝐵))
18	11, 14, 17	mp2an 708	. . . . . . . 8 ⊢ (𝐵 +𝑜 ((𝐴 +𝑜 𝐵) +𝑜 𝐴)) = (((𝐴 +𝑜 𝐵) +𝑜 𝐴) +𝑜 𝐵)
19	16, 18	sseqtri 3637	. . . . . . 7 ⊢ 𝐵 ⊆ (((𝐴 +𝑜 𝐵) +𝑜 𝐴) +𝑜 𝐵)
20		nnaass 7702	. . . . . . . . 9 ⊢ (((𝐴 +𝑜 𝐵) ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (((𝐴 +𝑜 𝐵) +𝑜 𝐴) +𝑜 𝐵) = ((𝐴 +𝑜 𝐵) +𝑜 (𝐴 +𝑜 𝐵)))
21	13, 12, 11, 20	mp3an 1424	. . . . . . . 8 ⊢ (((𝐴 +𝑜 𝐵) +𝑜 𝐴) +𝑜 𝐵) = ((𝐴 +𝑜 𝐵) +𝑜 (𝐴 +𝑜 𝐵))
22		nnm2 7729	. . . . . . . . 9 ⊢ ((𝐴 +𝑜 𝐵) ∈ ω → ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜) = ((𝐴 +𝑜 𝐵) +𝑜 (𝐴 +𝑜 𝐵)))
23	13, 22	ax-mp 5	. . . . . . . 8 ⊢ ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜) = ((𝐴 +𝑜 𝐵) +𝑜 (𝐴 +𝑜 𝐵))
24	21, 23	eqtr4i 2647	. . . . . . 7 ⊢ (((𝐴 +𝑜 𝐵) +𝑜 𝐴) +𝑜 𝐵) = ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜)
25	19, 24	sseqtri 3637	. . . . . 6 ⊢ 𝐵 ⊆ ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜)
26		2onn 7720	. . . . . . . 8 ⊢ 2𝑜 ∈ ω
27	13, 26	nnmcli 7695	. . . . . . 7 ⊢ ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜) ∈ ω
28	13, 13	nnmcli 7695	. . . . . . 7 ⊢ ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) ∈ ω
29		nnawordi 7701	. . . . . . 7 ⊢ ((𝐵 ∈ ω ∧ ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜) ∈ ω ∧ ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) ∈ ω) → (𝐵 ⊆ ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜) → (𝐵 +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵))) ⊆ (((𝐴 +𝑜 𝐵) ·𝑜 2𝑜) +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)))))
30	11, 27, 28, 29	mp3an 1424	. . . . . 6 ⊢ (𝐵 ⊆ ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜) → (𝐵 +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵))) ⊆ (((𝐴 +𝑜 𝐵) ·𝑜 2𝑜) +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵))))
31	25, 30	ax-mp 5	. . . . 5 ⊢ (𝐵 +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵))) ⊆ (((𝐴 +𝑜 𝐵) ·𝑜 2𝑜) +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)))
32		nnacom 7697	. . . . . 6 ⊢ ((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) ∈ ω ∧ 𝐵 ∈ ω) → (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (𝐵 +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵))))
33	28, 11, 32	mp2an 708	. . . . 5 ⊢ (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (𝐵 +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)))
34		nnacom 7697	. . . . . 6 ⊢ ((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) ∈ ω ∧ ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜) ∈ ω) → (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜)) = (((𝐴 +𝑜 𝐵) ·𝑜 2𝑜) +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵))))
35	28, 27, 34	mp2an 708	. . . . 5 ⊢ (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜)) = (((𝐴 +𝑜 𝐵) ·𝑜 2𝑜) +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)))
36	31, 33, 35	3sstr4i 3644	. . . 4 ⊢ (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ⊆ (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜))
37	13, 1	omopthlem1 7735	. . . 4 ⊢ ((𝐴 +𝑜 𝐵) ∈ 𝐶 → (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜)) ∈ (𝐶 ·𝑜 𝐶))
38	28, 11	nnacli 7694	. . . . . 6 ⊢ (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ∈ ω
39	38	nnoni 7072	. . . . 5 ⊢ (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ∈ On
40	2	nnoni 7072	. . . . 5 ⊢ (𝐶 ·𝑜 𝐶) ∈ On
41		ontr2 5772	. . . . 5 ⊢ (((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ∈ On ∧ (𝐶 ·𝑜 𝐶) ∈ On) → (((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ⊆ (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜)) ∧ (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜)) ∈ (𝐶 ·𝑜 𝐶)) → (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐶)))
42	39, 40, 41	mp2an 708	. . . 4 ⊢ (((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ⊆ (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜)) ∧ (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜)) ∈ (𝐶 ·𝑜 𝐶)) → (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐶))
43	36, 37, 42	sylancr 695	. . 3 ⊢ ((𝐴 +𝑜 𝐵) ∈ 𝐶 → (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐶))
44	10, 43	sseldi 3601	. 2 ⊢ ((𝐴 +𝑜 𝐵) ∈ 𝐶 → (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ∈ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷))
45	8, 44	nsyl3 133	1 ⊢ ((𝐴 +𝑜 𝐵) ∈ 𝐶 → ¬ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷) = (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ⊆ wss 3574  Oncon0 5723  (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:  omopthi  7737