Theorem omopthlem2 7623

Description: Lemma for omopthi 7624. (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 7582	. . . . . 6 ⊢ (𝐶 ·𝑜 𝐶) ∈ ω
3		omopthlem2.4	. . . . . 6 ⊢ 𝐷 ∈ ω
4	2, 3	nnacli 7581	. . . . 5 ⊢ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷) ∈ ω
5	4	nnoni 6964	. . . 4 ⊢ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷) ∈ On
6	5	onirri 5751	. . 3 ⊢ ¬ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷) ∈ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷)
7		eleq1 2676	. . 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 315	. 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 7596	. . . 4 ⊢ (((𝐶 ·𝑜 𝐶) ∈ ω ∧ 𝐷 ∈ ω) → (𝐶 ·𝑜 𝐶) ⊆ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷))
10	2, 3, 9	mp2an 704	. . 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 7581	. . . . . . . . . 10 ⊢ (𝐴 +𝑜 𝐵) ∈ ω
14	13, 12	nnacli 7581	. . . . . . . . 9 ⊢ ((𝐴 +𝑜 𝐵) +𝑜 𝐴) ∈ ω
15		nnaword1 7596	. . . . . . . . 9 ⊢ ((𝐵 ∈ ω ∧ ((𝐴 +𝑜 𝐵) +𝑜 𝐴) ∈ ω) → 𝐵 ⊆ (𝐵 +𝑜 ((𝐴 +𝑜 𝐵) +𝑜 𝐴)))
16	11, 14, 15	mp2an 704	. . . . . . . 8 ⊢ 𝐵 ⊆ (𝐵 +𝑜 ((𝐴 +𝑜 𝐵) +𝑜 𝐴))
17		nnacom 7584	. . . . . . . . 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 704	. . . . . . . 8 ⊢ (𝐵 +𝑜 ((𝐴 +𝑜 𝐵) +𝑜 𝐴)) = (((𝐴 +𝑜 𝐵) +𝑜 𝐴) +𝑜 𝐵)
19	16, 18	sseqtri 3600	. . . . . . 7 ⊢ 𝐵 ⊆ (((𝐴 +𝑜 𝐵) +𝑜 𝐴) +𝑜 𝐵)
20		nnaass 7589	. . . . . . . . 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 1416	. . . . . . . 8 ⊢ (((𝐴 +𝑜 𝐵) +𝑜 𝐴) +𝑜 𝐵) = ((𝐴 +𝑜 𝐵) +𝑜 (𝐴 +𝑜 𝐵))
22		nnm2 7616	. . . . . . . . 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 2635	. . . . . . 7 ⊢ (((𝐴 +𝑜 𝐵) +𝑜 𝐴) +𝑜 𝐵) = ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜)
25	19, 24	sseqtri 3600	. . . . . 6 ⊢ 𝐵 ⊆ ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜)
26		2onn 7607	. . . . . . . 8 ⊢ 2𝑜 ∈ ω
27	13, 26	nnmcli 7582	. . . . . . 7 ⊢ ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜) ∈ ω
28	13, 13	nnmcli 7582	. . . . . . 7 ⊢ ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) ∈ ω
29		nnawordi 7588	. . . . . . 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 1416	. . . . . 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 7584	. . . . . 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 704	. . . . 5 ⊢ (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (𝐵 +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)))
34		nnacom 7584	. . . . . 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 704	. . . . 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 3607	. . . 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 7622	. . . 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 7581	. . . . . 6 ⊢ (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ∈ ω
39	38	nnoni 6964	. . . . 5 ⊢ (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ∈ On
40	2	nnoni 6964	. . . . 5 ⊢ (𝐶 ·𝑜 𝐶) ∈ On
41		ontr2 5689	. . . . 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 704	. . . 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 694	. . 3 ⊢ ((𝐴 +𝑜 𝐵) ∈ 𝐶 → (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐶))
44	10, 43	sseldi 3566	. 2 ⊢ ((𝐴 +𝑜 𝐵) ∈ 𝐶 → (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ∈ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷))
45	8, 44	nsyl3 132	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 383   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  Oncon0 5640  (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:  omopthi  7624