Theorem omopthlem2 8285

Description: Lemma for omopthi 8286. (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 8243	. . . . . 6 ⊢ (𝐶 ·o 𝐶) ∈ ω
3		omopthlem2.4	. . . . . 6 ⊢ 𝐷 ∈ ω
4	2, 3	nnacli 8242	. . . . 5 ⊢ ((𝐶 ·o 𝐶) +o 𝐷) ∈ ω
5	4	nnoni 7589	. . . 4 ⊢ ((𝐶 ·o 𝐶) +o 𝐷) ∈ On
6	5	onirri 6299	. . 3 ⊢ ¬ ((𝐶 ·o 𝐶) +o 𝐷) ∈ ((𝐶 ·o 𝐶) +o 𝐷)
7		eleq1 2902	. . 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 328	. 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 8257	. . . 4 ⊢ (((𝐶 ·o 𝐶) ∈ ω ∧ 𝐷 ∈ ω) → (𝐶 ·o 𝐶) ⊆ ((𝐶 ·o 𝐶) +o 𝐷))
10	2, 3, 9	mp2an 690	. . 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 8242	. . . . . . . . . 10 ⊢ (𝐴 +o 𝐵) ∈ ω
14	13, 12	nnacli 8242	. . . . . . . . 9 ⊢ ((𝐴 +o 𝐵) +o 𝐴) ∈ ω
15		nnaword1 8257	. . . . . . . . 9 ⊢ ((𝐵 ∈ ω ∧ ((𝐴 +o 𝐵) +o 𝐴) ∈ ω) → 𝐵 ⊆ (𝐵 +o ((𝐴 +o 𝐵) +o 𝐴)))
16	11, 14, 15	mp2an 690	. . . . . . . 8 ⊢ 𝐵 ⊆ (𝐵 +o ((𝐴 +o 𝐵) +o 𝐴))
17		nnacom 8245	. . . . . . . . 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 690	. . . . . . . 8 ⊢ (𝐵 +o ((𝐴 +o 𝐵) +o 𝐴)) = (((𝐴 +o 𝐵) +o 𝐴) +o 𝐵)
19	16, 18	sseqtri 4005	. . . . . . 7 ⊢ 𝐵 ⊆ (((𝐴 +o 𝐵) +o 𝐴) +o 𝐵)
20		nnaass 8250	. . . . . . . . 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 1457	. . . . . . . 8 ⊢ (((𝐴 +o 𝐵) +o 𝐴) +o 𝐵) = ((𝐴 +o 𝐵) +o (𝐴 +o 𝐵))
22		nnm2 8278	. . . . . . . . 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 2849	. . . . . . 7 ⊢ (((𝐴 +o 𝐵) +o 𝐴) +o 𝐵) = ((𝐴 +o 𝐵) ·o 2o)
25	19, 24	sseqtri 4005	. . . . . 6 ⊢ 𝐵 ⊆ ((𝐴 +o 𝐵) ·o 2o)
26		2onn 8268	. . . . . . . 8 ⊢ 2o ∈ ω
27	13, 26	nnmcli 8243	. . . . . . 7 ⊢ ((𝐴 +o 𝐵) ·o 2o) ∈ ω
28	13, 13	nnmcli 8243	. . . . . . 7 ⊢ ((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) ∈ ω
29		nnawordi 8249	. . . . . . 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 1457	. . . . . 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 8245	. . . . . 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 690	. . . . 5 ⊢ (((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) = (𝐵 +o ((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)))
34		nnacom 8245	. . . . . 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 690	. . . . 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 4012	. . . 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 8284	. . . 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 8242	. . . . . 6 ⊢ (((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) ∈ ω
39	38	nnoni 7589	. . . . 5 ⊢ (((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) ∈ On
40	2	nnoni 7589	. . . . 5 ⊢ (𝐶 ·o 𝐶) ∈ On
41		ontr2 6240	. . . . 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 690	. . . 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 589	. . 3 ⊢ ((𝐴 +o 𝐵) ∈ 𝐶 → (((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) ∈ (𝐶 ·o 𝐶))
44	10, 43	sseldi 3967	. 2 ⊢ ((𝐴 +o 𝐵) ∈ 𝐶 → (((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) ∈ ((𝐶 ·o 𝐶) +o 𝐷))
45	8, 44	nsyl3 140	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 398   = wceq 1537   ∈ wcel 2114   ⊆ wss 3938  Oncon0 6193  (class class class)co 7158  ωcom 7582  2_oc2o 8098   +_o coa 8101   ·_o comu 8102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-2o 8105  df-oadd 8108  df-omul 8109

This theorem is referenced by:  omopthi  8286