Theorem omopthlem2 8653

Description: Lemma for omopthi 8654. (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 8608	. . . . . 6 ⊢ (𝐶 ·o 𝐶) ∈ ω
3		omopthlem2.4	. . . . . 6 ⊢ 𝐷 ∈ ω
4	2, 3	nnacli 8607	. . . . 5 ⊢ ((𝐶 ·o 𝐶) +o 𝐷) ∈ ω
5	4	nnoni 7873	. . . 4 ⊢ ((𝐶 ·o 𝐶) +o 𝐷) ∈ On
6	5	onirri 6470	. . 3 ⊢ ¬ ((𝐶 ·o 𝐶) +o 𝐷) ∈ ((𝐶 ·o 𝐶) +o 𝐷)
7		eleq1 2849	. . 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 329	. 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 8622	. . . 4 ⊢ (((𝐶 ·o 𝐶) ∈ ω ∧ 𝐷 ∈ ω) → (𝐶 ·o 𝐶) ⊆ ((𝐶 ·o 𝐶) +o 𝐷))
10	2, 3, 9	mp2an 705	. . 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 8607	. . . . . . . . . 10 ⊢ (𝐴 +o 𝐵) ∈ ω
14	13, 12	nnacli 8607	. . . . . . . . 9 ⊢ ((𝐴 +o 𝐵) +o 𝐴) ∈ ω
15		nnaword1 8622	. . . . . . . . 9 ⊢ ((𝐵 ∈ ω ∧ ((𝐴 +o 𝐵) +o 𝐴) ∈ ω) → 𝐵 ⊆ (𝐵 +o ((𝐴 +o 𝐵) +o 𝐴)))
16	11, 14, 15	mp2an 705	. . . . . . . 8 ⊢ 𝐵 ⊆ (𝐵 +o ((𝐴 +o 𝐵) +o 𝐴))
17		nnacom 8610	. . . . . . . . 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 705	. . . . . . . 8 ⊢ (𝐵 +o ((𝐴 +o 𝐵) +o 𝐴)) = (((𝐴 +o 𝐵) +o 𝐴) +o 𝐵)
19	16, 18	sseqtri 3979	. . . . . . 7 ⊢ 𝐵 ⊆ (((𝐴 +o 𝐵) +o 𝐴) +o 𝐵)
20		nnaass 8615	. . . . . . . . 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 1490	. . . . . . . 8 ⊢ (((𝐴 +o 𝐵) +o 𝐴) +o 𝐵) = ((𝐴 +o 𝐵) +o (𝐴 +o 𝐵))
22		nnm2 8646	. . . . . . . . 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 2787	. . . . . . 7 ⊢ (((𝐴 +o 𝐵) +o 𝐴) +o 𝐵) = ((𝐴 +o 𝐵) ·o 2o)
25	19, 24	sseqtri 3979	. . . . . 6 ⊢ 𝐵 ⊆ ((𝐴 +o 𝐵) ·o 2o)
26		2onn 8635	. . . . . . . 8 ⊢ 2o ∈ ω
27	13, 26	nnmcli 8608	. . . . . . 7 ⊢ ((𝐴 +o 𝐵) ·o 2o) ∈ ω
28	13, 13	nnmcli 8608	. . . . . . 7 ⊢ ((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) ∈ ω
29		nnawordi 8614	. . . . . . 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 1490	. . . . . 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 8610	. . . . . 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 705	. . . . 5 ⊢ (((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) = (𝐵 +o ((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)))
34		nnacom 8610	. . . . . 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 705	. . . . 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 3982	. . . 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 8652	. . . 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 8607	. . . . . 6 ⊢ (((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) ∈ ω
39	38	nnoni 7873	. . . . 5 ⊢ (((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) ∈ On
40	2	nnoni 7873	. . . . 5 ⊢ (𝐶 ·o 𝐶) ∈ On
41		ontr2 6404	. . . . 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 705	. . . 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 599	. . 3 ⊢ ((𝐴 +o 𝐵) ∈ 𝐶 → (((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) ∈ (𝐶 ·o 𝐶))
44	10, 43	sselid 3929	. 2 ⊢ ((𝐴 +o 𝐵) ∈ 𝐶 → (((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) ∈ ((𝐶 ·o 𝐶) +o 𝐷))
45	8, 44	nsyl3 139	1 ⊢ ((𝐴 +o 𝐵) ∈ 𝐶 → ¬ ((𝐶 ·o 𝐶) +o 𝐷) = (((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ⊆ wss 3899  Oncon0 6355  (class class class)co 7412  ωcom 7866  2_oc2o 8454   +_o coa 8457   ·_o comu 8458

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-2o 8461  df-oadd 8464  df-omul 8465

This theorem is used by:  omopthi  8654