Theorem omopthlem2 8598

Description: Lemma for omopthi 8599. (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 8553	. . . . . 6 ⊢ (𝐶 ·o 𝐶) ∈ ω
3		omopthlem2.4	. . . . . 6 ⊢ 𝐷 ∈ ω
4	2, 3	nnacli 8552	. . . . 5 ⊢ ((𝐶 ·o 𝐶) +o 𝐷) ∈ ω
5	4	nnoni 7825	. . . 4 ⊢ ((𝐶 ·o 𝐶) +o 𝐷) ∈ On
6	5	onirri 6439	. . 3 ⊢ ¬ ((𝐶 ·o 𝐶) +o 𝐷) ∈ ((𝐶 ·o 𝐶) +o 𝐷)
7		eleq1 2825	. . 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 326	. 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 8567	. . . 4 ⊢ (((𝐶 ·o 𝐶) ∈ ω ∧ 𝐷 ∈ ω) → (𝐶 ·o 𝐶) ⊆ ((𝐶 ·o 𝐶) +o 𝐷))
10	2, 3, 9	mp2an 693	. . 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 8552	. . . . . . . . . 10 ⊢ (𝐴 +o 𝐵) ∈ ω
14	13, 12	nnacli 8552	. . . . . . . . 9 ⊢ ((𝐴 +o 𝐵) +o 𝐴) ∈ ω
15		nnaword1 8567	. . . . . . . . 9 ⊢ ((𝐵 ∈ ω ∧ ((𝐴 +o 𝐵) +o 𝐴) ∈ ω) → 𝐵 ⊆ (𝐵 +o ((𝐴 +o 𝐵) +o 𝐴)))
16	11, 14, 15	mp2an 693	. . . . . . . 8 ⊢ 𝐵 ⊆ (𝐵 +o ((𝐴 +o 𝐵) +o 𝐴))
17		nnacom 8555	. . . . . . . . 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 693	. . . . . . . 8 ⊢ (𝐵 +o ((𝐴 +o 𝐵) +o 𝐴)) = (((𝐴 +o 𝐵) +o 𝐴) +o 𝐵)
19	16, 18	sseqtri 3984	. . . . . . 7 ⊢ 𝐵 ⊆ (((𝐴 +o 𝐵) +o 𝐴) +o 𝐵)
20		nnaass 8560	. . . . . . . . 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 1464	. . . . . . . 8 ⊢ (((𝐴 +o 𝐵) +o 𝐴) +o 𝐵) = ((𝐴 +o 𝐵) +o (𝐴 +o 𝐵))
22		nnm2 8591	. . . . . . . . 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 2763	. . . . . . 7 ⊢ (((𝐴 +o 𝐵) +o 𝐴) +o 𝐵) = ((𝐴 +o 𝐵) ·o 2o)
25	19, 24	sseqtri 3984	. . . . . 6 ⊢ 𝐵 ⊆ ((𝐴 +o 𝐵) ·o 2o)
26		2onn 8580	. . . . . . . 8 ⊢ 2o ∈ ω
27	13, 26	nnmcli 8553	. . . . . . 7 ⊢ ((𝐴 +o 𝐵) ·o 2o) ∈ ω
28	13, 13	nnmcli 8553	. . . . . . 7 ⊢ ((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) ∈ ω
29		nnawordi 8559	. . . . . . 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 1464	. . . . . 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 8555	. . . . . 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 693	. . . . 5 ⊢ (((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) = (𝐵 +o ((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)))
34		nnacom 8555	. . . . . 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 693	. . . . 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 3987	. . . 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 8597	. . . 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 8552	. . . . . 6 ⊢ (((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) ∈ ω
39	38	nnoni 7825	. . . . 5 ⊢ (((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) ∈ On
40	2	nnoni 7825	. . . . 5 ⊢ (𝐶 ·o 𝐶) ∈ On
41		ontr2 6373	. . . . 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 693	. . . 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 588	. . 3 ⊢ ((𝐴 +o 𝐵) ∈ 𝐶 → (((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) ∈ (𝐶 ·o 𝐶))
44	10, 43	sselid 3933	. 2 ⊢ ((𝐴 +o 𝐵) ∈ 𝐶 → (((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) ∈ ((𝐶 ·o 𝐶) +o 𝐷))
45	8, 44	nsyl3 138	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 395   = wceq 1542   ∈ wcel 2114   ⊆ wss 3903  Oncon0 6325  (class class class)co 7368  ωcom 7818  2_oc2o 8401   +_o coa 8404   ·_o comu 8405

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-oadd 8411  df-omul 8412

This theorem is referenced by:  omopthi  8599