Theorem oaomoencom 43501

Description: Ordinal addition, multiplication, and exponentiation do not generally commute. Theorem 4.1 of [Schloeder] p. 11. (Contributed by RP, 30-Jan-2025.)

Assertion

Ref	Expression
oaomoencom	⊢ (∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 +o 𝑏) = (𝑏 +o 𝑎) ∧ ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ·o 𝑏) = (𝑏 ·o 𝑎) ∧ ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ↑o 𝑏) = (𝑏 ↑o 𝑎))

Distinct variable group:   𝑎,𝑏

Proof of Theorem oaomoencom

Step	Hyp	Ref	Expression
1		oancom 9558	. . . 4 ⊢ (1o +o ω) ≠ (ω +o 1o)
2	1	neii 2932	. . 3 ⊢ ¬ (1o +o ω) = (ω +o 1o)
3		1on 8407	. . . 4 ⊢ 1o ∈ On
4		omelon 9553	. . . . 5 ⊢ ω ∈ On
5		oveq2 7364	. . . . . . . 8 ⊢ (𝑏 = ω → (1o +o 𝑏) = (1o +o ω))
6		oveq1 7363	. . . . . . . 8 ⊢ (𝑏 = ω → (𝑏 +o 1o) = (ω +o 1o))
7	5, 6	eqeq12d 2750	. . . . . . 7 ⊢ (𝑏 = ω → ((1o +o 𝑏) = (𝑏 +o 1o) ↔ (1o +o ω) = (ω +o 1o)))
8	7	notbid 318	. . . . . 6 ⊢ (𝑏 = ω → (¬ (1o +o 𝑏) = (𝑏 +o 1o) ↔ ¬ (1o +o ω) = (ω +o 1o)))
9	8	rspcev 3574	. . . . 5 ⊢ ((ω ∈ On ∧ ¬ (1o +o ω) = (ω +o 1o)) → ∃𝑏 ∈ On ¬ (1o +o 𝑏) = (𝑏 +o 1o))
10	4, 9	mpan 690	. . . 4 ⊢ (¬ (1o +o ω) = (ω +o 1o) → ∃𝑏 ∈ On ¬ (1o +o 𝑏) = (𝑏 +o 1o))
11		oveq1 7363	. . . . . . . 8 ⊢ (𝑎 = 1o → (𝑎 +o 𝑏) = (1o +o 𝑏))
12		oveq2 7364	. . . . . . . 8 ⊢ (𝑎 = 1o → (𝑏 +o 𝑎) = (𝑏 +o 1o))
13	11, 12	eqeq12d 2750	. . . . . . 7 ⊢ (𝑎 = 1o → ((𝑎 +o 𝑏) = (𝑏 +o 𝑎) ↔ (1o +o 𝑏) = (𝑏 +o 1o)))
14	13	notbid 318	. . . . . 6 ⊢ (𝑎 = 1o → (¬ (𝑎 +o 𝑏) = (𝑏 +o 𝑎) ↔ ¬ (1o +o 𝑏) = (𝑏 +o 1o)))
15	14	rexbidv 3158	. . . . 5 ⊢ (𝑎 = 1o → (∃𝑏 ∈ On ¬ (𝑎 +o 𝑏) = (𝑏 +o 𝑎) ↔ ∃𝑏 ∈ On ¬ (1o +o 𝑏) = (𝑏 +o 1o)))
16	15	rspcev 3574	. . . 4 ⊢ ((1o ∈ On ∧ ∃𝑏 ∈ On ¬ (1o +o 𝑏) = (𝑏 +o 1o)) → ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 +o 𝑏) = (𝑏 +o 𝑎))
17	3, 10, 16	sylancr 587	. . 3 ⊢ (¬ (1o +o ω) = (ω +o 1o) → ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 +o 𝑏) = (𝑏 +o 𝑎))
18	2, 17	ax-mp 5	. 2 ⊢ ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 +o 𝑏) = (𝑏 +o 𝑎)
19	4, 4	pm3.2i 470	. . . . . . 7 ⊢ (ω ∈ On ∧ ω ∈ On)
20		peano1 7829	. . . . . . 7 ⊢ ∅ ∈ ω
21	19, 20	pm3.2i 470	. . . . . 6 ⊢ ((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω)
22		oaord1 8476	. . . . . . 7 ⊢ ((ω ∈ On ∧ ω ∈ On) → (∅ ∈ ω ↔ ω ∈ (ω +o ω)))
23	22	biimpa 476	. . . . . 6 ⊢ (((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω) → ω ∈ (ω +o ω))
24		elneq 9503	. . . . . 6 ⊢ (ω ∈ (ω +o ω) → ω ≠ (ω +o ω))
25	21, 23, 24	mp2b 10	. . . . 5 ⊢ ω ≠ (ω +o ω)
26		2omomeqom 43487	. . . . . 6 ⊢ (2o ·o ω) = ω
27		df-2o 8396	. . . . . . . 8 ⊢ 2o = suc 1o
28	27	oveq2i 7367	. . . . . . 7 ⊢ (ω ·o 2o) = (ω ·o suc 1o)
29		omsuc 8451	. . . . . . . 8 ⊢ ((ω ∈ On ∧ 1o ∈ On) → (ω ·o suc 1o) = ((ω ·o 1o) +o ω))
30	4, 3, 29	mp2an 692	. . . . . . 7 ⊢ (ω ·o suc 1o) = ((ω ·o 1o) +o ω)
31		om1 8467	. . . . . . . . 9 ⊢ (ω ∈ On → (ω ·o 1o) = ω)
32	4, 31	ax-mp 5	. . . . . . . 8 ⊢ (ω ·o 1o) = ω
33	32	oveq1i 7366	. . . . . . 7 ⊢ ((ω ·o 1o) +o ω) = (ω +o ω)
34	28, 30, 33	3eqtri 2761	. . . . . 6 ⊢ (ω ·o 2o) = (ω +o ω)
35	26, 34	neeq12i 2996	. . . . 5 ⊢ ((2o ·o ω) ≠ (ω ·o 2o) ↔ ω ≠ (ω +o ω))
36	25, 35	mpbir 231	. . . 4 ⊢ (2o ·o ω) ≠ (ω ·o 2o)
37	36	neii 2932	. . 3 ⊢ ¬ (2o ·o ω) = (ω ·o 2o)
38		2on 8408	. . . 4 ⊢ 2o ∈ On
39		oveq2 7364	. . . . . . . 8 ⊢ (𝑏 = ω → (2o ·o 𝑏) = (2o ·o ω))
40		oveq1 7363	. . . . . . . 8 ⊢ (𝑏 = ω → (𝑏 ·o 2o) = (ω ·o 2o))
41	39, 40	eqeq12d 2750	. . . . . . 7 ⊢ (𝑏 = ω → ((2o ·o 𝑏) = (𝑏 ·o 2o) ↔ (2o ·o ω) = (ω ·o 2o)))
42	41	notbid 318	. . . . . 6 ⊢ (𝑏 = ω → (¬ (2o ·o 𝑏) = (𝑏 ·o 2o) ↔ ¬ (2o ·o ω) = (ω ·o 2o)))
43	42	rspcev 3574	. . . . 5 ⊢ ((ω ∈ On ∧ ¬ (2o ·o ω) = (ω ·o 2o)) → ∃𝑏 ∈ On ¬ (2o ·o 𝑏) = (𝑏 ·o 2o))
44	4, 43	mpan 690	. . . 4 ⊢ (¬ (2o ·o ω) = (ω ·o 2o) → ∃𝑏 ∈ On ¬ (2o ·o 𝑏) = (𝑏 ·o 2o))
45		oveq1 7363	. . . . . . . 8 ⊢ (𝑎 = 2o → (𝑎 ·o 𝑏) = (2o ·o 𝑏))
46		oveq2 7364	. . . . . . . 8 ⊢ (𝑎 = 2o → (𝑏 ·o 𝑎) = (𝑏 ·o 2o))
47	45, 46	eqeq12d 2750	. . . . . . 7 ⊢ (𝑎 = 2o → ((𝑎 ·o 𝑏) = (𝑏 ·o 𝑎) ↔ (2o ·o 𝑏) = (𝑏 ·o 2o)))
48	47	notbid 318	. . . . . 6 ⊢ (𝑎 = 2o → (¬ (𝑎 ·o 𝑏) = (𝑏 ·o 𝑎) ↔ ¬ (2o ·o 𝑏) = (𝑏 ·o 2o)))
49	48	rexbidv 3158	. . . . 5 ⊢ (𝑎 = 2o → (∃𝑏 ∈ On ¬ (𝑎 ·o 𝑏) = (𝑏 ·o 𝑎) ↔ ∃𝑏 ∈ On ¬ (2o ·o 𝑏) = (𝑏 ·o 2o)))
50	49	rspcev 3574	. . . 4 ⊢ ((2o ∈ On ∧ ∃𝑏 ∈ On ¬ (2o ·o 𝑏) = (𝑏 ·o 2o)) → ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ·o 𝑏) = (𝑏 ·o 𝑎))
51	38, 44, 50	sylancr 587	. . 3 ⊢ (¬ (2o ·o ω) = (ω ·o 2o) → ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ·o 𝑏) = (𝑏 ·o 𝑎))
52	37, 51	ax-mp 5	. 2 ⊢ ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ·o 𝑏) = (𝑏 ·o 𝑎)
53		1onn 8566	. . . . . . 7 ⊢ 1o ∈ ω
54	21, 53	pm3.2i 470	. . . . . 6 ⊢ (((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω) ∧ 1o ∈ ω)
55	4, 31	mp1i 13	. . . . . . 7 ⊢ ((((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω) ∧ 1o ∈ ω) → (ω ·o 1o) = ω)
56		omordi 8491	. . . . . . . 8 ⊢ (((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω) → (1o ∈ ω → (ω ·o 1o) ∈ (ω ·o ω)))
57	56	imp 406	. . . . . . 7 ⊢ ((((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω) ∧ 1o ∈ ω) → (ω ·o 1o) ∈ (ω ·o ω))
58	55, 57	eqeltrrd 2835	. . . . . 6 ⊢ ((((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω) ∧ 1o ∈ ω) → ω ∈ (ω ·o ω))
59		elneq 9503	. . . . . 6 ⊢ (ω ∈ (ω ·o ω) → ω ≠ (ω ·o ω))
60	54, 58, 59	mp2b 10	. . . . 5 ⊢ ω ≠ (ω ·o ω)
61		2onn 8568	. . . . . . 7 ⊢ 2o ∈ ω
62		1oex 8405	. . . . . . . . 9 ⊢ 1o ∈ V
63	62	prid2 4718	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o}
64		df2o3 8403	. . . . . . . 8 ⊢ 2o = {∅, 1o}
65	63, 64	eleqtrri 2833	. . . . . . 7 ⊢ 1o ∈ 2o
66		nnoeomeqom 43496	. . . . . . 7 ⊢ ((2o ∈ ω ∧ 1o ∈ 2o) → (2o ↑o ω) = ω)
67	61, 65, 66	mp2an 692	. . . . . 6 ⊢ (2o ↑o ω) = ω
68	27	oveq2i 7367	. . . . . . 7 ⊢ (ω ↑o 2o) = (ω ↑o suc 1o)
69		oesuc 8452	. . . . . . . 8 ⊢ ((ω ∈ On ∧ 1o ∈ On) → (ω ↑o suc 1o) = ((ω ↑o 1o) ·o ω))
70	4, 3, 69	mp2an 692	. . . . . . 7 ⊢ (ω ↑o suc 1o) = ((ω ↑o 1o) ·o ω)
71		oe1 8469	. . . . . . . . 9 ⊢ (ω ∈ On → (ω ↑o 1o) = ω)
72	4, 71	ax-mp 5	. . . . . . . 8 ⊢ (ω ↑o 1o) = ω
73	72	oveq1i 7366	. . . . . . 7 ⊢ ((ω ↑o 1o) ·o ω) = (ω ·o ω)
74	68, 70, 73	3eqtri 2761	. . . . . 6 ⊢ (ω ↑o 2o) = (ω ·o ω)
75	67, 74	neeq12i 2996	. . . . 5 ⊢ ((2o ↑o ω) ≠ (ω ↑o 2o) ↔ ω ≠ (ω ·o ω))
76	60, 75	mpbir 231	. . . 4 ⊢ (2o ↑o ω) ≠ (ω ↑o 2o)
77	76	neii 2932	. . 3 ⊢ ¬ (2o ↑o ω) = (ω ↑o 2o)
78		oveq2 7364	. . . . . . . 8 ⊢ (𝑏 = ω → (2o ↑o 𝑏) = (2o ↑o ω))
79		oveq1 7363	. . . . . . . 8 ⊢ (𝑏 = ω → (𝑏 ↑o 2o) = (ω ↑o 2o))
80	78, 79	eqeq12d 2750	. . . . . . 7 ⊢ (𝑏 = ω → ((2o ↑o 𝑏) = (𝑏 ↑o 2o) ↔ (2o ↑o ω) = (ω ↑o 2o)))
81	80	notbid 318	. . . . . 6 ⊢ (𝑏 = ω → (¬ (2o ↑o 𝑏) = (𝑏 ↑o 2o) ↔ ¬ (2o ↑o ω) = (ω ↑o 2o)))
82	81	rspcev 3574	. . . . 5 ⊢ ((ω ∈ On ∧ ¬ (2o ↑o ω) = (ω ↑o 2o)) → ∃𝑏 ∈ On ¬ (2o ↑o 𝑏) = (𝑏 ↑o 2o))
83	4, 82	mpan 690	. . . 4 ⊢ (¬ (2o ↑o ω) = (ω ↑o 2o) → ∃𝑏 ∈ On ¬ (2o ↑o 𝑏) = (𝑏 ↑o 2o))
84		oveq1 7363	. . . . . . . 8 ⊢ (𝑎 = 2o → (𝑎 ↑o 𝑏) = (2o ↑o 𝑏))
85		oveq2 7364	. . . . . . . 8 ⊢ (𝑎 = 2o → (𝑏 ↑o 𝑎) = (𝑏 ↑o 2o))
86	84, 85	eqeq12d 2750	. . . . . . 7 ⊢ (𝑎 = 2o → ((𝑎 ↑o 𝑏) = (𝑏 ↑o 𝑎) ↔ (2o ↑o 𝑏) = (𝑏 ↑o 2o)))
87	86	notbid 318	. . . . . 6 ⊢ (𝑎 = 2o → (¬ (𝑎 ↑o 𝑏) = (𝑏 ↑o 𝑎) ↔ ¬ (2o ↑o 𝑏) = (𝑏 ↑o 2o)))
88	87	rexbidv 3158	. . . . 5 ⊢ (𝑎 = 2o → (∃𝑏 ∈ On ¬ (𝑎 ↑o 𝑏) = (𝑏 ↑o 𝑎) ↔ ∃𝑏 ∈ On ¬ (2o ↑o 𝑏) = (𝑏 ↑o 2o)))
89	88	rspcev 3574	. . . 4 ⊢ ((2o ∈ On ∧ ∃𝑏 ∈ On ¬ (2o ↑o 𝑏) = (𝑏 ↑o 2o)) → ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ↑o 𝑏) = (𝑏 ↑o 𝑎))
90	38, 83, 89	sylancr 587	. . 3 ⊢ (¬ (2o ↑o ω) = (ω ↑o 2o) → ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ↑o 𝑏) = (𝑏 ↑o 𝑎))
91	77, 90	ax-mp 5	. 2 ⊢ ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ↑o 𝑏) = (𝑏 ↑o 𝑎)
92	18, 52, 91	3pm3.2i 1340	1 ⊢ (∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 +o 𝑏) = (𝑏 +o 𝑎) ∧ ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ·o 𝑏) = (𝑏 ·o 𝑎) ∧ ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ↑o 𝑏) = (𝑏 ↑o 𝑎))

Syntax hints:  ¬ wn 3   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2930  ∃wrex 3058  ∅c0 4283  {cpr 4580  Oncon0 6315  suc csuc 6317  (class class class)co 7356  ωcom 7806  1_oc1o 8388  2_oc2o 8389   +_o coa 8392   ·_o comu 8393   ↑_o coe 8394

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678  ax-reg 9495  ax-inf2 9548

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-oadd 8399  df-omul 8400  df-oexp 8401

This theorem is referenced by: (None)