Theorem oaomoencom 43341

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 9665	. . . 4 ⊢ (1o +o ω) ≠ (ω +o 1o)
2	1	neii 2934	. . 3 ⊢ ¬ (1o +o ω) = (ω +o 1o)
3		1on 8492	. . . 4 ⊢ 1o ∈ On
4		omelon 9660	. . . . 5 ⊢ ω ∈ On
5		oveq2 7413	. . . . . . . 8 ⊢ (𝑏 = ω → (1o +o 𝑏) = (1o +o ω))
6		oveq1 7412	. . . . . . . 8 ⊢ (𝑏 = ω → (𝑏 +o 1o) = (ω +o 1o))
7	5, 6	eqeq12d 2751	. . . . . . 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 3601	. . . . 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 7412	. . . . . . . 8 ⊢ (𝑎 = 1o → (𝑎 +o 𝑏) = (1o +o 𝑏))
12		oveq2 7413	. . . . . . . 8 ⊢ (𝑎 = 1o → (𝑏 +o 𝑎) = (𝑏 +o 1o))
13	11, 12	eqeq12d 2751	. . . . . . 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 3164	. . . . 5 ⊢ (𝑎 = 1o → (∃𝑏 ∈ On ¬ (𝑎 +o 𝑏) = (𝑏 +o 𝑎) ↔ ∃𝑏 ∈ On ¬ (1o +o 𝑏) = (𝑏 +o 1o)))
16	15	rspcev 3601	. . . 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 7884	. . . . . . 7 ⊢ ∅ ∈ ω
21	19, 20	pm3.2i 470	. . . . . 6 ⊢ ((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω)
22		oaord1 8563	. . . . . . 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 9612	. . . . . 6 ⊢ (ω ∈ (ω +o ω) → ω ≠ (ω +o ω))
25	21, 23, 24	mp2b 10	. . . . 5 ⊢ ω ≠ (ω +o ω)
26		2omomeqom 43327	. . . . . 6 ⊢ (2o ·o ω) = ω
27		df-2o 8481	. . . . . . . 8 ⊢ 2o = suc 1o
28	27	oveq2i 7416	. . . . . . 7 ⊢ (ω ·o 2o) = (ω ·o suc 1o)
29		omsuc 8538	. . . . . . . 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 8554	. . . . . . . . 9 ⊢ (ω ∈ On → (ω ·o 1o) = ω)
32	4, 31	ax-mp 5	. . . . . . . 8 ⊢ (ω ·o 1o) = ω
33	32	oveq1i 7415	. . . . . . 7 ⊢ ((ω ·o 1o) +o ω) = (ω +o ω)
34	28, 30, 33	3eqtri 2762	. . . . . 6 ⊢ (ω ·o 2o) = (ω +o ω)
35	26, 34	neeq12i 2998	. . . . 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 2934	. . 3 ⊢ ¬ (2o ·o ω) = (ω ·o 2o)
38		2on 8494	. . . 4 ⊢ 2o ∈ On
39		oveq2 7413	. . . . . . . 8 ⊢ (𝑏 = ω → (2o ·o 𝑏) = (2o ·o ω))
40		oveq1 7412	. . . . . . . 8 ⊢ (𝑏 = ω → (𝑏 ·o 2o) = (ω ·o 2o))
41	39, 40	eqeq12d 2751	. . . . . . 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 3601	. . . . 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 7412	. . . . . . . 8 ⊢ (𝑎 = 2o → (𝑎 ·o 𝑏) = (2o ·o 𝑏))
46		oveq2 7413	. . . . . . . 8 ⊢ (𝑎 = 2o → (𝑏 ·o 𝑎) = (𝑏 ·o 2o))
47	45, 46	eqeq12d 2751	. . . . . . 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 3164	. . . . 5 ⊢ (𝑎 = 2o → (∃𝑏 ∈ On ¬ (𝑎 ·o 𝑏) = (𝑏 ·o 𝑎) ↔ ∃𝑏 ∈ On ¬ (2o ·o 𝑏) = (𝑏 ·o 2o)))
50	49	rspcev 3601	. . . 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 8652	. . . . . . 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 8578	. . . . . . . 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 9612	. . . . . 6 ⊢ (ω ∈ (ω ·o ω) → ω ≠ (ω ·o ω))
60	54, 58, 59	mp2b 10	. . . . 5 ⊢ ω ≠ (ω ·o ω)
61		2onn 8654	. . . . . . 7 ⊢ 2o ∈ ω
62		1oex 8490	. . . . . . . . 9 ⊢ 1o ∈ V
63	62	prid2 4739	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o}
64		df2o3 8488	. . . . . . . 8 ⊢ 2o = {∅, 1o}
65	63, 64	eleqtrri 2833	. . . . . . 7 ⊢ 1o ∈ 2o
66		nnoeomeqom 43336	. . . . . . 7 ⊢ ((2o ∈ ω ∧ 1o ∈ 2o) → (2o ↑o ω) = ω)
67	61, 65, 66	mp2an 692	. . . . . 6 ⊢ (2o ↑o ω) = ω
68	27	oveq2i 7416	. . . . . . 7 ⊢ (ω ↑o 2o) = (ω ↑o suc 1o)
69		oesuc 8539	. . . . . . . 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 8556	. . . . . . . . 9 ⊢ (ω ∈ On → (ω ↑o 1o) = ω)
72	4, 71	ax-mp 5	. . . . . . . 8 ⊢ (ω ↑o 1o) = ω
73	72	oveq1i 7415	. . . . . . 7 ⊢ ((ω ↑o 1o) ·o ω) = (ω ·o ω)
74	68, 70, 73	3eqtri 2762	. . . . . 6 ⊢ (ω ↑o 2o) = (ω ·o ω)
75	67, 74	neeq12i 2998	. . . . 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 2934	. . 3 ⊢ ¬ (2o ↑o ω) = (ω ↑o 2o)
78		oveq2 7413	. . . . . . . 8 ⊢ (𝑏 = ω → (2o ↑o 𝑏) = (2o ↑o ω))
79		oveq1 7412	. . . . . . . 8 ⊢ (𝑏 = ω → (𝑏 ↑o 2o) = (ω ↑o 2o))
80	78, 79	eqeq12d 2751	. . . . . . 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 3601	. . . . 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 7412	. . . . . . . 8 ⊢ (𝑎 = 2o → (𝑎 ↑o 𝑏) = (2o ↑o 𝑏))
85		oveq2 7413	. . . . . . . 8 ⊢ (𝑎 = 2o → (𝑏 ↑o 𝑎) = (𝑏 ↑o 2o))
86	84, 85	eqeq12d 2751	. . . . . . 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 3164	. . . . 5 ⊢ (𝑎 = 2o → (∃𝑏 ∈ On ¬ (𝑎 ↑o 𝑏) = (𝑏 ↑o 𝑎) ↔ ∃𝑏 ∈ On ¬ (2o ↑o 𝑏) = (𝑏 ↑o 2o)))
89	88	rspcev 3601	. . . 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 1540   ∈ wcel 2108   ≠ wne 2932  ∃wrex 3060  ∅c0 4308  {cpr 4603  Oncon0 6352  suc csuc 6354  (class class class)co 7405  ωcom 7861  1_oc1o 8473  2_oc2o 8474   +_o coa 8477   ·_o comu 8478   ↑_o coe 8479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729  ax-reg 9606  ax-inf2 9655

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-omul 8485  df-oexp 8486

This theorem is referenced by: (None)