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Theorem oaomoencom 43330
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 ¬ (𝑎o 𝑏) = (𝑏o 𝑎))
Distinct variable group:   𝑎,𝑏

Proof of Theorem oaomoencom
StepHypRef Expression
1 oancom 9691 . . . 4 (1o +o ω) ≠ (ω +o 1o)
21neii 2942 . . 3 ¬ (1o +o ω) = (ω +o 1o)
3 1on 8518 . . . 4 1o ∈ On
4 omelon 9686 . . . . 5 ω ∈ On
5 oveq2 7439 . . . . . . . 8 (𝑏 = ω → (1o +o 𝑏) = (1o +o ω))
6 oveq1 7438 . . . . . . . 8 (𝑏 = ω → (𝑏 +o 1o) = (ω +o 1o))
75, 6eqeq12d 2753 . . . . . . 7 (𝑏 = ω → ((1o +o 𝑏) = (𝑏 +o 1o) ↔ (1o +o ω) = (ω +o 1o)))
87notbid 318 . . . . . 6 (𝑏 = ω → (¬ (1o +o 𝑏) = (𝑏 +o 1o) ↔ ¬ (1o +o ω) = (ω +o 1o)))
98rspcev 3622 . . . . 5 ((ω ∈ On ∧ ¬ (1o +o ω) = (ω +o 1o)) → ∃𝑏 ∈ On ¬ (1o +o 𝑏) = (𝑏 +o 1o))
104, 9mpan 690 . . . 4 (¬ (1o +o ω) = (ω +o 1o) → ∃𝑏 ∈ On ¬ (1o +o 𝑏) = (𝑏 +o 1o))
11 oveq1 7438 . . . . . . . 8 (𝑎 = 1o → (𝑎 +o 𝑏) = (1o +o 𝑏))
12 oveq2 7439 . . . . . . . 8 (𝑎 = 1o → (𝑏 +o 𝑎) = (𝑏 +o 1o))
1311, 12eqeq12d 2753 . . . . . . 7 (𝑎 = 1o → ((𝑎 +o 𝑏) = (𝑏 +o 𝑎) ↔ (1o +o 𝑏) = (𝑏 +o 1o)))
1413notbid 318 . . . . . 6 (𝑎 = 1o → (¬ (𝑎 +o 𝑏) = (𝑏 +o 𝑎) ↔ ¬ (1o +o 𝑏) = (𝑏 +o 1o)))
1514rexbidv 3179 . . . . 5 (𝑎 = 1o → (∃𝑏 ∈ On ¬ (𝑎 +o 𝑏) = (𝑏 +o 𝑎) ↔ ∃𝑏 ∈ On ¬ (1o +o 𝑏) = (𝑏 +o 1o)))
1615rspcev 3622 . . . 4 ((1o ∈ On ∧ ∃𝑏 ∈ On ¬ (1o +o 𝑏) = (𝑏 +o 1o)) → ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 +o 𝑏) = (𝑏 +o 𝑎))
173, 10, 16sylancr 587 . . 3 (¬ (1o +o ω) = (ω +o 1o) → ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 +o 𝑏) = (𝑏 +o 𝑎))
182, 17ax-mp 5 . 2 𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 +o 𝑏) = (𝑏 +o 𝑎)
194, 4pm3.2i 470 . . . . . . 7 (ω ∈ On ∧ ω ∈ On)
20 peano1 7910 . . . . . . 7 ∅ ∈ ω
2119, 20pm3.2i 470 . . . . . 6 ((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω)
22 oaord1 8589 . . . . . . 7 ((ω ∈ On ∧ ω ∈ On) → (∅ ∈ ω ↔ ω ∈ (ω +o ω)))
2322biimpa 476 . . . . . 6 (((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω) → ω ∈ (ω +o ω))
24 elneq 9638 . . . . . 6 (ω ∈ (ω +o ω) → ω ≠ (ω +o ω))
2521, 23, 24mp2b 10 . . . . 5 ω ≠ (ω +o ω)
26 2omomeqom 43316 . . . . . 6 (2o ·o ω) = ω
27 df-2o 8507 . . . . . . . 8 2o = suc 1o
2827oveq2i 7442 . . . . . . 7 (ω ·o 2o) = (ω ·o suc 1o)
29 omsuc 8564 . . . . . . . 8 ((ω ∈ On ∧ 1o ∈ On) → (ω ·o suc 1o) = ((ω ·o 1o) +o ω))
304, 3, 29mp2an 692 . . . . . . 7 (ω ·o suc 1o) = ((ω ·o 1o) +o ω)
31 om1 8580 . . . . . . . . 9 (ω ∈ On → (ω ·o 1o) = ω)
324, 31ax-mp 5 . . . . . . . 8 (ω ·o 1o) = ω
3332oveq1i 7441 . . . . . . 7 ((ω ·o 1o) +o ω) = (ω +o ω)
3428, 30, 333eqtri 2769 . . . . . 6 (ω ·o 2o) = (ω +o ω)
3526, 34neeq12i 3007 . . . . 5 ((2o ·o ω) ≠ (ω ·o 2o) ↔ ω ≠ (ω +o ω))
3625, 35mpbir 231 . . . 4 (2o ·o ω) ≠ (ω ·o 2o)
3736neii 2942 . . 3 ¬ (2o ·o ω) = (ω ·o 2o)
38 2on 8520 . . . 4 2o ∈ On
39 oveq2 7439 . . . . . . . 8 (𝑏 = ω → (2o ·o 𝑏) = (2o ·o ω))
40 oveq1 7438 . . . . . . . 8 (𝑏 = ω → (𝑏 ·o 2o) = (ω ·o 2o))
4139, 40eqeq12d 2753 . . . . . . 7 (𝑏 = ω → ((2o ·o 𝑏) = (𝑏 ·o 2o) ↔ (2o ·o ω) = (ω ·o 2o)))
4241notbid 318 . . . . . 6 (𝑏 = ω → (¬ (2o ·o 𝑏) = (𝑏 ·o 2o) ↔ ¬ (2o ·o ω) = (ω ·o 2o)))
4342rspcev 3622 . . . . 5 ((ω ∈ On ∧ ¬ (2o ·o ω) = (ω ·o 2o)) → ∃𝑏 ∈ On ¬ (2o ·o 𝑏) = (𝑏 ·o 2o))
444, 43mpan 690 . . . 4 (¬ (2o ·o ω) = (ω ·o 2o) → ∃𝑏 ∈ On ¬ (2o ·o 𝑏) = (𝑏 ·o 2o))
45 oveq1 7438 . . . . . . . 8 (𝑎 = 2o → (𝑎 ·o 𝑏) = (2o ·o 𝑏))
46 oveq2 7439 . . . . . . . 8 (𝑎 = 2o → (𝑏 ·o 𝑎) = (𝑏 ·o 2o))
4745, 46eqeq12d 2753 . . . . . . 7 (𝑎 = 2o → ((𝑎 ·o 𝑏) = (𝑏 ·o 𝑎) ↔ (2o ·o 𝑏) = (𝑏 ·o 2o)))
4847notbid 318 . . . . . 6 (𝑎 = 2o → (¬ (𝑎 ·o 𝑏) = (𝑏 ·o 𝑎) ↔ ¬ (2o ·o 𝑏) = (𝑏 ·o 2o)))
4948rexbidv 3179 . . . . 5 (𝑎 = 2o → (∃𝑏 ∈ On ¬ (𝑎 ·o 𝑏) = (𝑏 ·o 𝑎) ↔ ∃𝑏 ∈ On ¬ (2o ·o 𝑏) = (𝑏 ·o 2o)))
5049rspcev 3622 . . . 4 ((2o ∈ On ∧ ∃𝑏 ∈ On ¬ (2o ·o 𝑏) = (𝑏 ·o 2o)) → ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ·o 𝑏) = (𝑏 ·o 𝑎))
5138, 44, 50sylancr 587 . . 3 (¬ (2o ·o ω) = (ω ·o 2o) → ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ·o 𝑏) = (𝑏 ·o 𝑎))
5237, 51ax-mp 5 . 2 𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ·o 𝑏) = (𝑏 ·o 𝑎)
53 1onn 8678 . . . . . . 7 1o ∈ ω
5421, 53pm3.2i 470 . . . . . 6 (((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω) ∧ 1o ∈ ω)
554, 31mp1i 13 . . . . . . 7 ((((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω) ∧ 1o ∈ ω) → (ω ·o 1o) = ω)
56 omordi 8604 . . . . . . . 8 (((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω) → (1o ∈ ω → (ω ·o 1o) ∈ (ω ·o ω)))
5756imp 406 . . . . . . 7 ((((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω) ∧ 1o ∈ ω) → (ω ·o 1o) ∈ (ω ·o ω))
5855, 57eqeltrrd 2842 . . . . . 6 ((((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω) ∧ 1o ∈ ω) → ω ∈ (ω ·o ω))
59 elneq 9638 . . . . . 6 (ω ∈ (ω ·o ω) → ω ≠ (ω ·o ω))
6054, 58, 59mp2b 10 . . . . 5 ω ≠ (ω ·o ω)
61 2onn 8680 . . . . . . 7 2o ∈ ω
62 1oex 8516 . . . . . . . . 9 1o ∈ V
6362prid2 4763 . . . . . . . 8 1o ∈ {∅, 1o}
64 df2o3 8514 . . . . . . . 8 2o = {∅, 1o}
6563, 64eleqtrri 2840 . . . . . . 7 1o ∈ 2o
66 nnoeomeqom 43325 . . . . . . 7 ((2o ∈ ω ∧ 1o ∈ 2o) → (2oo ω) = ω)
6761, 65, 66mp2an 692 . . . . . 6 (2oo ω) = ω
6827oveq2i 7442 . . . . . . 7 (ω ↑o 2o) = (ω ↑o suc 1o)
69 oesuc 8565 . . . . . . . 8 ((ω ∈ On ∧ 1o ∈ On) → (ω ↑o suc 1o) = ((ω ↑o 1o) ·o ω))
704, 3, 69mp2an 692 . . . . . . 7 (ω ↑o suc 1o) = ((ω ↑o 1o) ·o ω)
71 oe1 8582 . . . . . . . . 9 (ω ∈ On → (ω ↑o 1o) = ω)
724, 71ax-mp 5 . . . . . . . 8 (ω ↑o 1o) = ω
7372oveq1i 7441 . . . . . . 7 ((ω ↑o 1o) ·o ω) = (ω ·o ω)
7468, 70, 733eqtri 2769 . . . . . 6 (ω ↑o 2o) = (ω ·o ω)
7567, 74neeq12i 3007 . . . . 5 ((2oo ω) ≠ (ω ↑o 2o) ↔ ω ≠ (ω ·o ω))
7660, 75mpbir 231 . . . 4 (2oo ω) ≠ (ω ↑o 2o)
7776neii 2942 . . 3 ¬ (2oo ω) = (ω ↑o 2o)
78 oveq2 7439 . . . . . . . 8 (𝑏 = ω → (2oo 𝑏) = (2oo ω))
79 oveq1 7438 . . . . . . . 8 (𝑏 = ω → (𝑏o 2o) = (ω ↑o 2o))
8078, 79eqeq12d 2753 . . . . . . 7 (𝑏 = ω → ((2oo 𝑏) = (𝑏o 2o) ↔ (2oo ω) = (ω ↑o 2o)))
8180notbid 318 . . . . . 6 (𝑏 = ω → (¬ (2oo 𝑏) = (𝑏o 2o) ↔ ¬ (2oo ω) = (ω ↑o 2o)))
8281rspcev 3622 . . . . 5 ((ω ∈ On ∧ ¬ (2oo ω) = (ω ↑o 2o)) → ∃𝑏 ∈ On ¬ (2oo 𝑏) = (𝑏o 2o))
834, 82mpan 690 . . . 4 (¬ (2oo ω) = (ω ↑o 2o) → ∃𝑏 ∈ On ¬ (2oo 𝑏) = (𝑏o 2o))
84 oveq1 7438 . . . . . . . 8 (𝑎 = 2o → (𝑎o 𝑏) = (2oo 𝑏))
85 oveq2 7439 . . . . . . . 8 (𝑎 = 2o → (𝑏o 𝑎) = (𝑏o 2o))
8684, 85eqeq12d 2753 . . . . . . 7 (𝑎 = 2o → ((𝑎o 𝑏) = (𝑏o 𝑎) ↔ (2oo 𝑏) = (𝑏o 2o)))
8786notbid 318 . . . . . 6 (𝑎 = 2o → (¬ (𝑎o 𝑏) = (𝑏o 𝑎) ↔ ¬ (2oo 𝑏) = (𝑏o 2o)))
8887rexbidv 3179 . . . . 5 (𝑎 = 2o → (∃𝑏 ∈ On ¬ (𝑎o 𝑏) = (𝑏o 𝑎) ↔ ∃𝑏 ∈ On ¬ (2oo 𝑏) = (𝑏o 2o)))
8988rspcev 3622 . . . 4 ((2o ∈ On ∧ ∃𝑏 ∈ On ¬ (2oo 𝑏) = (𝑏o 2o)) → ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎o 𝑏) = (𝑏o 𝑎))
9038, 83, 89sylancr 587 . . 3 (¬ (2oo ω) = (ω ↑o 2o) → ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎o 𝑏) = (𝑏o 𝑎))
9177, 90ax-mp 5 . 2 𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎o 𝑏) = (𝑏o 𝑎)
9218, 52, 913pm3.2i 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 ¬ (𝑎o 𝑏) = (𝑏o 𝑎))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wrex 3070  c0 4333  {cpr 4628  Oncon0 6384  suc csuc 6386  (class class class)co 7431  ωcom 7887  1oc1o 8499  2oc2o 8500   +o coa 8503   ·o comu 8504  o coe 8505
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 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-reg 9632  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511  df-oexp 8512
This theorem is referenced by: (None)
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