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Theorem oaomoencom 43307
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 9689 . . . 4 (1o +o ω) ≠ (ω +o 1o)
21neii 2940 . . 3 ¬ (1o +o ω) = (ω +o 1o)
3 1on 8517 . . . 4 1o ∈ On
4 omelon 9684 . . . . 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 2751 . . . . . . 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 2751 . . . . . . 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 3177 . . . . 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 7911 . . . . . . 7 ∅ ∈ ω
2119, 20pm3.2i 470 . . . . . 6 ((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω)
22 oaord1 8588 . . . . . . 7 ((ω ∈ On ∧ ω ∈ On) → (∅ ∈ ω ↔ ω ∈ (ω +o ω)))
2322biimpa 476 . . . . . 6 (((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω) → ω ∈ (ω +o ω))
24 elneq 9636 . . . . . 6 (ω ∈ (ω +o ω) → ω ≠ (ω +o ω))
2521, 23, 24mp2b 10 . . . . 5 ω ≠ (ω +o ω)
26 2omomeqom 43293 . . . . . 6 (2o ·o ω) = ω
27 df-2o 8506 . . . . . . . 8 2o = suc 1o
2827oveq2i 7442 . . . . . . 7 (ω ·o 2o) = (ω ·o suc 1o)
29 omsuc 8563 . . . . . . . 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 8579 . . . . . . . . 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 2767 . . . . . 6 (ω ·o 2o) = (ω +o ω)
3526, 34neeq12i 3005 . . . . 5 ((2o ·o ω) ≠ (ω ·o 2o) ↔ ω ≠ (ω +o ω))
3625, 35mpbir 231 . . . 4 (2o ·o ω) ≠ (ω ·o 2o)
3736neii 2940 . . 3 ¬ (2o ·o ω) = (ω ·o 2o)
38 2on 8519 . . . 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 2751 . . . . . . 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 2751 . . . . . . 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 3177 . . . . 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 8677 . . . . . . 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 8603 . . . . . . . 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 2840 . . . . . 6 ((((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω) ∧ 1o ∈ ω) → ω ∈ (ω ·o ω))
59 elneq 9636 . . . . . 6 (ω ∈ (ω ·o ω) → ω ≠ (ω ·o ω))
6054, 58, 59mp2b 10 . . . . 5 ω ≠ (ω ·o ω)
61 2onn 8679 . . . . . . 7 2o ∈ ω
62 1oex 8515 . . . . . . . . 9 1o ∈ V
6362prid2 4768 . . . . . . . 8 1o ∈ {∅, 1o}
64 df2o3 8513 . . . . . . . 8 2o = {∅, 1o}
6563, 64eleqtrri 2838 . . . . . . 7 1o ∈ 2o
66 nnoeomeqom 43302 . . . . . . 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 8564 . . . . . . . 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 8581 . . . . . . . . 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 2767 . . . . . 6 (ω ↑o 2o) = (ω ·o ω)
7567, 74neeq12i 3005 . . . . 5 ((2oo ω) ≠ (ω ↑o 2o) ↔ ω ≠ (ω ·o ω))
7660, 75mpbir 231 . . . 4 (2oo ω) ≠ (ω ↑o 2o)
7776neii 2940 . . 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 2751 . . . . . . 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 2751 . . . . . . 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 3177 . . . . 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 1338 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 1086   = wceq 1537  wcel 2106  wne 2938  wrex 3068  c0 4339  {cpr 4633  Oncon0 6386  suc csuc 6388  (class class class)co 7431  ωcom 7887  1oc1o 8498  2oc2o 8499   +o coa 8502   ·o comu 8503  o coe 8504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-omul 8510  df-oexp 8511
This theorem is referenced by: (None)
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