Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  omcl3g Structured version   Visualization version   GIF version

Theorem omcl3g 41855
Description: Closure law for ordinal multiplication. (Contributed by RP, 14-Jan-2025.)
Assertion
Ref Expression
omcl3g (((𝐴𝐶𝐵𝐶) ∧ (𝐶 ∈ 3o ∨ (𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On))) → (𝐴 ·o 𝐵) ∈ 𝐶)

Proof of Theorem omcl3g
StepHypRef Expression
1 eltpi 4684 . . . . 5 (𝐶 ∈ {∅, 1o, 2o} → (𝐶 = ∅ ∨ 𝐶 = 1o𝐶 = 2o))
2 df-3o 8450 . . . . . 6 3o = suc 2o
3 df2o3 8456 . . . . . . . 8 2o = {∅, 1o}
43uneq1i 4155 . . . . . . 7 (2o ∪ {2o}) = ({∅, 1o} ∪ {2o})
5 df-suc 6359 . . . . . . 7 suc 2o = (2o ∪ {2o})
6 df-tp 4627 . . . . . . 7 {∅, 1o, 2o} = ({∅, 1o} ∪ {2o})
74, 5, 63eqtr4i 2769 . . . . . 6 suc 2o = {∅, 1o, 2o}
82, 7eqtri 2759 . . . . 5 3o = {∅, 1o, 2o}
91, 8eleq2s 2850 . . . 4 (𝐶 ∈ 3o → (𝐶 = ∅ ∨ 𝐶 = 1o𝐶 = 2o))
10 orc 865 . . . . . . 7 (𝐶 = ∅ → (𝐶 = ∅ ∨ (𝐶 = (ω ↑o (ω ↑o 𝐶)) ∧ 𝐶 ∈ On)))
11 omcl2 41854 . . . . . . 7 (((𝐴𝐶𝐵𝐶) ∧ (𝐶 = ∅ ∨ (𝐶 = (ω ↑o (ω ↑o 𝐶)) ∧ 𝐶 ∈ On))) → (𝐴 ·o 𝐵) ∈ 𝐶)
1210, 11sylan2 593 . . . . . 6 (((𝐴𝐶𝐵𝐶) ∧ 𝐶 = ∅) → (𝐴 ·o 𝐵) ∈ 𝐶)
1312ex 413 . . . . 5 ((𝐴𝐶𝐵𝐶) → (𝐶 = ∅ → (𝐴 ·o 𝐵) ∈ 𝐶))
14 el1o 8477 . . . . . . . . 9 (𝐴 ∈ 1o𝐴 = ∅)
15 el1o 8477 . . . . . . . . 9 (𝐵 ∈ 1o𝐵 = ∅)
16 oveq12 7402 . . . . . . . . . 10 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) = (∅ ·o ∅))
17 0elon 6407 . . . . . . . . . . . 12 ∅ ∈ On
18 om0 8499 . . . . . . . . . . . 12 (∅ ∈ On → (∅ ·o ∅) = ∅)
1917, 18ax-mp 5 . . . . . . . . . . 11 (∅ ·o ∅) = ∅
20 0lt1o 8486 . . . . . . . . . . 11 ∅ ∈ 1o
2119, 20eqeltri 2828 . . . . . . . . . 10 (∅ ·o ∅) ∈ 1o
2216, 21eqeltrdi 2840 . . . . . . . . 9 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) ∈ 1o)
2314, 15, 22syl2anb 598 . . . . . . . 8 ((𝐴 ∈ 1o𝐵 ∈ 1o) → (𝐴 ·o 𝐵) ∈ 1o)
2423a1i 11 . . . . . . 7 (𝐶 = 1o → ((𝐴 ∈ 1o𝐵 ∈ 1o) → (𝐴 ·o 𝐵) ∈ 1o))
25 eleq2 2821 . . . . . . . 8 (𝐶 = 1o → (𝐴𝐶𝐴 ∈ 1o))
26 eleq2 2821 . . . . . . . 8 (𝐶 = 1o → (𝐵𝐶𝐵 ∈ 1o))
2725, 26anbi12d 631 . . . . . . 7 (𝐶 = 1o → ((𝐴𝐶𝐵𝐶) ↔ (𝐴 ∈ 1o𝐵 ∈ 1o)))
28 eleq2 2821 . . . . . . 7 (𝐶 = 1o → ((𝐴 ·o 𝐵) ∈ 𝐶 ↔ (𝐴 ·o 𝐵) ∈ 1o))
2924, 27, 283imtr4d 293 . . . . . 6 (𝐶 = 1o → ((𝐴𝐶𝐵𝐶) → (𝐴 ·o 𝐵) ∈ 𝐶))
3029com12 32 . . . . 5 ((𝐴𝐶𝐵𝐶) → (𝐶 = 1o → (𝐴 ·o 𝐵) ∈ 𝐶))
31 elpri 4644 . . . . . . . . . 10 (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o))
3231, 3eleq2s 2850 . . . . . . . . 9 (𝐴 ∈ 2o → (𝐴 = ∅ ∨ 𝐴 = 1o))
33 elpri 4644 . . . . . . . . . 10 (𝐵 ∈ {∅, 1o} → (𝐵 = ∅ ∨ 𝐵 = 1o))
3433, 3eleq2s 2850 . . . . . . . . 9 (𝐵 ∈ 2o → (𝐵 = ∅ ∨ 𝐵 = 1o))
35 0ex 5300 . . . . . . . . . . . . 13 ∅ ∈ V
3635prid1 4759 . . . . . . . . . . . 12 ∅ ∈ {∅, 1o}
3736, 19, 33eltr4i 2845 . . . . . . . . . . 11 (∅ ·o ∅) ∈ 2o
3816, 37eqeltrdi 2840 . . . . . . . . . 10 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) ∈ 2o)
39 oveq12 7402 . . . . . . . . . . 11 ((𝐴 = 1o𝐵 = ∅) → (𝐴 ·o 𝐵) = (1o ·o ∅))
40 1on 8460 . . . . . . . . . . . . 13 1o ∈ On
41 om0 8499 . . . . . . . . . . . . 13 (1o ∈ On → (1o ·o ∅) = ∅)
4240, 41ax-mp 5 . . . . . . . . . . . 12 (1o ·o ∅) = ∅
4336, 42, 33eltr4i 2845 . . . . . . . . . . 11 (1o ·o ∅) ∈ 2o
4439, 43eqeltrdi 2840 . . . . . . . . . 10 ((𝐴 = 1o𝐵 = ∅) → (𝐴 ·o 𝐵) ∈ 2o)
45 oveq12 7402 . . . . . . . . . . 11 ((𝐴 = ∅ ∧ 𝐵 = 1o) → (𝐴 ·o 𝐵) = (∅ ·o 1o))
46 om0r 8521 . . . . . . . . . . . . 13 (1o ∈ On → (∅ ·o 1o) = ∅)
4740, 46ax-mp 5 . . . . . . . . . . . 12 (∅ ·o 1o) = ∅
4836, 47, 33eltr4i 2845 . . . . . . . . . . 11 (∅ ·o 1o) ∈ 2o
4945, 48eqeltrdi 2840 . . . . . . . . . 10 ((𝐴 = ∅ ∧ 𝐵 = 1o) → (𝐴 ·o 𝐵) ∈ 2o)
50 oveq12 7402 . . . . . . . . . . 11 ((𝐴 = 1o𝐵 = 1o) → (𝐴 ·o 𝐵) = (1o ·o 1o))
51 1oex 8458 . . . . . . . . . . . . 13 1o ∈ V
5251prid2 4760 . . . . . . . . . . . 12 1o ∈ {∅, 1o}
53 om1 8525 . . . . . . . . . . . . 13 (1o ∈ On → (1o ·o 1o) = 1o)
5440, 53ax-mp 5 . . . . . . . . . . . 12 (1o ·o 1o) = 1o
5552, 54, 33eltr4i 2845 . . . . . . . . . . 11 (1o ·o 1o) ∈ 2o
5650, 55eqeltrdi 2840 . . . . . . . . . 10 ((𝐴 = 1o𝐵 = 1o) → (𝐴 ·o 𝐵) ∈ 2o)
5738, 44, 49, 56ccase 1036 . . . . . . . . 9 (((𝐴 = ∅ ∨ 𝐴 = 1o) ∧ (𝐵 = ∅ ∨ 𝐵 = 1o)) → (𝐴 ·o 𝐵) ∈ 2o)
5832, 34, 57syl2an 596 . . . . . . . 8 ((𝐴 ∈ 2o𝐵 ∈ 2o) → (𝐴 ·o 𝐵) ∈ 2o)
5958a1i 11 . . . . . . 7 (𝐶 = 2o → ((𝐴 ∈ 2o𝐵 ∈ 2o) → (𝐴 ·o 𝐵) ∈ 2o))
60 eleq2 2821 . . . . . . . 8 (𝐶 = 2o → (𝐴𝐶𝐴 ∈ 2o))
61 eleq2 2821 . . . . . . . 8 (𝐶 = 2o → (𝐵𝐶𝐵 ∈ 2o))
6260, 61anbi12d 631 . . . . . . 7 (𝐶 = 2o → ((𝐴𝐶𝐵𝐶) ↔ (𝐴 ∈ 2o𝐵 ∈ 2o)))
63 eleq2 2821 . . . . . . 7 (𝐶 = 2o → ((𝐴 ·o 𝐵) ∈ 𝐶 ↔ (𝐴 ·o 𝐵) ∈ 2o))
6459, 62, 633imtr4d 293 . . . . . 6 (𝐶 = 2o → ((𝐴𝐶𝐵𝐶) → (𝐴 ·o 𝐵) ∈ 𝐶))
6564com12 32 . . . . 5 ((𝐴𝐶𝐵𝐶) → (𝐶 = 2o → (𝐴 ·o 𝐵) ∈ 𝐶))
6613, 30, 653jaod 1428 . . . 4 ((𝐴𝐶𝐵𝐶) → ((𝐶 = ∅ ∨ 𝐶 = 1o𝐶 = 2o) → (𝐴 ·o 𝐵) ∈ 𝐶))
679, 66syl5 34 . . 3 ((𝐴𝐶𝐵𝐶) → (𝐶 ∈ 3o → (𝐴 ·o 𝐵) ∈ 𝐶))
68 olc 866 . . . . 5 ((𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On) → (𝐶 = ∅ ∨ (𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On)))
69 omcl2 41854 . . . . 5 (((𝐴𝐶𝐵𝐶) ∧ (𝐶 = ∅ ∨ (𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On))) → (𝐴 ·o 𝐵) ∈ 𝐶)
7068, 69sylan2 593 . . . 4 (((𝐴𝐶𝐵𝐶) ∧ (𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On)) → (𝐴 ·o 𝐵) ∈ 𝐶)
7170ex 413 . . 3 ((𝐴𝐶𝐵𝐶) → ((𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On) → (𝐴 ·o 𝐵) ∈ 𝐶))
7267, 71jaod 857 . 2 ((𝐴𝐶𝐵𝐶) → ((𝐶 ∈ 3o ∨ (𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On)) → (𝐴 ·o 𝐵) ∈ 𝐶))
7372imp 407 1 (((𝐴𝐶𝐵𝐶) ∧ (𝐶 ∈ 3o ∨ (𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On))) → (𝐴 ·o 𝐵) ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 845  w3o 1086   = wceq 1541  wcel 2106  cun 3942  c0 4318  {csn 4622  {cpr 4624  {ctp 4626  Oncon0 6353  suc csuc 6355  (class class class)co 7393  ωcom 7838  1oc1o 8441  2oc2o 8442  3oc3o 8443   ·o comu 8446  o coe 8447
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7708  ax-reg 9569  ax-inf2 9618
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-tp 4627  df-op 4629  df-ot 4631  df-uni 4902  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-1st 7957  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-1o 8448  df-2o 8449  df-3o 8450  df-oadd 8452  df-omul 8453  df-oexp 8454
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator