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Theorem omcl 8157
Description: Closure law for ordinal multiplication. Proposition 8.16 of [TakeutiZaring] p. 57. (Contributed by NM, 3-Aug-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
omcl ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)

Proof of Theorem omcl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7157 . . . 4 (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
21eleq1d 2900 . . 3 (𝑥 = ∅ → ((𝐴 ·o 𝑥) ∈ On ↔ (𝐴 ·o ∅) ∈ On))
3 oveq2 7157 . . . 4 (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
43eleq1d 2900 . . 3 (𝑥 = 𝑦 → ((𝐴 ·o 𝑥) ∈ On ↔ (𝐴 ·o 𝑦) ∈ On))
5 oveq2 7157 . . . 4 (𝑥 = suc 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o suc 𝑦))
65eleq1d 2900 . . 3 (𝑥 = suc 𝑦 → ((𝐴 ·o 𝑥) ∈ On ↔ (𝐴 ·o suc 𝑦) ∈ On))
7 oveq2 7157 . . . 4 (𝑥 = 𝐵 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐵))
87eleq1d 2900 . . 3 (𝑥 = 𝐵 → ((𝐴 ·o 𝑥) ∈ On ↔ (𝐴 ·o 𝐵) ∈ On))
9 om0 8138 . . . 4 (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
10 0elon 6231 . . . 4 ∅ ∈ On
119, 10eqeltrdi 2924 . . 3 (𝐴 ∈ On → (𝐴 ·o ∅) ∈ On)
12 oacl 8156 . . . . . . 7 (((𝐴 ·o 𝑦) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·o 𝑦) +o 𝐴) ∈ On)
1312expcom 417 . . . . . 6 (𝐴 ∈ On → ((𝐴 ·o 𝑦) ∈ On → ((𝐴 ·o 𝑦) +o 𝐴) ∈ On))
1413adantr 484 . . . . 5 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·o 𝑦) ∈ On → ((𝐴 ·o 𝑦) +o 𝐴) ∈ On))
15 omsuc 8147 . . . . . 6 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
1615eleq1d 2900 . . . . 5 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·o suc 𝑦) ∈ On ↔ ((𝐴 ·o 𝑦) +o 𝐴) ∈ On))
1714, 16sylibrd 262 . . . 4 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·o 𝑦) ∈ On → (𝐴 ·o suc 𝑦) ∈ On))
1817expcom 417 . . 3 (𝑦 ∈ On → (𝐴 ∈ On → ((𝐴 ·o 𝑦) ∈ On → (𝐴 ·o suc 𝑦) ∈ On)))
19 vex 3483 . . . . . 6 𝑥 ∈ V
20 iunon 7972 . . . . . 6 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (𝐴 ·o 𝑦) ∈ On) → 𝑦𝑥 (𝐴 ·o 𝑦) ∈ On)
2119, 20mpan 689 . . . . 5 (∀𝑦𝑥 (𝐴 ·o 𝑦) ∈ On → 𝑦𝑥 (𝐴 ·o 𝑦) ∈ On)
22 omlim 8154 . . . . . . 7 ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 ·o 𝑥) = 𝑦𝑥 (𝐴 ·o 𝑦))
2319, 22mpanr1 702 . . . . . 6 ((𝐴 ∈ On ∧ Lim 𝑥) → (𝐴 ·o 𝑥) = 𝑦𝑥 (𝐴 ·o 𝑦))
2423eleq1d 2900 . . . . 5 ((𝐴 ∈ On ∧ Lim 𝑥) → ((𝐴 ·o 𝑥) ∈ On ↔ 𝑦𝑥 (𝐴 ·o 𝑦) ∈ On))
2521, 24syl5ibr 249 . . . 4 ((𝐴 ∈ On ∧ Lim 𝑥) → (∀𝑦𝑥 (𝐴 ·o 𝑦) ∈ On → (𝐴 ·o 𝑥) ∈ On))
2625expcom 417 . . 3 (Lim 𝑥 → (𝐴 ∈ On → (∀𝑦𝑥 (𝐴 ·o 𝑦) ∈ On → (𝐴 ·o 𝑥) ∈ On)))
272, 4, 6, 8, 11, 18, 26tfinds3 7573 . 2 (𝐵 ∈ On → (𝐴 ∈ On → (𝐴 ·o 𝐵) ∈ On))
2827impcom 411 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3133  Vcvv 3480  c0 4276   ciun 4905  Oncon0 6178  Lim wlim 6179  suc csuc 6180  (class class class)co 7149   +o coa 8095   ·o comu 8096
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-oadd 8102  df-omul 8103
This theorem is referenced by:  oecl  8158  omordi  8188  omord2  8189  omcan  8191  omword  8192  omwordri  8194  om00  8197  om00el  8198  omlimcl  8200  odi  8201  omass  8202  oneo  8203  omeulem1  8204  omeulem2  8205  omopth2  8206  oeoelem  8220  oeoe  8221  oeeui  8224  oaabs2  8268  omxpenlem  8614  omxpen  8615  cantnfle  9131  cantnflt  9132  cantnflem1d  9148  cantnflem1  9149  cantnflem3  9151  cantnflem4  9152  cnfcomlem  9159  xpnum  9377  infxpenc  9442  dfac12lem2  9568
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