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Theorem om0r 8576
Description: Ordinal multiplication with zero. Proposition 8.18(1) of [TakeutiZaring] p. 63. (Contributed by NM, 3-Aug-2004.)
Assertion
Ref Expression
om0r (𝐴 ∈ On → (∅ ·o 𝐴) = ∅)

Proof of Theorem om0r
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7439 . . 3 (𝑥 = ∅ → (∅ ·o 𝑥) = (∅ ·o ∅))
21eqeq1d 2737 . 2 (𝑥 = ∅ → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o ∅) = ∅))
3 oveq2 7439 . . 3 (𝑥 = 𝑦 → (∅ ·o 𝑥) = (∅ ·o 𝑦))
43eqeq1d 2737 . 2 (𝑥 = 𝑦 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o 𝑦) = ∅))
5 oveq2 7439 . . 3 (𝑥 = suc 𝑦 → (∅ ·o 𝑥) = (∅ ·o suc 𝑦))
65eqeq1d 2737 . 2 (𝑥 = suc 𝑦 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o suc 𝑦) = ∅))
7 oveq2 7439 . . 3 (𝑥 = 𝐴 → (∅ ·o 𝑥) = (∅ ·o 𝐴))
87eqeq1d 2737 . 2 (𝑥 = 𝐴 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o 𝐴) = ∅))
9 0elon 6440 . . 3 ∅ ∈ On
10 om0 8554 . . 3 (∅ ∈ On → (∅ ·o ∅) = ∅)
119, 10ax-mp 5 . 2 (∅ ·o ∅) = ∅
12 oveq1 7438 . . 3 ((∅ ·o 𝑦) = ∅ → ((∅ ·o 𝑦) +o ∅) = (∅ +o ∅))
13 omsuc 8563 . . . . 5 ((∅ ∈ On ∧ 𝑦 ∈ On) → (∅ ·o suc 𝑦) = ((∅ ·o 𝑦) +o ∅))
149, 13mpan 690 . . . 4 (𝑦 ∈ On → (∅ ·o suc 𝑦) = ((∅ ·o 𝑦) +o ∅))
15 oa0 8553 . . . . . . 7 (∅ ∈ On → (∅ +o ∅) = ∅)
169, 15ax-mp 5 . . . . . 6 (∅ +o ∅) = ∅
1716eqcomi 2744 . . . . 5 ∅ = (∅ +o ∅)
1817a1i 11 . . . 4 (𝑦 ∈ On → ∅ = (∅ +o ∅))
1914, 18eqeq12d 2751 . . 3 (𝑦 ∈ On → ((∅ ·o suc 𝑦) = ∅ ↔ ((∅ ·o 𝑦) +o ∅) = (∅ +o ∅)))
2012, 19imbitrrid 246 . 2 (𝑦 ∈ On → ((∅ ·o 𝑦) = ∅ → (∅ ·o suc 𝑦) = ∅))
21 iuneq2 5016 . . . 4 (∀𝑦𝑥 (∅ ·o 𝑦) = ∅ → 𝑦𝑥 (∅ ·o 𝑦) = 𝑦𝑥 ∅)
22 iun0 5067 . . . 4 𝑦𝑥 ∅ = ∅
2321, 22eqtrdi 2791 . . 3 (∀𝑦𝑥 (∅ ·o 𝑦) = ∅ → 𝑦𝑥 (∅ ·o 𝑦) = ∅)
24 vex 3482 . . . . 5 𝑥 ∈ V
25 omlim 8570 . . . . . 6 ((∅ ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∅ ·o 𝑥) = 𝑦𝑥 (∅ ·o 𝑦))
269, 25mpan 690 . . . . 5 ((𝑥 ∈ V ∧ Lim 𝑥) → (∅ ·o 𝑥) = 𝑦𝑥 (∅ ·o 𝑦))
2724, 26mpan 690 . . . 4 (Lim 𝑥 → (∅ ·o 𝑥) = 𝑦𝑥 (∅ ·o 𝑦))
2827eqeq1d 2737 . . 3 (Lim 𝑥 → ((∅ ·o 𝑥) = ∅ ↔ 𝑦𝑥 (∅ ·o 𝑦) = ∅))
2923, 28imbitrrid 246 . 2 (Lim 𝑥 → (∀𝑦𝑥 (∅ ·o 𝑦) = ∅ → (∅ ·o 𝑥) = ∅))
302, 4, 6, 8, 11, 20, 29tfinds 7881 1 (𝐴 ∈ On → (∅ ·o 𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  c0 4339   ciun 4996  Oncon0 6386  Lim wlim 6387  suc csuc 6388  (class class class)co 7431   +o coa 8502   ·o comu 8503
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
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-oadd 8509  df-omul 8510
This theorem is referenced by:  omord  8605  omwordi  8608  om00  8612  odi  8616  omass  8617  oeoa  8634  omxpenlem  9112  onmcl  43321  omcl2  43323  omcl3g  43324
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