Theorem om0x 3126

Description: Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. Unlike om0 3125, this version works whether or not A is an ordinal. However it since it is an artifact of our particular function value definition outside the domain, we will not use it in order to be conventional and present it only as a curiosity.

Assertion

Ref	Expression
om0x	⊢ (A ·o ∅) = ∅

Proof of Theorem om0x

Step	Hyp	Ref	Expression
1		omv 3120	. . 3 ⊢ ((A ∈ On ∧ ∅ ∈ On) → (A ·o ∅) = (rec({⟨x, y⟩∣y = (x +o A)}, ∅) ‘∅))
2		0ex 1745	. . . 4 ⊢ ∅ ∈ V
3	2	rdgzer 2979	. . 3 ⊢ (rec({⟨x, y⟩∣y = (x +o A)}, ∅) ‘∅) = ∅
4	1, 3	syl6eq 1140	. 2 ⊢ ((A ∈ On ∧ ∅ ∈ On) → (A ·o ∅) = ∅)
5		fnom 3118	. . . 4 ⊢ ·o Fn (On × On)
6		fndm 2723	. . . 4 ⊢ ( ·o Fn (On × On) → dom ·o = (On × On))
7	5, 6	ax-mp 6	. . 3 ⊢ dom ·o = (On × On)
8	2, 7	ndmopr 3059	. 2 ⊢ (¬ (A ∈ On ∧ ∅ ∈ On) → (A ·o ∅) = ∅)
9	4, 8	pm2.61i 110	1 ⊢ (A ·o ∅) = ∅

Syntax hints:   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∅c0 1707  {copab 2055  Oncon0 2199   × cxp 2408  dom cdm 2410   Fn wfn 2417   ‘cfv 2422  reccrdg 2969  (class class class)co 3001   +_o coa 3101   ·_o comu 3102

This theorem is referenced by:  om0r 3142  om1r 3145

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-omul 3107

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