Theorem om0x 4096

Description: Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. Unlike om0 4094, this version works whether or not A is an ordinal. However, 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		om0 4094	. . 3 |- (A e. On -> (A .o (/)) = (/))
2	1	adantr 389	. 2 |- ((A e. On /\ (/) e. On) -> (A .o (/)) = (/))
3		0ex 2679	. . 3 |- (/) e. V
4		fnom 4087	. . . 4 |- .o Fn (On X. On)
5		fndm 3527	. . . 4 |- ( .o Fn (On X. On) -> dom .o = (On X. On))
6	4, 5	ax-mp 7	. . 3 |- dom .o = (On X. On)
7	3, 6	ndmopr 3985	. 2 |- (-. (A e. On /\ (/) e. On) -> (A .o (/)) = (/))
8	2, 7	pm2.61i 126	1 |- (A .o (/)) = (/)

Syntax hints:   /\ wa 223   = wceq 1099   e. wcel 1105  (/)c0 2251  Oncon0 2911   X. cxp 3131  dom cdm 3133   Fn wfn 3140  (class class class)co 3902   .o comu 4069

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-rep 2661  ax-sep 2671  ax-nul 2678  ax-pow 2710  ax-pr 2747  ax-un 2830

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 773  df-3an 774  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-ral 1625  df-rex 1626  df-rab 1628  df-v 1787  df-sbc 1913  df-csb 1973  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-if 2333  df-pw 2373  df-sn 2383  df-pr 2384  df-tp 2386  df-op 2387  df-uni 2472  df-iun 2536  df-br 2588  df-opab 2635  df-tr 2649  df-eprel 2794  df-id 2797  df-po 2804  df-so 2814  df-fr 2880  df-we 2897  df-ord 2914  df-on 2915  df-lim 2916  df-suc 2917  df-xp 3147  df-rel 3148  df-cnv 3149  df-co 3150  df-dm 3151  df-rn 3152  df-res 3153  df-ima 3154  df-fun 3155  df-fn 3156  df-f 3157  df-fv 3161  df-rdg 3871  df-opr 3904  df-oprab 3905  df-1st 4017  df-2nd 4018  df-omul 4074

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