Theorem om0x 4216

Description: Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. Unlike om0 4214, 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 4214	. . 3 |- (A e. On -> (A .o (/)) = (/))
2	1	adantr 398	. 2 |- ((A e. On /\ (/) e. On) -> (A .o (/)) = (/))
3		0ex 2766	. . 3 |- (/) e. V
4		fnom 4207	. . . 4 |- .o Fn (On X. On)
5		fndm 3644	. . . 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 4103	. 2 |- (-. (A e. On /\ (/) e. On) -> (A .o (/)) = (/))
8	2, 7	pm2.61i 132	1 |- (A .o (/)) = (/)

Syntax hints:   /\ wa 230   = wceq 997   e. wcel 999  (/)c0 2331  Oncon0 3005   X. cxp 3225  dom cdm 3227   Fn wfn 3234  (class class class)co 4021   .o comu 4189

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-9 1006  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-rep 2748  ax-sep 2758  ax-nul 2765  ax-pow 2798  ax-pr 2835  ax-un 2922

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-3or 788  df-3an 789  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-ral 1696  df-rex 1697  df-rab 1699  df-v 1859  df-sbc 1989  df-csb 2052  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-nul 2332  df-if 2414  df-pw 2454  df-sn 2464  df-pr 2465  df-tp 2467  df-op 2468  df-uni 2558  df-iun 2622  df-br 2675  df-opab 2722  df-tr 2736  df-eprel 2888  df-id 2891  df-po 2896  df-so 2906  df-fr 2974  df-we 2991  df-ord 3008  df-on 3009  df-lim 3010  df-suc 3011  df-xp 3241  df-rel 3242  df-cnv 3243  df-co 3244  df-dm 3245  df-rn 3246  df-res 3247  df-ima 3248  df-fun 3249  df-fn 3250  df-f 3251  df-fv 3255  df-rdg 3990  df-opr 4023  df-oprab 4024  df-1st 4137  df-2nd 4138  df-omul 4194

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