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Theorem om0x 8469
Description: Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. Unlike om0 8467, this version works whether or not ๐ด 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. (Contributed by NM, 1-Feb-1996.) (New usage is discouraged.)
Assertion
Ref Expression
om0x (๐ด ยทo โˆ…) = โˆ…

Proof of Theorem om0x
StepHypRef Expression
1 om0 8467 . . 3 (๐ด โˆˆ On โ†’ (๐ด ยทo โˆ…) = โˆ…)
21adantr 482 . 2 ((๐ด โˆˆ On โˆง โˆ… โˆˆ On) โ†’ (๐ด ยทo โˆ…) = โˆ…)
3 fnom 8459 . . . 4 ยทo Fn (On ร— On)
43fndmi 6610 . . 3 dom ยทo = (On ร— On)
54ndmov 7542 . 2 (ยฌ (๐ด โˆˆ On โˆง โˆ… โˆˆ On) โ†’ (๐ด ยทo โˆ…) = โˆ…)
62, 5pm2.61i 182 1 (๐ด ยทo โˆ…) = โˆ…
Colors of variables: wff setvar class
Syntax hints:   โˆง wa 397   = wceq 1542   โˆˆ wcel 2107  โˆ…c0 4286   ร— cxp 5635  Oncon0 6321  (class class class)co 7361   ยทo comu 8414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-omul 8421
This theorem is referenced by: (None)
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