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Mirrors > Home > MPE Home > Th. List > om0x | Structured version Visualization version GIF version |
Description: Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. Unlike om0 8000, 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. (Contributed by NM, 1-Feb-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
om0x | ⊢ (𝐴 ·o ∅) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | om0 8000 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = ∅) | |
2 | 1 | adantr 481 | . 2 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 ·o ∅) = ∅) |
3 | fnom 7992 | . . . 4 ⊢ ·o Fn (On × On) | |
4 | fndm 6332 | . . . 4 ⊢ ( ·o Fn (On × On) → dom ·o = (On × On)) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ dom ·o = (On × On) |
6 | 5 | ndmov 7195 | . 2 ⊢ (¬ (𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 ·o ∅) = ∅) |
7 | 2, 6 | pm2.61i 183 | 1 ⊢ (𝐴 ·o ∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1525 ∈ wcel 2083 ∅c0 4217 × cxp 5448 dom cdm 5450 Oncon0 6073 Fn wfn 6227 (class class class)co 7023 ·o comu 7958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-ral 3112 df-rex 3113 df-reu 3114 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-omul 7965 |
This theorem is referenced by: (None) |
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