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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 8174, 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 8174 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = ∅) | |
2 | 1 | adantr 484 | . 2 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 ·o ∅) = ∅) |
3 | fnom 8166 | . . . 4 ⊢ ·o Fn (On × On) | |
4 | 3 | fndmi 6442 | . . 3 ⊢ dom ·o = (On × On) |
5 | 4 | ndmov 7349 | . 2 ⊢ (¬ (𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 ·o ∅) = ∅) |
6 | 2, 5 | pm2.61i 185 | 1 ⊢ (𝐴 ·o ∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1542 ∈ wcel 2113 ∅c0 4212 × cxp 5524 Oncon0 6173 (class class class)co 7171 ·o comu 8130 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5168 ax-nul 5175 ax-pr 5297 ax-un 7480 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3683 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-ov 7174 df-oprab 7175 df-mpo 7176 df-om 7601 df-1st 7715 df-2nd 7716 df-wrecs 7977 df-recs 8038 df-rdg 8076 df-omul 8137 |
This theorem is referenced by: (None) |
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