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Mirrors > Home > MPE Home > Th. List > om0 | Structured version Visualization version GIF version |
Description: Ordinal multiplication with zero. Definition 8.15(a) of [TakeutiZaring] p. 62. Definition 2.5 of [Schloeder] p. 4. See om0x 8519 for a way to remove the antecedent ๐ด โ On. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
om0 | โข (๐ด โ On โ (๐ด ยทo โ ) = โ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elon 6419 | . . 3 โข โ โ On | |
2 | omv 8512 | . . 3 โข ((๐ด โ On โง โ โ On) โ (๐ด ยทo โ ) = (rec((๐ฅ โ V โฆ (๐ฅ +o ๐ด)), โ )โโ )) | |
3 | 1, 2 | mpan2 690 | . 2 โข (๐ด โ On โ (๐ด ยทo โ ) = (rec((๐ฅ โ V โฆ (๐ฅ +o ๐ด)), โ )โโ )) |
4 | 0ex 5308 | . . 3 โข โ โ V | |
5 | 4 | rdg0 8421 | . 2 โข (rec((๐ฅ โ V โฆ (๐ฅ +o ๐ด)), โ )โโ ) = โ |
6 | 3, 5 | eqtrdi 2789 | 1 โข (๐ด โ On โ (๐ด ยทo โ ) = โ ) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1542 โ wcel 2107 Vcvv 3475 โ c0 4323 โฆ cmpt 5232 Oncon0 6365 โcfv 6544 (class class class)co 7409 reccrdg 8409 +o coa 8463 ยทo comu 8464 |
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 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
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 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-omul 8471 |
This theorem is referenced by: om0x 8519 oesuclem 8525 omcl 8536 om0r 8539 om1 8542 om1r 8543 omwordri 8572 om00 8575 odi 8579 omass 8580 omeulem1 8582 oen0 8586 oeoa 8597 oeoelem 8598 oeeui 8602 nnm0 8605 nnm0r 8610 nneob 8655 cantnfle 9666 cantnfp1 9676 fin1a2lem6 10400 onexlimgt 41992 om0suclim 42026 oaabsb 42044 dflim5 42079 onmcl 42081 omcl3g 42084 |
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