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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 8469 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 6375 | . . 3 โข โ โ On | |
2 | omv 8462 | . . 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 5268 | . . 3 โข โ โ V | |
5 | 4 | rdg0 8371 | . 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 3447 โ c0 4286 โฆ cmpt 5192 Oncon0 6321 โcfv 6500 (class class class)co 7361 reccrdg 8359 +o coa 8413 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-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-omul 8421 |
This theorem is referenced by: om0x 8469 oesuclem 8475 omcl 8486 om0r 8489 om1 8493 om1r 8494 omwordri 8523 om00 8526 odi 8530 omass 8531 omeulem1 8533 oen0 8537 oeoa 8548 oeoelem 8549 oeeui 8553 nnm0 8556 nnm0r 8561 nneob 8606 cantnfle 9615 cantnfp1 9625 fin1a2lem6 10349 onexlimgt 41624 om0suclim 41658 oaabsb 41676 dflim5 41711 onmcl 41713 omcl3g 41716 |
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