![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > om1 | Structured version Visualization version GIF version |
Description: Ordinal multiplication with 1. Proposition 8.18(2) of [TakeutiZaring] p. 63. Lemma 2.15 of [Schloeder] p. 5. (Contributed by NM, 29-Oct-1995.) |
Ref | Expression |
---|---|
om1 | โข (๐ด โ On โ (๐ด ยทo 1o) = ๐ด) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-1o 8461 | . . . 4 โข 1o = suc โ | |
2 | 1 | oveq2i 7412 | . . 3 โข (๐ด ยทo 1o) = (๐ด ยทo suc โ ) |
3 | peano1 7872 | . . . 4 โข โ โ ฯ | |
4 | onmsuc 8524 | . . . 4 โข ((๐ด โ On โง โ โ ฯ) โ (๐ด ยทo suc โ ) = ((๐ด ยทo โ ) +o ๐ด)) | |
5 | 3, 4 | mpan2 688 | . . 3 โข (๐ด โ On โ (๐ด ยทo suc โ ) = ((๐ด ยทo โ ) +o ๐ด)) |
6 | 2, 5 | eqtrid 2776 | . 2 โข (๐ด โ On โ (๐ด ยทo 1o) = ((๐ด ยทo โ ) +o ๐ด)) |
7 | om0 8512 | . . 3 โข (๐ด โ On โ (๐ด ยทo โ ) = โ ) | |
8 | 7 | oveq1d 7416 | . 2 โข (๐ด โ On โ ((๐ด ยทo โ ) +o ๐ด) = (โ +o ๐ด)) |
9 | oa0r 8533 | . 2 โข (๐ด โ On โ (โ +o ๐ด) = ๐ด) | |
10 | 6, 8, 9 | 3eqtrd 2768 | 1 โข (๐ด โ On โ (๐ด ยทo 1o) = ๐ด) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1533 โ wcel 2098 โ c0 4314 Oncon0 6354 suc csuc 6356 (class class class)co 7401 ฯcom 7848 1oc1o 8454 +o coa 8458 ยทo comu 8459 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-oadd 8465 df-omul 8466 |
This theorem is referenced by: oe1m 8540 omword1 8568 oeordi 8582 oeoalem 8591 oeoa 8592 oeeui 8597 oaabs2 8644 infxpenc 10009 om1om1r 42523 oaabsb 42533 oaomoencom 42556 cantnfresb 42563 omabs2 42571 omcl3g 42573 om2 42644 |
Copyright terms: Public domain | W3C validator |