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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 8416 | . . . 4 โข 1o = suc โ | |
2 | 1 | oveq2i 7372 | . . 3 โข (๐ด ยทo 1o) = (๐ด ยทo suc โ ) |
3 | peano1 7829 | . . . 4 โข โ โ ฯ | |
4 | onmsuc 8479 | . . . 4 โข ((๐ด โ On โง โ โ ฯ) โ (๐ด ยทo suc โ ) = ((๐ด ยทo โ ) +o ๐ด)) | |
5 | 3, 4 | mpan2 690 | . . 3 โข (๐ด โ On โ (๐ด ยทo suc โ ) = ((๐ด ยทo โ ) +o ๐ด)) |
6 | 2, 5 | eqtrid 2785 | . 2 โข (๐ด โ On โ (๐ด ยทo 1o) = ((๐ด ยทo โ ) +o ๐ด)) |
7 | om0 8467 | . . 3 โข (๐ด โ On โ (๐ด ยทo โ ) = โ ) | |
8 | 7 | oveq1d 7376 | . 2 โข (๐ด โ On โ ((๐ด ยทo โ ) +o ๐ด) = (โ +o ๐ด)) |
9 | oa0r 8488 | . 2 โข (๐ด โ On โ (โ +o ๐ด) = ๐ด) | |
10 | 6, 8, 9 | 3eqtrd 2777 | 1 โข (๐ด โ On โ (๐ด ยทo 1o) = ๐ด) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1542 โ wcel 2107 โ c0 4286 Oncon0 6321 suc csuc 6323 (class class class)co 7361 ฯcom 7806 1oc1o 8409 +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-rep 5246 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-1o 8416 df-oadd 8420 df-omul 8421 |
This theorem is referenced by: oe1m 8496 omword1 8524 oeordi 8538 oeoalem 8547 oeoa 8548 oeeui 8553 oaabs2 8599 infxpenc 9962 om1om1r 41666 oaabsb 41676 oaomoencom 41699 cantnfresb 41706 omabs2 41714 omcl3g 41716 om2 41768 |
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