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Mathbox for Richard Penner |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > om2 | Structured version Visualization version GIF version |
Description: Two ways to double an ordinal. (Contributed by RP, 3-Jan-2025.) |
Ref | Expression |
---|---|
om2 | โข (๐ด โ On โ (๐ด +o ๐ด) = (๐ด ยทo 2o)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2o 8462 | . . 3 โข 2o = suc 1o | |
2 | 1 | oveq2i 7412 | . 2 โข (๐ด ยทo 2o) = (๐ด ยทo suc 1o) |
3 | 1on 8473 | . . . 4 โข 1o โ On | |
4 | omsuc 8521 | . . . 4 โข ((๐ด โ On โง 1o โ On) โ (๐ด ยทo suc 1o) = ((๐ด ยทo 1o) +o ๐ด)) | |
5 | 3, 4 | mpan2 688 | . . 3 โข (๐ด โ On โ (๐ด ยทo suc 1o) = ((๐ด ยทo 1o) +o ๐ด)) |
6 | om1 8537 | . . . 4 โข (๐ด โ On โ (๐ด ยทo 1o) = ๐ด) | |
7 | 6 | oveq1d 7416 | . . 3 โข (๐ด โ On โ ((๐ด ยทo 1o) +o ๐ด) = (๐ด +o ๐ด)) |
8 | 5, 7 | eqtrd 2764 | . 2 โข (๐ด โ On โ (๐ด ยทo suc 1o) = (๐ด +o ๐ด)) |
9 | 2, 8 | eqtr2id 2777 | 1 โข (๐ด โ On โ (๐ด +o ๐ด) = (๐ด ยทo 2o)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1533 โ wcel 2098 Oncon0 6354 suc csuc 6356 (class class class)co 7401 1oc1o 8454 2oc2o 8455 +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-2o 8462 df-oadd 8465 df-omul 8466 |
This theorem is referenced by: oaltom 42645 |
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