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Mathbox for Richard Penner |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > oe2 | Structured version Visualization version GIF version |
Description: Two ways to square an ordinal. (Contributed by RP, 3-Jan-2025.) |
Ref | Expression |
---|---|
oe2 | โข (๐ด โ On โ (๐ด ยทo ๐ด) = (๐ด โo 2o)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2o 8481 | . . 3 โข 2o = suc 1o | |
2 | 1 | oveq2i 7425 | . 2 โข (๐ด โo 2o) = (๐ด โo suc 1o) |
3 | 1on 8492 | . . . 4 โข 1o โ On | |
4 | oesuc 8541 | . . . 4 โข ((๐ด โ On โง 1o โ On) โ (๐ด โo suc 1o) = ((๐ด โo 1o) ยทo ๐ด)) | |
5 | 3, 4 | mpan2 690 | . . 3 โข (๐ด โ On โ (๐ด โo suc 1o) = ((๐ด โo 1o) ยทo ๐ด)) |
6 | oe1 8558 | . . . 4 โข (๐ด โ On โ (๐ด โo 1o) = ๐ด) | |
7 | 6 | oveq1d 7429 | . . 3 โข (๐ด โ On โ ((๐ด โo 1o) ยทo ๐ด) = (๐ด ยทo ๐ด)) |
8 | 5, 7 | eqtrd 2767 | . 2 โข (๐ด โ On โ (๐ด โo suc 1o) = (๐ด ยทo ๐ด)) |
9 | 2, 8 | eqtr2id 2780 | 1 โข (๐ด โ On โ (๐ด ยทo ๐ด) = (๐ด โo 2o)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1534 โ wcel 2099 Oncon0 6363 suc csuc 6365 (class class class)co 7414 1oc1o 8473 2oc2o 8474 ยทo comu 8478 โo coe 8479 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-omul 8485 df-oexp 8486 |
This theorem is referenced by: omltoe 42809 |
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