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Mirrors > Home > MPE Home > Th. List > oe1 | Structured version Visualization version GIF version |
Description: Ordinal exponentiation with an exponent of 1. Lemma 2.16 of [Schloeder] p. 6. (Contributed by NM, 2-Jan-2005.) |
Ref | Expression |
---|---|
oe1 | ⊢ (𝐴 ∈ On → (𝐴 ↑o 1o) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-1o 8493 | . . . 4 ⊢ 1o = suc ∅ | |
2 | 1 | oveq2i 7437 | . . 3 ⊢ (𝐴 ↑o 1o) = (𝐴 ↑o suc ∅) |
3 | peano1 7900 | . . . 4 ⊢ ∅ ∈ ω | |
4 | onesuc 8557 | . . . 4 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ ω) → (𝐴 ↑o suc ∅) = ((𝐴 ↑o ∅) ·o 𝐴)) | |
5 | 3, 4 | mpan2 689 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 ↑o suc ∅) = ((𝐴 ↑o ∅) ·o 𝐴)) |
6 | 2, 5 | eqtrid 2780 | . 2 ⊢ (𝐴 ∈ On → (𝐴 ↑o 1o) = ((𝐴 ↑o ∅) ·o 𝐴)) |
7 | oe0 8549 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o) | |
8 | 7 | oveq1d 7441 | . 2 ⊢ (𝐴 ∈ On → ((𝐴 ↑o ∅) ·o 𝐴) = (1o ·o 𝐴)) |
9 | om1r 8570 | . 2 ⊢ (𝐴 ∈ On → (1o ·o 𝐴) = 𝐴) | |
10 | 6, 8, 9 | 3eqtrd 2772 | 1 ⊢ (𝐴 ∈ On → (𝐴 ↑o 1o) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∅c0 4326 Oncon0 6374 suc csuc 6376 (class class class)co 7426 ωcom 7876 1oc1o 8486 ·o comu 8491 ↑o coe 8492 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-oadd 8497 df-omul 8498 df-oexp 8499 |
This theorem is referenced by: omabs 8678 cnfcom3lem 9734 infxpenc2 10053 oege1 42766 oaomoencom 42777 oenassex 42778 omabs2 42792 oe2 42867 |
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