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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. (Contributed by NM, 2-Jan-2005.) |
Ref | Expression |
---|---|
oe1 | ⊢ (𝐴 ∈ On → (𝐴 ↑o 1o) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-1o 8105 | . . . 4 ⊢ 1o = suc ∅ | |
2 | 1 | oveq2i 7170 | . . 3 ⊢ (𝐴 ↑o 1o) = (𝐴 ↑o suc ∅) |
3 | peano1 7604 | . . . 4 ⊢ ∅ ∈ ω | |
4 | onesuc 8158 | . . . 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 | syl5eq 2871 | . 2 ⊢ (𝐴 ∈ On → (𝐴 ↑o 1o) = ((𝐴 ↑o ∅) ·o 𝐴)) |
7 | oe0 8150 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o) | |
8 | 7 | oveq1d 7174 | . 2 ⊢ (𝐴 ∈ On → ((𝐴 ↑o ∅) ·o 𝐴) = (1o ·o 𝐴)) |
9 | om1r 8172 | . 2 ⊢ (𝐴 ∈ On → (1o ·o 𝐴) = 𝐴) | |
10 | 6, 8, 9 | 3eqtrd 2863 | 1 ⊢ (𝐴 ∈ On → (𝐴 ↑o 1o) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ∅c0 4294 Oncon0 6194 suc csuc 6196 (class class class)co 7159 ωcom 7583 1oc1o 8098 ·o comu 8103 ↑o coe 8104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-omul 8110 df-oexp 8111 |
This theorem is referenced by: omabs 8277 cnfcom3lem 9169 infxpenc2 9451 |
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