![]() |
Mathbox for ML |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > finxp3o | Structured version Visualization version GIF version |
Description: The value of Cartesian exponentiation at three. (Contributed by ML, 24-Oct-2020.) |
Ref | Expression |
---|---|
finxp3o | ⊢ (𝑈↑↑3o) = ((𝑈 × 𝑈) × 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3o 7828 | . . 3 ⊢ 3o = suc 2o | |
2 | finxpeq2 33762 | . . 3 ⊢ (3o = suc 2o → (𝑈↑↑3o) = (𝑈↑↑suc 2o)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (𝑈↑↑3o) = (𝑈↑↑suc 2o) |
4 | 2onn 7987 | . . 3 ⊢ 2o ∈ ω | |
5 | 2on0 7836 | . . 3 ⊢ 2o ≠ ∅ | |
6 | finxpsuc 33773 | . . 3 ⊢ ((2o ∈ ω ∧ 2o ≠ ∅) → (𝑈↑↑suc 2o) = ((𝑈↑↑2o) × 𝑈)) | |
7 | 4, 5, 6 | mp2an 683 | . 2 ⊢ (𝑈↑↑suc 2o) = ((𝑈↑↑2o) × 𝑈) |
8 | finxp2o 33774 | . . 3 ⊢ (𝑈↑↑2o) = (𝑈 × 𝑈) | |
9 | 8 | xpeq1i 5368 | . 2 ⊢ ((𝑈↑↑2o) × 𝑈) = ((𝑈 × 𝑈) × 𝑈) |
10 | 3, 7, 9 | 3eqtri 2853 | 1 ⊢ (𝑈↑↑3o) = ((𝑈 × 𝑈) × 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1656 ∈ wcel 2164 ≠ wne 2999 ∅c0 4144 × cxp 5340 suc csuc 5965 ωcom 7326 2oc2o 7820 3oc3o 7821 ↑↑cfinxp 33758 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-2o 7827 df-3o 7828 df-oadd 7830 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-finxp 33759 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |