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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 8469 | . . 3 ⊢ 3o = suc 2o | |
2 | finxpeq2 36775 | . . 3 ⊢ (3o = suc 2o → (𝑈↑↑3o) = (𝑈↑↑suc 2o)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (𝑈↑↑3o) = (𝑈↑↑suc 2o) |
4 | 2onn 8643 | . . 3 ⊢ 2o ∈ ω | |
5 | 2on0 8483 | . . 3 ⊢ 2o ≠ ∅ | |
6 | finxpsuc 36786 | . . 3 ⊢ ((2o ∈ ω ∧ 2o ≠ ∅) → (𝑈↑↑suc 2o) = ((𝑈↑↑2o) × 𝑈)) | |
7 | 4, 5, 6 | mp2an 689 | . 2 ⊢ (𝑈↑↑suc 2o) = ((𝑈↑↑2o) × 𝑈) |
8 | finxp2o 36787 | . . 3 ⊢ (𝑈↑↑2o) = (𝑈 × 𝑈) | |
9 | 8 | xpeq1i 5695 | . 2 ⊢ ((𝑈↑↑2o) × 𝑈) = ((𝑈 × 𝑈) × 𝑈) |
10 | 3, 7, 9 | 3eqtri 2758 | 1 ⊢ (𝑈↑↑3o) = ((𝑈 × 𝑈) × 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∅c0 4317 × cxp 5667 suc csuc 6360 ωcom 7852 2oc2o 8461 3oc3o 8462 ↑↑cfinxp 36771 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-3o 8469 df-oadd 8471 df-en 8942 df-fin 8945 df-finxp 36772 |
This theorem is referenced by: (None) |
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