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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 7449 | . . 3 ⊢ 3𝑜 = suc 2𝑜 | |
2 | finxpeq2 32400 | . . 3 ⊢ (3𝑜 = suc 2𝑜 → (𝑈↑↑3𝑜) = (𝑈↑↑suc 2𝑜)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (𝑈↑↑3𝑜) = (𝑈↑↑suc 2𝑜) |
4 | 2onn 7607 | . . 3 ⊢ 2𝑜 ∈ ω | |
5 | 2on0 7456 | . . 3 ⊢ 2𝑜 ≠ ∅ | |
6 | finxpsuc 32411 | . . 3 ⊢ ((2𝑜 ∈ ω ∧ 2𝑜 ≠ ∅) → (𝑈↑↑suc 2𝑜) = ((𝑈↑↑2𝑜) × 𝑈)) | |
7 | 4, 5, 6 | mp2an 704 | . 2 ⊢ (𝑈↑↑suc 2𝑜) = ((𝑈↑↑2𝑜) × 𝑈) |
8 | finxp2o 32412 | . . 3 ⊢ (𝑈↑↑2𝑜) = (𝑈 × 𝑈) | |
9 | 8 | xpeq1i 5059 | . 2 ⊢ ((𝑈↑↑2𝑜) × 𝑈) = ((𝑈 × 𝑈) × 𝑈) |
10 | 3, 7, 9 | 3eqtri 2636 | 1 ⊢ (𝑈↑↑3𝑜) = ((𝑈 × 𝑈) × 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 × cxp 5036 suc csuc 5642 ωcom 6957 2𝑜c2o 7441 3𝑜c3o 7442 ↑↑cfinxp 32396 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-reg 8380 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-3o 7449 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-finxp 32397 |
This theorem is referenced by: (None) |
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