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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrnval | Structured version Visualization version GIF version |
Description: The n-dimensional Euclidean space. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.) |
Ref | Expression |
---|---|
rrnval.1 | ⊢ 𝑋 = (ℝ ↑𝑚 𝐼) |
Ref | Expression |
---|---|
rrnval | ⊢ (𝐼 ∈ Fin → (ℝn‘𝐼) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 6557 | . . . 4 ⊢ (𝑖 = 𝐼 → (ℝ ↑𝑚 𝑖) = (ℝ ↑𝑚 𝐼)) | |
2 | rrnval.1 | . . . 4 ⊢ 𝑋 = (ℝ ↑𝑚 𝐼) | |
3 | 1, 2 | syl6eqr 2662 | . . 3 ⊢ (𝑖 = 𝐼 → (ℝ ↑𝑚 𝑖) = 𝑋) |
4 | sumeq1 14267 | . . . 4 ⊢ (𝑖 = 𝐼 → Σ𝑘 ∈ 𝑖 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2) = Σ𝑘 ∈ 𝐼 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2)) | |
5 | 4 | fveq2d 6107 | . . 3 ⊢ (𝑖 = 𝐼 → (√‘Σ𝑘 ∈ 𝑖 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2)) = (√‘Σ𝑘 ∈ 𝐼 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2))) |
6 | 3, 3, 5 | mpt2eq123dv 6615 | . 2 ⊢ (𝑖 = 𝐼 → (𝑥 ∈ (ℝ ↑𝑚 𝑖), 𝑦 ∈ (ℝ ↑𝑚 𝑖) ↦ (√‘Σ𝑘 ∈ 𝑖 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2))) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2)))) |
7 | df-rrn 32795 | . 2 ⊢ ℝn = (𝑖 ∈ Fin ↦ (𝑥 ∈ (ℝ ↑𝑚 𝑖), 𝑦 ∈ (ℝ ↑𝑚 𝑖) ↦ (√‘Σ𝑘 ∈ 𝑖 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2)))) | |
8 | fvrn0 6126 | . . . . 5 ⊢ (√‘Σ𝑘 ∈ 𝐼 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2)) ∈ (ran √ ∪ {∅}) | |
9 | 8 | rgen2w 2909 | . . . 4 ⊢ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (√‘Σ𝑘 ∈ 𝐼 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2)) ∈ (ran √ ∪ {∅}) |
10 | eqid 2610 | . . . . 5 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2))) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2))) | |
11 | 10 | fmpt2 7126 | . . . 4 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (√‘Σ𝑘 ∈ 𝐼 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2)) ∈ (ran √ ∪ {∅}) ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2))):(𝑋 × 𝑋)⟶(ran √ ∪ {∅})) |
12 | 9, 11 | mpbi 219 | . . 3 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2))):(𝑋 × 𝑋)⟶(ran √ ∪ {∅}) |
13 | ovex 6577 | . . . . 5 ⊢ (ℝ ↑𝑚 𝐼) ∈ V | |
14 | 2, 13 | eqeltri 2684 | . . . 4 ⊢ 𝑋 ∈ V |
15 | 14, 14 | xpex 6860 | . . 3 ⊢ (𝑋 × 𝑋) ∈ V |
16 | cnex 9896 | . . . . 5 ⊢ ℂ ∈ V | |
17 | sqrtf 13951 | . . . . . 6 ⊢ √:ℂ⟶ℂ | |
18 | frn 5966 | . . . . . 6 ⊢ (√:ℂ⟶ℂ → ran √ ⊆ ℂ) | |
19 | 17, 18 | ax-mp 5 | . . . . 5 ⊢ ran √ ⊆ ℂ |
20 | 16, 19 | ssexi 4731 | . . . 4 ⊢ ran √ ∈ V |
21 | p0ex 4779 | . . . 4 ⊢ {∅} ∈ V | |
22 | 20, 21 | unex 6854 | . . 3 ⊢ (ran √ ∪ {∅}) ∈ V |
23 | fex2 7014 | . . 3 ⊢ (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2))):(𝑋 × 𝑋)⟶(ran √ ∪ {∅}) ∧ (𝑋 × 𝑋) ∈ V ∧ (ran √ ∪ {∅}) ∈ V) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2))) ∈ V) | |
24 | 12, 15, 22, 23 | mp3an 1416 | . 2 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2))) ∈ V |
25 | 6, 7, 24 | fvmpt 6191 | 1 ⊢ (𝐼 ∈ Fin → (ℝn‘𝐼) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ∪ cun 3538 ⊆ wss 3540 ∅c0 3874 {csn 4125 × cxp 5036 ran crn 5039 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ↑𝑚 cmap 7744 Fincfn 7841 ℂcc 9813 ℝcr 9814 − cmin 10145 2c2 10947 ↑cexp 12722 √csqrt 13821 Σcsu 14264 ℝncrrn 32794 |
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-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 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-nel 2783 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-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-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-sum 14265 df-rrn 32795 |
This theorem is referenced by: rrnmval 32797 rrnmet 32798 |
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