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Mirrors > Home > MPE Home > Th. List > Mathboxes > k0004ss2 | Structured version Visualization version GIF version |
Description: The topological simplex of dimension 𝑁 is a subset of the base set of a real vector space of dimension (𝑁 + 1). (Contributed by RP, 29-Mar-2021.) |
Ref | Expression |
---|---|
k0004.a | ⊢ 𝐴 = (𝑛 ∈ ℕ0 ↦ {𝑡 ∈ ((0[,]1) ↑𝑚 (1...(𝑛 + 1))) ∣ Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡‘𝑘) = 1}) |
Ref | Expression |
---|---|
k0004ss2 | ⊢ (𝑁 ∈ ℕ0 → (𝐴‘𝑁) ⊆ (Base‘(ℝ^‘(1...(𝑁 + 1))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | k0004.a | . . 3 ⊢ 𝐴 = (𝑛 ∈ ℕ0 ↦ {𝑡 ∈ ((0[,]1) ↑𝑚 (1...(𝑛 + 1))) ∣ Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡‘𝑘) = 1}) | |
2 | 1 | k0004ss1 37469 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝐴‘𝑁) ⊆ (ℝ ↑𝑚 (1...(𝑁 + 1)))) |
3 | ssid 3587 | . . . . 5 ⊢ (ℝ ↑𝑚 (1...(𝑁 + 1))) ⊆ (ℝ ↑𝑚 (1...(𝑁 + 1))) | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (ℝ ↑𝑚 (1...(𝑁 + 1))) ⊆ (ℝ ↑𝑚 (1...(𝑁 + 1)))) |
5 | elmapi 7765 | . . . . . 6 ⊢ (𝑣 ∈ (ℝ ↑𝑚 (1...(𝑁 + 1))) → 𝑣:(1...(𝑁 + 1))⟶ℝ) | |
6 | 5 | adantl 481 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑣 ∈ (ℝ ↑𝑚 (1...(𝑁 + 1)))) → 𝑣:(1...(𝑁 + 1))⟶ℝ) |
7 | fzfid 12634 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑣 ∈ (ℝ ↑𝑚 (1...(𝑁 + 1)))) → (1...(𝑁 + 1)) ∈ Fin) | |
8 | 0red 9920 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑣 ∈ (ℝ ↑𝑚 (1...(𝑁 + 1)))) → 0 ∈ ℝ) | |
9 | 6, 7, 8 | fdmfifsupp 8168 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑣 ∈ (ℝ ↑𝑚 (1...(𝑁 + 1)))) → 𝑣 finSupp 0) |
10 | 4, 9 | ssrabdv 3644 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (ℝ ↑𝑚 (1...(𝑁 + 1))) ⊆ {𝑣 ∈ (ℝ ↑𝑚 (1...(𝑁 + 1))) ∣ 𝑣 finSupp 0}) |
11 | ovex 6577 | . . . 4 ⊢ (1...(𝑁 + 1)) ∈ V | |
12 | eqid 2610 | . . . . 5 ⊢ (ℝ^‘(1...(𝑁 + 1))) = (ℝ^‘(1...(𝑁 + 1))) | |
13 | eqid 2610 | . . . . 5 ⊢ (Base‘(ℝ^‘(1...(𝑁 + 1)))) = (Base‘(ℝ^‘(1...(𝑁 + 1)))) | |
14 | 12, 13 | rrxbase 22984 | . . . 4 ⊢ ((1...(𝑁 + 1)) ∈ V → (Base‘(ℝ^‘(1...(𝑁 + 1)))) = {𝑣 ∈ (ℝ ↑𝑚 (1...(𝑁 + 1))) ∣ 𝑣 finSupp 0}) |
15 | 11, 14 | ax-mp 5 | . . 3 ⊢ (Base‘(ℝ^‘(1...(𝑁 + 1)))) = {𝑣 ∈ (ℝ ↑𝑚 (1...(𝑁 + 1))) ∣ 𝑣 finSupp 0} |
16 | 10, 15 | syl6sseqr 3615 | . 2 ⊢ (𝑁 ∈ ℕ0 → (ℝ ↑𝑚 (1...(𝑁 + 1))) ⊆ (Base‘(ℝ^‘(1...(𝑁 + 1))))) |
17 | 2, 16 | sstrd 3578 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝐴‘𝑁) ⊆ (Base‘(ℝ^‘(1...(𝑁 + 1))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 ⊆ wss 3540 class class class wbr 4583 ↦ cmpt 4643 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 finSupp cfsupp 8158 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 ℕ0cn0 11169 [,]cicc 12049 ...cfz 12197 Σcsu 14264 Basecbs 15695 ℝ^crrx 22979 |
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-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 ax-addf 9894 ax-mulf 9895 |
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-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-supp 7183 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 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-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-icc 12053 df-fz 12198 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-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-0g 15925 df-prds 15931 df-pws 15933 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-subg 17414 df-cmn 18018 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-drng 18572 df-field 18573 df-subrg 18601 df-sra 18993 df-rgmod 18994 df-cnfld 19568 df-refld 19770 df-dsmm 19895 df-frlm 19910 df-tng 22199 df-tch 22777 df-rrx 22981 |
This theorem is referenced by: (None) |
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