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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 40011 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝐴‘𝑁) ⊆ (ℝ ↑𝑚 (1...(𝑁 + 1)))) |
3 | ssidd 3915 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (ℝ ↑𝑚 (1...(𝑁 + 1))) ⊆ (ℝ ↑𝑚 (1...(𝑁 + 1)))) | |
4 | elmapi 8283 | . . . . . 6 ⊢ (𝑣 ∈ (ℝ ↑𝑚 (1...(𝑁 + 1))) → 𝑣:(1...(𝑁 + 1))⟶ℝ) | |
5 | 4 | adantl 482 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑣 ∈ (ℝ ↑𝑚 (1...(𝑁 + 1)))) → 𝑣:(1...(𝑁 + 1))⟶ℝ) |
6 | fzfid 13196 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑣 ∈ (ℝ ↑𝑚 (1...(𝑁 + 1)))) → (1...(𝑁 + 1)) ∈ Fin) | |
7 | 0red 10495 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑣 ∈ (ℝ ↑𝑚 (1...(𝑁 + 1)))) → 0 ∈ ℝ) | |
8 | 5, 6, 7 | fdmfifsupp 8694 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑣 ∈ (ℝ ↑𝑚 (1...(𝑁 + 1)))) → 𝑣 finSupp 0) |
9 | 3, 8 | ssrabdv 3975 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (ℝ ↑𝑚 (1...(𝑁 + 1))) ⊆ {𝑣 ∈ (ℝ ↑𝑚 (1...(𝑁 + 1))) ∣ 𝑣 finSupp 0}) |
10 | ovex 7053 | . . . 4 ⊢ (1...(𝑁 + 1)) ∈ V | |
11 | eqid 2795 | . . . . 5 ⊢ (ℝ^‘(1...(𝑁 + 1))) = (ℝ^‘(1...(𝑁 + 1))) | |
12 | eqid 2795 | . . . . 5 ⊢ (Base‘(ℝ^‘(1...(𝑁 + 1)))) = (Base‘(ℝ^‘(1...(𝑁 + 1)))) | |
13 | 11, 12 | rrxbase 23679 | . . . 4 ⊢ ((1...(𝑁 + 1)) ∈ V → (Base‘(ℝ^‘(1...(𝑁 + 1)))) = {𝑣 ∈ (ℝ ↑𝑚 (1...(𝑁 + 1))) ∣ 𝑣 finSupp 0}) |
14 | 10, 13 | ax-mp 5 | . . 3 ⊢ (Base‘(ℝ^‘(1...(𝑁 + 1)))) = {𝑣 ∈ (ℝ ↑𝑚 (1...(𝑁 + 1))) ∣ 𝑣 finSupp 0} |
15 | 9, 14 | syl6sseqr 3943 | . 2 ⊢ (𝑁 ∈ ℕ0 → (ℝ ↑𝑚 (1...(𝑁 + 1))) ⊆ (Base‘(ℝ^‘(1...(𝑁 + 1))))) |
16 | 2, 15 | sstrd 3903 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝐴‘𝑁) ⊆ (Base‘(ℝ^‘(1...(𝑁 + 1))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 {crab 3109 Vcvv 3437 ⊆ wss 3863 class class class wbr 4966 ↦ cmpt 5045 ⟶wf 6226 ‘cfv 6230 (class class class)co 7021 ↑𝑚 cmap 8261 finSupp cfsupp 8684 ℝcr 10387 0cc0 10388 1c1 10389 + caddc 10391 ℕ0cn0 11750 [,]cicc 12596 ...cfz 12747 Σcsu 14881 Basecbs 16317 ℝ^crrx 23674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-cnex 10444 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465 ax-pre-sup 10466 ax-addf 10467 ax-mulf 10468 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-int 4787 df-iun 4831 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-om 7442 df-1st 7550 df-2nd 7551 df-supp 7687 df-tpos 7748 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-1o 7958 df-oadd 7962 df-er 8144 df-map 8263 df-ixp 8316 df-en 8363 df-dom 8364 df-sdom 8365 df-fin 8366 df-fsupp 8685 df-sup 8757 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-div 11151 df-nn 11492 df-2 11553 df-3 11554 df-4 11555 df-5 11556 df-6 11557 df-7 11558 df-8 11559 df-9 11560 df-n0 11751 df-z 11835 df-dec 11953 df-uz 12099 df-rp 12245 df-icc 12600 df-fz 12748 df-seq 13225 df-exp 13285 df-cj 14297 df-re 14298 df-im 14299 df-sqrt 14433 df-abs 14434 df-sum 14882 df-struct 16319 df-ndx 16320 df-slot 16321 df-base 16323 df-sets 16324 df-ress 16325 df-plusg 16412 df-mulr 16413 df-starv 16414 df-sca 16415 df-vsca 16416 df-ip 16417 df-tset 16418 df-ple 16419 df-ds 16421 df-unif 16422 df-hom 16423 df-cco 16424 df-0g 16549 df-prds 16555 df-pws 16557 df-mgm 17686 df-sgrp 17728 df-mnd 17739 df-grp 17869 df-minusg 17870 df-subg 18035 df-cmn 18640 df-mgp 18935 df-ur 18947 df-ring 18994 df-cring 18995 df-oppr 19068 df-dvdsr 19086 df-unit 19087 df-invr 19117 df-dvr 19128 df-drng 19199 df-field 19200 df-subrg 19228 df-sra 19639 df-rgmod 19640 df-cnfld 20233 df-refld 20436 df-dsmm 20563 df-frlm 20578 df-tng 22882 df-tcph 23461 df-rrx 23676 |
This theorem is referenced by: (None) |
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