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Mirrors > Home > MPE Home > Th. List > ehl0base | Structured version Visualization version GIF version |
Description: The base of the Euclidean space of dimension 0 consists only of one element, the empty set. (Contributed by AV, 12-Feb-2023.) |
Ref | Expression |
---|---|
ehl0base.e | ⊢ 𝐸 = (𝔼hil‘0) |
Ref | Expression |
---|---|
ehl0base | ⊢ (Base‘𝐸) = {∅} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nn0 12539 | . . 3 ⊢ 0 ∈ ℕ0 | |
2 | ehl0base.e | . . . . 5 ⊢ 𝐸 = (𝔼hil‘0) | |
3 | 2 | ehlbase 25434 | . . . 4 ⊢ (0 ∈ ℕ0 → (ℝ ↑m (1...0)) = (Base‘𝐸)) |
4 | 3 | eqcomd 2732 | . . 3 ⊢ (0 ∈ ℕ0 → (Base‘𝐸) = (ℝ ↑m (1...0))) |
5 | 1, 4 | ax-mp 5 | . 2 ⊢ (Base‘𝐸) = (ℝ ↑m (1...0)) |
6 | fz10 13576 | . . 3 ⊢ (1...0) = ∅ | |
7 | 6 | oveq2i 7435 | . 2 ⊢ (ℝ ↑m (1...0)) = (ℝ ↑m ∅) |
8 | reex 11249 | . . 3 ⊢ ℝ ∈ V | |
9 | mapdm0 8871 | . . 3 ⊢ (ℝ ∈ V → (ℝ ↑m ∅) = {∅}) | |
10 | 8, 9 | ax-mp 5 | . 2 ⊢ (ℝ ↑m ∅) = {∅} |
11 | 5, 7, 10 | 3eqtri 2758 | 1 ⊢ (Base‘𝐸) = {∅} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 Vcvv 3462 ∅c0 4325 {csn 4633 ‘cfv 6554 (class class class)co 7424 ↑m cmap 8855 ℝcr 11157 0cc0 11158 1c1 11159 ℕ0cn0 12524 ...cfz 13538 Basecbs 17213 𝔼hilcehl 25403 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 ax-addf 11237 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-supp 8175 df-tpos 8241 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-map 8857 df-ixp 8927 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9406 df-sup 9485 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-uz 12875 df-rp 13029 df-fz 13539 df-seq 14022 df-exp 14082 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-starv 17281 df-sca 17282 df-vsca 17283 df-ip 17284 df-tset 17285 df-ple 17286 df-ds 17288 df-unif 17289 df-hom 17290 df-cco 17291 df-0g 17456 df-prds 17462 df-pws 17464 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-grp 18931 df-minusg 18932 df-subg 19117 df-cmn 19780 df-abl 19781 df-mgp 20118 df-rng 20136 df-ur 20165 df-ring 20218 df-cring 20219 df-oppr 20316 df-dvdsr 20339 df-unit 20340 df-invr 20370 df-dvr 20383 df-subrng 20528 df-subrg 20553 df-drng 20709 df-field 20710 df-sra 21151 df-rgmod 21152 df-cnfld 21344 df-refld 21601 df-dsmm 21730 df-frlm 21745 df-tng 24584 df-tcph 25188 df-rrx 25404 df-ehl 25405 |
This theorem is referenced by: ehl0 25436 |
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