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Mirrors > Home > MPE Home > Th. List > ehl0 | Structured version Visualization version GIF version |
Description: The Euclidean space of dimension 0 consists of the neutral element only. (Contributed by AV, 12-Feb-2023.) |
Ref | Expression |
---|---|
ehl0base.e | ⊢ 𝐸 = (𝔼hil‘0) |
ehl0base.0 | ⊢ 0 = (0g‘𝐸) |
Ref | Expression |
---|---|
ehl0 | ⊢ (Base‘𝐸) = { 0 } |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ehl0base.e | . . 3 ⊢ 𝐸 = (𝔼hil‘0) | |
2 | 1 | ehl0base 24924 | . 2 ⊢ (Base‘𝐸) = {∅} |
3 | ehl0base.0 | . . . . . 6 ⊢ 0 = (0g‘𝐸) | |
4 | ovex 7438 | . . . . . . 7 ⊢ (1...0) ∈ V | |
5 | 0nn0 12483 | . . . . . . . . 9 ⊢ 0 ∈ ℕ0 | |
6 | 1 | ehlval 24922 | . . . . . . . . 9 ⊢ (0 ∈ ℕ0 → 𝐸 = (ℝ^‘(1...0))) |
7 | 5, 6 | ax-mp 5 | . . . . . . . 8 ⊢ 𝐸 = (ℝ^‘(1...0)) |
8 | fz10 13518 | . . . . . . . . . 10 ⊢ (1...0) = ∅ | |
9 | 8 | xpeq1i 5701 | . . . . . . . . 9 ⊢ ((1...0) × {0}) = (∅ × {0}) |
10 | 9 | eqcomi 2741 | . . . . . . . 8 ⊢ (∅ × {0}) = ((1...0) × {0}) |
11 | 7, 10 | rrx0 24905 | . . . . . . 7 ⊢ ((1...0) ∈ V → (0g‘𝐸) = (∅ × {0})) |
12 | 4, 11 | ax-mp 5 | . . . . . 6 ⊢ (0g‘𝐸) = (∅ × {0}) |
13 | 3, 12 | eqtri 2760 | . . . . 5 ⊢ 0 = (∅ × {0}) |
14 | 0xp 5772 | . . . . 5 ⊢ (∅ × {0}) = ∅ | |
15 | 13, 14 | eqtri 2760 | . . . 4 ⊢ 0 = ∅ |
16 | 15 | eqcomi 2741 | . . 3 ⊢ ∅ = 0 |
17 | 16 | sneqi 4638 | . 2 ⊢ {∅} = { 0 } |
18 | 2, 17 | eqtri 2760 | 1 ⊢ (Base‘𝐸) = { 0 } |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∅c0 4321 {csn 4627 × cxp 5673 ‘cfv 6540 (class class class)co 7405 0cc0 11106 1c1 11107 ℕ0cn0 12468 ...cfz 13480 Basecbs 17140 0gc0g 17381 ℝ^crrx 24891 𝔼hilcehl 24892 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-rp 12971 df-fz 13481 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-0g 17383 df-prds 17389 df-pws 17391 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-sbg 18820 df-subg 18997 df-cmn 19644 df-mgp 19982 df-ur 19999 df-ring 20051 df-cring 20052 df-oppr 20142 df-dvdsr 20163 df-unit 20164 df-invr 20194 df-dvr 20207 df-drng 20309 df-field 20310 df-subrg 20353 df-lmod 20465 df-lss 20535 df-sra 20777 df-rgmod 20778 df-cnfld 20937 df-refld 21149 df-dsmm 21278 df-frlm 21293 df-tng 24084 df-tcph 24677 df-rrx 24893 df-ehl 24894 |
This theorem is referenced by: (None) |
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