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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 24783 | . 2 ⊢ (Base‘𝐸) = {∅} |
3 | ehl0base.0 | . . . . . 6 ⊢ 0 = (0g‘𝐸) | |
4 | ovex 7391 | . . . . . . 7 ⊢ (1...0) ∈ V | |
5 | 0nn0 12429 | . . . . . . . . 9 ⊢ 0 ∈ ℕ0 | |
6 | 1 | ehlval 24781 | . . . . . . . . 9 ⊢ (0 ∈ ℕ0 → 𝐸 = (ℝ^‘(1...0))) |
7 | 5, 6 | ax-mp 5 | . . . . . . . 8 ⊢ 𝐸 = (ℝ^‘(1...0)) |
8 | fz10 13463 | . . . . . . . . . 10 ⊢ (1...0) = ∅ | |
9 | 8 | xpeq1i 5660 | . . . . . . . . 9 ⊢ ((1...0) × {0}) = (∅ × {0}) |
10 | 9 | eqcomi 2746 | . . . . . . . 8 ⊢ (∅ × {0}) = ((1...0) × {0}) |
11 | 7, 10 | rrx0 24764 | . . . . . . 7 ⊢ ((1...0) ∈ V → (0g‘𝐸) = (∅ × {0})) |
12 | 4, 11 | ax-mp 5 | . . . . . 6 ⊢ (0g‘𝐸) = (∅ × {0}) |
13 | 3, 12 | eqtri 2765 | . . . . 5 ⊢ 0 = (∅ × {0}) |
14 | 0xp 5731 | . . . . 5 ⊢ (∅ × {0}) = ∅ | |
15 | 13, 14 | eqtri 2765 | . . . 4 ⊢ 0 = ∅ |
16 | 15 | eqcomi 2746 | . . 3 ⊢ ∅ = 0 |
17 | 16 | sneqi 4598 | . 2 ⊢ {∅} = { 0 } |
18 | 2, 17 | eqtri 2765 | 1 ⊢ (Base‘𝐸) = { 0 } |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 Vcvv 3446 ∅c0 4283 {csn 4587 × cxp 5632 ‘cfv 6497 (class class class)co 7358 0cc0 11052 1c1 11053 ℕ0cn0 12414 ...cfz 13425 Basecbs 17084 0gc0g 17322 ℝ^crrx 24750 𝔼hilcehl 24751 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-pre-sup 11130 ax-addf 11131 ax-mulf 11132 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-tpos 8158 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-map 8768 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9307 df-sup 9379 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-7 12222 df-8 12223 df-9 12224 df-n0 12415 df-z 12501 df-dec 12620 df-uz 12765 df-rp 12917 df-fz 13426 df-seq 13908 df-exp 13969 df-cj 14985 df-re 14986 df-im 14987 df-sqrt 15121 df-abs 15122 df-struct 17020 df-sets 17037 df-slot 17055 df-ndx 17067 df-base 17085 df-ress 17114 df-plusg 17147 df-mulr 17148 df-starv 17149 df-sca 17150 df-vsca 17151 df-ip 17152 df-tset 17153 df-ple 17154 df-ds 17156 df-unif 17157 df-hom 17158 df-cco 17159 df-0g 17324 df-prds 17330 df-pws 17332 df-mgm 18498 df-sgrp 18547 df-mnd 18558 df-grp 18752 df-minusg 18753 df-sbg 18754 df-subg 18926 df-cmn 19565 df-mgp 19898 df-ur 19915 df-ring 19967 df-cring 19968 df-oppr 20050 df-dvdsr 20071 df-unit 20072 df-invr 20102 df-dvr 20113 df-drng 20188 df-field 20189 df-subrg 20223 df-lmod 20327 df-lss 20396 df-sra 20636 df-rgmod 20637 df-cnfld 20800 df-refld 21012 df-dsmm 21141 df-frlm 21156 df-tng 23943 df-tcph 24536 df-rrx 24752 df-ehl 24753 |
This theorem is referenced by: (None) |
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