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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 24012 | . 2 ⊢ (Base‘𝐸) = {∅} |
3 | ehl0base.0 | . . . . . 6 ⊢ 0 = (0g‘𝐸) | |
4 | ovex 7182 | . . . . . . 7 ⊢ (1...0) ∈ V | |
5 | 0nn0 11906 | . . . . . . . . 9 ⊢ 0 ∈ ℕ0 | |
6 | 1 | ehlval 24010 | . . . . . . . . 9 ⊢ (0 ∈ ℕ0 → 𝐸 = (ℝ^‘(1...0))) |
7 | 5, 6 | ax-mp 5 | . . . . . . . 8 ⊢ 𝐸 = (ℝ^‘(1...0)) |
8 | fz10 12925 | . . . . . . . . . 10 ⊢ (1...0) = ∅ | |
9 | 8 | xpeq1i 5574 | . . . . . . . . 9 ⊢ ((1...0) × {0}) = (∅ × {0}) |
10 | 9 | eqcomi 2829 | . . . . . . . 8 ⊢ (∅ × {0}) = ((1...0) × {0}) |
11 | 7, 10 | rrx0 23993 | . . . . . . 7 ⊢ ((1...0) ∈ V → (0g‘𝐸) = (∅ × {0})) |
12 | 4, 11 | ax-mp 5 | . . . . . 6 ⊢ (0g‘𝐸) = (∅ × {0}) |
13 | 3, 12 | eqtri 2843 | . . . . 5 ⊢ 0 = (∅ × {0}) |
14 | 0xp 5642 | . . . . 5 ⊢ (∅ × {0}) = ∅ | |
15 | 13, 14 | eqtri 2843 | . . . 4 ⊢ 0 = ∅ |
16 | 15 | eqcomi 2829 | . . 3 ⊢ ∅ = 0 |
17 | 16 | sneqi 4571 | . 2 ⊢ {∅} = { 0 } |
18 | 2, 17 | eqtri 2843 | 1 ⊢ (Base‘𝐸) = { 0 } |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2113 Vcvv 3491 ∅c0 4284 {csn 4560 × cxp 5546 ‘cfv 6348 (class class class)co 7149 0cc0 10530 1c1 10531 ℕ0cn0 11891 ...cfz 12889 Basecbs 16476 0gc0g 16706 ℝ^crrx 23979 𝔼hilcehl 23980 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 ax-addf 10609 ax-mulf 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-supp 7824 df-tpos 7885 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-ixp 8455 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-fsupp 8827 df-sup 8899 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-rp 12384 df-fz 12890 df-seq 13367 df-exp 13427 df-cj 14451 df-re 14452 df-im 14453 df-sqrt 14587 df-abs 14588 df-struct 16478 df-ndx 16479 df-slot 16480 df-base 16482 df-sets 16483 df-ress 16484 df-plusg 16571 df-mulr 16572 df-starv 16573 df-sca 16574 df-vsca 16575 df-ip 16576 df-tset 16577 df-ple 16578 df-ds 16580 df-unif 16581 df-hom 16582 df-cco 16583 df-0g 16708 df-prds 16714 df-pws 16716 df-mgm 17845 df-sgrp 17894 df-mnd 17905 df-grp 18099 df-minusg 18100 df-sbg 18101 df-subg 18269 df-cmn 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19366 df-dvdsr 19384 df-unit 19385 df-invr 19415 df-dvr 19426 df-drng 19497 df-field 19498 df-subrg 19526 df-lmod 19629 df-lss 19697 df-sra 19937 df-rgmod 19938 df-cnfld 20539 df-refld 20742 df-dsmm 20869 df-frlm 20884 df-tng 23187 df-tcph 23766 df-rrx 23981 df-ehl 23982 |
This theorem is referenced by: (None) |
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