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Mirrors > Home > HSE Home > Th. List > h1did | Structured version Visualization version GIF version |
Description: A generating vector belongs to the 1-dimensional subspace it generates. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.) |
Ref | Expression |
---|---|
h1did | ⊢ (𝐴 ∈ ℋ → 𝐴 ∈ (⊥‘(⊥‘{𝐴}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snssi 4773 | . . 3 ⊢ (𝐴 ∈ ℋ → {𝐴} ⊆ ℋ) | |
2 | ococss 30277 | . . 3 ⊢ ({𝐴} ⊆ ℋ → {𝐴} ⊆ (⊥‘(⊥‘{𝐴}))) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ ℋ → {𝐴} ⊆ (⊥‘(⊥‘{𝐴}))) |
4 | snssg 4749 | . 2 ⊢ (𝐴 ∈ ℋ → (𝐴 ∈ (⊥‘(⊥‘{𝐴})) ↔ {𝐴} ⊆ (⊥‘(⊥‘{𝐴})))) | |
5 | 3, 4 | mpbird 257 | 1 ⊢ (𝐴 ∈ ℋ → 𝐴 ∈ (⊥‘(⊥‘{𝐴}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ⊆ wss 3915 {csn 4591 ‘cfv 6501 ℋchba 29903 ⊥cort 29914 |
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-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-hilex 29983 ax-hfvadd 29984 ax-hv0cl 29987 ax-hfvmul 29989 ax-hvmul0 29994 ax-hfi 30063 ax-his1 30066 ax-his2 30067 ax-his3 30068 |
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 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-po 5550 df-so 5551 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-2 12223 df-cj 14991 df-re 14992 df-im 14993 df-sh 30191 df-oc 30236 |
This theorem is referenced by: h1dn0 30536 h1de2bi 30538 h1de2ctlem 30539 spansnid 30547 |
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