![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > hilid | Structured version Visualization version GIF version |
Description: The group identity element of Hilbert space vector addition is the zero vector. (Contributed by NM, 16-Apr-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hilid | ⊢ (GId‘ +ℎ ) = 0ℎ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hilablo 28620 | . . . 4 ⊢ +ℎ ∈ AbelOp | |
2 | ablogrpo 28007 | . . . 4 ⊢ ( +ℎ ∈ AbelOp → +ℎ ∈ GrpOp) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ +ℎ ∈ GrpOp |
4 | ax-hfvadd 28460 | . . . . . 6 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | |
5 | 4 | fdmi 6395 | . . . . 5 ⊢ dom +ℎ = ( ℋ × ℋ) |
6 | 3, 5 | grporn 27981 | . . . 4 ⊢ ℋ = ran +ℎ |
7 | eqid 2794 | . . . 4 ⊢ (GId‘ +ℎ ) = (GId‘ +ℎ ) | |
8 | 6, 7 | grpoidval 27973 | . . 3 ⊢ ( +ℎ ∈ GrpOp → (GId‘ +ℎ ) = (℩𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 +ℎ 𝑥) = 𝑥)) |
9 | 3, 8 | ax-mp 5 | . 2 ⊢ (GId‘ +ℎ ) = (℩𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 +ℎ 𝑥) = 𝑥) |
10 | hvaddid2 28483 | . . . 4 ⊢ (𝑥 ∈ ℋ → (0ℎ +ℎ 𝑥) = 𝑥) | |
11 | 10 | rgen 3114 | . . 3 ⊢ ∀𝑥 ∈ ℋ (0ℎ +ℎ 𝑥) = 𝑥 |
12 | ax-hv0cl 28463 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
13 | 6 | grpoideu 27969 | . . . . 5 ⊢ ( +ℎ ∈ GrpOp → ∃!𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 +ℎ 𝑥) = 𝑥) |
14 | 3, 13 | ax-mp 5 | . . . 4 ⊢ ∃!𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 +ℎ 𝑥) = 𝑥 |
15 | oveq1 7026 | . . . . . . 7 ⊢ (𝑦 = 0ℎ → (𝑦 +ℎ 𝑥) = (0ℎ +ℎ 𝑥)) | |
16 | 15 | eqeq1d 2796 | . . . . . 6 ⊢ (𝑦 = 0ℎ → ((𝑦 +ℎ 𝑥) = 𝑥 ↔ (0ℎ +ℎ 𝑥) = 𝑥)) |
17 | 16 | ralbidv 3163 | . . . . 5 ⊢ (𝑦 = 0ℎ → (∀𝑥 ∈ ℋ (𝑦 +ℎ 𝑥) = 𝑥 ↔ ∀𝑥 ∈ ℋ (0ℎ +ℎ 𝑥) = 𝑥)) |
18 | 17 | riota2 7002 | . . . 4 ⊢ ((0ℎ ∈ ℋ ∧ ∃!𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 +ℎ 𝑥) = 𝑥) → (∀𝑥 ∈ ℋ (0ℎ +ℎ 𝑥) = 𝑥 ↔ (℩𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 +ℎ 𝑥) = 𝑥) = 0ℎ)) |
19 | 12, 14, 18 | mp2an 688 | . . 3 ⊢ (∀𝑥 ∈ ℋ (0ℎ +ℎ 𝑥) = 𝑥 ↔ (℩𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 +ℎ 𝑥) = 𝑥) = 0ℎ) |
20 | 11, 19 | mpbi 231 | . 2 ⊢ (℩𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 +ℎ 𝑥) = 𝑥) = 0ℎ |
21 | 9, 20 | eqtri 2818 | 1 ⊢ (GId‘ +ℎ ) = 0ℎ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1522 ∈ wcel 2080 ∀wral 3104 ∃!wreu 3106 × cxp 5444 ‘cfv 6228 ℩crio 6979 (class class class)co 7019 GrpOpcgr 27949 GIdcgi 27950 AbelOpcablo 28004 ℋchba 28379 +ℎ cva 28380 0ℎc0v 28384 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-rep 5084 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 ax-resscn 10443 ax-1cn 10444 ax-icn 10445 ax-addcl 10446 ax-addrcl 10447 ax-mulcl 10448 ax-mulrcl 10449 ax-mulcom 10450 ax-addass 10451 ax-mulass 10452 ax-distr 10453 ax-i2m1 10454 ax-1ne0 10455 ax-1rid 10456 ax-rnegex 10457 ax-rrecex 10458 ax-cnre 10459 ax-pre-lttri 10460 ax-pre-lttrn 10461 ax-pre-ltadd 10462 ax-hilex 28459 ax-hfvadd 28460 ax-hvcom 28461 ax-hvass 28462 ax-hv0cl 28463 ax-hvaddid 28464 ax-hfvmul 28465 ax-hvmulid 28466 ax-hvdistr2 28469 ax-hvmul0 28470 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-nel 3090 df-ral 3109 df-rex 3110 df-reu 3111 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-op 4481 df-uni 4748 df-iun 4829 df-br 4965 df-opab 5027 df-mpt 5044 df-id 5351 df-po 5365 df-so 5366 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-f1 6233 df-fo 6234 df-f1o 6235 df-fv 6236 df-riota 6980 df-ov 7022 df-oprab 7023 df-mpo 7024 df-er 8142 df-en 8361 df-dom 8362 df-sdom 8363 df-pnf 10526 df-mnf 10527 df-ltxr 10529 df-sub 10721 df-neg 10722 df-grpo 27953 df-gid 27954 df-ablo 28005 df-hvsub 28431 |
This theorem is referenced by: hhnv 28625 hh0v 28628 hhssabloilem 28721 |
Copyright terms: Public domain | W3C validator |