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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 29888 | . . . 4 ⊢ +ℎ ∈ AbelOp | |
2 | ablogrpo 29275 | . . . 4 ⊢ ( +ℎ ∈ AbelOp → +ℎ ∈ GrpOp) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ +ℎ ∈ GrpOp |
4 | ax-hfvadd 29728 | . . . . . 6 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | |
5 | 4 | fdmi 6676 | . . . . 5 ⊢ dom +ℎ = ( ℋ × ℋ) |
6 | 3, 5 | grporn 29249 | . . . 4 ⊢ ℋ = ran +ℎ |
7 | eqid 2738 | . . . 4 ⊢ (GId‘ +ℎ ) = (GId‘ +ℎ ) | |
8 | 6, 7 | grpoidval 29241 | . . 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 29751 | . . . 4 ⊢ (𝑥 ∈ ℋ → (0ℎ +ℎ 𝑥) = 𝑥) | |
11 | 10 | rgen 3065 | . . 3 ⊢ ∀𝑥 ∈ ℋ (0ℎ +ℎ 𝑥) = 𝑥 |
12 | ax-hv0cl 29731 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
13 | 6 | grpoideu 29237 | . . . . 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 7357 | . . . . . . 7 ⊢ (𝑦 = 0ℎ → (𝑦 +ℎ 𝑥) = (0ℎ +ℎ 𝑥)) | |
16 | 15 | eqeq1d 2740 | . . . . . 6 ⊢ (𝑦 = 0ℎ → ((𝑦 +ℎ 𝑥) = 𝑥 ↔ (0ℎ +ℎ 𝑥) = 𝑥)) |
17 | 16 | ralbidv 3173 | . . . . 5 ⊢ (𝑦 = 0ℎ → (∀𝑥 ∈ ℋ (𝑦 +ℎ 𝑥) = 𝑥 ↔ ∀𝑥 ∈ ℋ (0ℎ +ℎ 𝑥) = 𝑥)) |
18 | 17 | riota2 7332 | . . . 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 691 | . . 3 ⊢ (∀𝑥 ∈ ℋ (0ℎ +ℎ 𝑥) = 𝑥 ↔ (℩𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 +ℎ 𝑥) = 𝑥) = 0ℎ) |
20 | 11, 19 | mpbi 229 | . 2 ⊢ (℩𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 +ℎ 𝑥) = 𝑥) = 0ℎ |
21 | 9, 20 | eqtri 2766 | 1 ⊢ (GId‘ +ℎ ) = 0ℎ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1542 ∈ wcel 2107 ∀wral 3063 ∃!wreu 3350 × cxp 5629 ‘cfv 6492 ℩crio 7305 (class class class)co 7350 GrpOpcgr 29217 GIdcgi 29218 AbelOpcablo 29272 ℋchba 29647 +ℎ cva 29648 0ℎc0v 29652 |
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 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-hilex 29727 ax-hfvadd 29728 ax-hvcom 29729 ax-hvass 29730 ax-hv0cl 29731 ax-hvaddid 29732 ax-hfvmul 29733 ax-hvmulid 29734 ax-hvdistr2 29737 ax-hvmul0 29738 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5529 df-po 5543 df-so 5544 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-pnf 11125 df-mnf 11126 df-ltxr 11128 df-sub 11321 df-neg 11322 df-grpo 29221 df-gid 29222 df-ablo 29273 df-hvsub 29699 |
This theorem is referenced by: hhnv 29893 hh0v 29896 hhssabloilem 29989 |
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