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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 30847 | . . . 4 ⊢ +ℎ ∈ AbelOp | |
2 | ablogrpo 30234 | . . . 4 ⊢ ( +ℎ ∈ AbelOp → +ℎ ∈ GrpOp) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ +ℎ ∈ GrpOp |
4 | ax-hfvadd 30687 | . . . . . 6 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | |
5 | 4 | fdmi 6729 | . . . . 5 ⊢ dom +ℎ = ( ℋ × ℋ) |
6 | 3, 5 | grporn 30208 | . . . 4 ⊢ ℋ = ran +ℎ |
7 | eqid 2731 | . . . 4 ⊢ (GId‘ +ℎ ) = (GId‘ +ℎ ) | |
8 | 6, 7 | grpoidval 30200 | . . 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 | hvaddlid 30710 | . . . 4 ⊢ (𝑥 ∈ ℋ → (0ℎ +ℎ 𝑥) = 𝑥) | |
11 | 10 | rgen 3062 | . . 3 ⊢ ∀𝑥 ∈ ℋ (0ℎ +ℎ 𝑥) = 𝑥 |
12 | ax-hv0cl 30690 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
13 | 6 | grpoideu 30196 | . . . . 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 7419 | . . . . . . 7 ⊢ (𝑦 = 0ℎ → (𝑦 +ℎ 𝑥) = (0ℎ +ℎ 𝑥)) | |
16 | 15 | eqeq1d 2733 | . . . . . 6 ⊢ (𝑦 = 0ℎ → ((𝑦 +ℎ 𝑥) = 𝑥 ↔ (0ℎ +ℎ 𝑥) = 𝑥)) |
17 | 16 | ralbidv 3176 | . . . . 5 ⊢ (𝑦 = 0ℎ → (∀𝑥 ∈ ℋ (𝑦 +ℎ 𝑥) = 𝑥 ↔ ∀𝑥 ∈ ℋ (0ℎ +ℎ 𝑥) = 𝑥)) |
18 | 17 | riota2 7394 | . . . 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 689 | . . 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 2759 | 1 ⊢ (GId‘ +ℎ ) = 0ℎ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1540 ∈ wcel 2105 ∀wral 3060 ∃!wreu 3373 × cxp 5674 ‘cfv 6543 ℩crio 7367 (class class class)co 7412 GrpOpcgr 30176 GIdcgi 30177 AbelOpcablo 30231 ℋchba 30606 +ℎ cva 30607 0ℎc0v 30611 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-hilex 30686 ax-hfvadd 30687 ax-hvcom 30688 ax-hvass 30689 ax-hv0cl 30690 ax-hvaddid 30691 ax-hfvmul 30692 ax-hvmulid 30693 ax-hvdistr2 30696 ax-hvmul0 30697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-ltxr 11260 df-sub 11453 df-neg 11454 df-grpo 30180 df-gid 30181 df-ablo 30232 df-hvsub 30658 |
This theorem is referenced by: hhnv 30852 hh0v 30855 hhssabloilem 30948 |
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