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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 28935 | . . . 4 ⊢ +ℎ ∈ AbelOp | |
2 | ablogrpo 28322 | . . . 4 ⊢ ( +ℎ ∈ AbelOp → +ℎ ∈ GrpOp) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ +ℎ ∈ GrpOp |
4 | ax-hfvadd 28775 | . . . . . 6 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | |
5 | 4 | fdmi 6517 | . . . . 5 ⊢ dom +ℎ = ( ℋ × ℋ) |
6 | 3, 5 | grporn 28296 | . . . 4 ⊢ ℋ = ran +ℎ |
7 | eqid 2820 | . . . 4 ⊢ (GId‘ +ℎ ) = (GId‘ +ℎ ) | |
8 | 6, 7 | grpoidval 28288 | . . 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 28798 | . . . 4 ⊢ (𝑥 ∈ ℋ → (0ℎ +ℎ 𝑥) = 𝑥) | |
11 | 10 | rgen 3147 | . . 3 ⊢ ∀𝑥 ∈ ℋ (0ℎ +ℎ 𝑥) = 𝑥 |
12 | ax-hv0cl 28778 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
13 | 6 | grpoideu 28284 | . . . . 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 7156 | . . . . . . 7 ⊢ (𝑦 = 0ℎ → (𝑦 +ℎ 𝑥) = (0ℎ +ℎ 𝑥)) | |
16 | 15 | eqeq1d 2822 | . . . . . 6 ⊢ (𝑦 = 0ℎ → ((𝑦 +ℎ 𝑥) = 𝑥 ↔ (0ℎ +ℎ 𝑥) = 𝑥)) |
17 | 16 | ralbidv 3196 | . . . . 5 ⊢ (𝑦 = 0ℎ → (∀𝑥 ∈ ℋ (𝑦 +ℎ 𝑥) = 𝑥 ↔ ∀𝑥 ∈ ℋ (0ℎ +ℎ 𝑥) = 𝑥)) |
18 | 17 | riota2 7132 | . . . 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 690 | . . 3 ⊢ (∀𝑥 ∈ ℋ (0ℎ +ℎ 𝑥) = 𝑥 ↔ (℩𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 +ℎ 𝑥) = 𝑥) = 0ℎ) |
20 | 11, 19 | mpbi 232 | . 2 ⊢ (℩𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 +ℎ 𝑥) = 𝑥) = 0ℎ |
21 | 9, 20 | eqtri 2843 | 1 ⊢ (GId‘ +ℎ ) = 0ℎ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1536 ∈ wcel 2113 ∀wral 3137 ∃!wreu 3139 × cxp 5546 ‘cfv 6348 ℩crio 7106 (class class class)co 7149 GrpOpcgr 28264 GIdcgi 28265 AbelOpcablo 28319 ℋchba 28694 +ℎ cva 28695 0ℎc0v 28699 |
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-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-hilex 28774 ax-hfvadd 28775 ax-hvcom 28776 ax-hvass 28777 ax-hv0cl 28778 ax-hvaddid 28779 ax-hfvmul 28780 ax-hvmulid 28781 ax-hvdistr2 28784 ax-hvmul0 28785 |
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-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-po 5467 df-so 5468 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-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-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-pnf 10670 df-mnf 10671 df-ltxr 10673 df-sub 10865 df-neg 10866 df-grpo 28268 df-gid 28269 df-ablo 28320 df-hvsub 28746 |
This theorem is referenced by: hhnv 28940 hh0v 28943 hhssabloilem 29036 |
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