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Mirrors > Home > HSE Home > Th. List > hhssabloi | Structured version Visualization version GIF version |
Description: Abelian group property of subspace addition. (Contributed by NM, 9-Apr-2008.) (Revised by Mario Carneiro, 23-Dec-2013.) (Proof shortened by AV, 27-Aug-2021.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hhssabl.1 | ⊢ 𝐻 ∈ Sℋ |
Ref | Expression |
---|---|
hhssabloi | ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) ∈ AbelOp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hhssabl.1 | . . . 4 ⊢ 𝐻 ∈ Sℋ | |
2 | 1 | hhssabloilem 29044 | . . 3 ⊢ ( +ℎ ∈ GrpOp ∧ ( +ℎ ↾ (𝐻 × 𝐻)) ∈ GrpOp ∧ ( +ℎ ↾ (𝐻 × 𝐻)) ⊆ +ℎ ) |
3 | 2 | simp2i 1137 | . 2 ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) ∈ GrpOp |
4 | 1 | shssii 28996 | . . . . 5 ⊢ 𝐻 ⊆ ℋ |
5 | xpss12 5534 | . . . . 5 ⊢ ((𝐻 ⊆ ℋ ∧ 𝐻 ⊆ ℋ) → (𝐻 × 𝐻) ⊆ ( ℋ × ℋ)) | |
6 | 4, 4, 5 | mp2an 691 | . . . 4 ⊢ (𝐻 × 𝐻) ⊆ ( ℋ × ℋ) |
7 | ax-hfvadd 28783 | . . . . 5 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | |
8 | 7 | fdmi 6498 | . . . 4 ⊢ dom +ℎ = ( ℋ × ℋ) |
9 | 6, 8 | sseqtrri 3952 | . . 3 ⊢ (𝐻 × 𝐻) ⊆ dom +ℎ |
10 | ssdmres 5841 | . . 3 ⊢ ((𝐻 × 𝐻) ⊆ dom +ℎ ↔ dom ( +ℎ ↾ (𝐻 × 𝐻)) = (𝐻 × 𝐻)) | |
11 | 9, 10 | mpbi 233 | . 2 ⊢ dom ( +ℎ ↾ (𝐻 × 𝐻)) = (𝐻 × 𝐻) |
12 | 1 | sheli 28997 | . . . 4 ⊢ (𝑥 ∈ 𝐻 → 𝑥 ∈ ℋ) |
13 | 1 | sheli 28997 | . . . 4 ⊢ (𝑦 ∈ 𝐻 → 𝑦 ∈ ℋ) |
14 | ax-hvcom 28784 | . . . 4 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 +ℎ 𝑦) = (𝑦 +ℎ 𝑥)) | |
15 | 12, 13, 14 | syl2an 598 | . . 3 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥 +ℎ 𝑦) = (𝑦 +ℎ 𝑥)) |
16 | ovres 7294 | . . 3 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥( +ℎ ↾ (𝐻 × 𝐻))𝑦) = (𝑥 +ℎ 𝑦)) | |
17 | ovres 7294 | . . . 4 ⊢ ((𝑦 ∈ 𝐻 ∧ 𝑥 ∈ 𝐻) → (𝑦( +ℎ ↾ (𝐻 × 𝐻))𝑥) = (𝑦 +ℎ 𝑥)) | |
18 | 17 | ancoms 462 | . . 3 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑦( +ℎ ↾ (𝐻 × 𝐻))𝑥) = (𝑦 +ℎ 𝑥)) |
19 | 15, 16, 18 | 3eqtr4d 2843 | . 2 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥( +ℎ ↾ (𝐻 × 𝐻))𝑦) = (𝑦( +ℎ ↾ (𝐻 × 𝐻))𝑥)) |
20 | 3, 11, 19 | isabloi 28334 | 1 ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) ∈ AbelOp |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 × cxp 5517 dom cdm 5519 ↾ cres 5521 (class class class)co 7135 GrpOpcgr 28272 AbelOpcablo 28327 ℋchba 28702 +ℎ cva 28703 Sℋ csh 28711 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-hilex 28782 ax-hfvadd 28783 ax-hvcom 28784 ax-hvass 28785 ax-hv0cl 28786 ax-hvaddid 28787 ax-hfvmul 28788 ax-hvmulid 28789 ax-hvmulass 28790 ax-hvdistr1 28791 ax-hvdistr2 28792 ax-hvmul0 28793 ax-hfi 28862 ax-his1 28865 ax-his2 28866 ax-his3 28867 ax-his4 28868 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-grpo 28276 df-gid 28277 df-ginv 28278 df-ablo 28328 df-vc 28342 df-nv 28375 df-va 28378 df-ba 28379 df-sm 28380 df-0v 28381 df-nmcv 28383 df-hnorm 28751 df-hba 28752 df-hvsub 28754 df-sh 28990 |
This theorem is referenced by: hhssablo 29046 hhssnv 29047 |
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