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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 28118 | . . 3 ⊢ ( +ℎ ∈ GrpOp ∧ ( +ℎ ↾ (𝐻 × 𝐻)) ∈ GrpOp ∧ ( +ℎ ↾ (𝐻 × 𝐻)) ⊆ +ℎ ) |
3 | 2 | simp2i 1071 | . 2 ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) ∈ GrpOp |
4 | 1 | shssii 28070 | . . . . 5 ⊢ 𝐻 ⊆ ℋ |
5 | xpss12 5225 | . . . . 5 ⊢ ((𝐻 ⊆ ℋ ∧ 𝐻 ⊆ ℋ) → (𝐻 × 𝐻) ⊆ ( ℋ × ℋ)) | |
6 | 4, 4, 5 | mp2an 708 | . . . 4 ⊢ (𝐻 × 𝐻) ⊆ ( ℋ × ℋ) |
7 | ax-hfvadd 27857 | . . . . 5 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | |
8 | 7 | fdmi 6052 | . . . 4 ⊢ dom +ℎ = ( ℋ × ℋ) |
9 | 6, 8 | sseqtr4i 3638 | . . 3 ⊢ (𝐻 × 𝐻) ⊆ dom +ℎ |
10 | ssdmres 5420 | . . 3 ⊢ ((𝐻 × 𝐻) ⊆ dom +ℎ ↔ dom ( +ℎ ↾ (𝐻 × 𝐻)) = (𝐻 × 𝐻)) | |
11 | 9, 10 | mpbi 220 | . 2 ⊢ dom ( +ℎ ↾ (𝐻 × 𝐻)) = (𝐻 × 𝐻) |
12 | 1 | sheli 28071 | . . . 4 ⊢ (𝑥 ∈ 𝐻 → 𝑥 ∈ ℋ) |
13 | 1 | sheli 28071 | . . . 4 ⊢ (𝑦 ∈ 𝐻 → 𝑦 ∈ ℋ) |
14 | ax-hvcom 27858 | . . . 4 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 +ℎ 𝑦) = (𝑦 +ℎ 𝑥)) | |
15 | 12, 13, 14 | syl2an 494 | . . 3 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥 +ℎ 𝑦) = (𝑦 +ℎ 𝑥)) |
16 | ovres 6800 | . . 3 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥( +ℎ ↾ (𝐻 × 𝐻))𝑦) = (𝑥 +ℎ 𝑦)) | |
17 | ovres 6800 | . . . 4 ⊢ ((𝑦 ∈ 𝐻 ∧ 𝑥 ∈ 𝐻) → (𝑦( +ℎ ↾ (𝐻 × 𝐻))𝑥) = (𝑦 +ℎ 𝑥)) | |
18 | 17 | ancoms 469 | . . 3 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑦( +ℎ ↾ (𝐻 × 𝐻))𝑥) = (𝑦 +ℎ 𝑥)) |
19 | 15, 16, 18 | 3eqtr4d 2666 | . 2 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥( +ℎ ↾ (𝐻 × 𝐻))𝑦) = (𝑦( +ℎ ↾ (𝐻 × 𝐻))𝑥)) |
20 | 3, 11, 19 | isabloi 27405 | 1 ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) ∈ AbelOp |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 × cxp 5112 dom cdm 5114 ↾ cres 5116 (class class class)co 6650 GrpOpcgr 27343 AbelOpcablo 27398 ℋchil 27776 +ℎ cva 27777 Sℋ csh 27785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-hilex 27856 ax-hfvadd 27857 ax-hvcom 27858 ax-hvass 27859 ax-hv0cl 27860 ax-hvaddid 27861 ax-hfvmul 27862 ax-hvmulid 27863 ax-hvmulass 27864 ax-hvdistr1 27865 ax-hvdistr2 27866 ax-hvmul0 27867 ax-hfi 27936 ax-his1 27939 ax-his2 27940 ax-his3 27941 ax-his4 27942 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-grpo 27347 df-gid 27348 df-ginv 27349 df-ablo 27399 df-vc 27414 df-nv 27447 df-va 27450 df-ba 27451 df-sm 27452 df-0v 27453 df-nmcv 27455 df-hnorm 27825 df-hba 27826 df-hvsub 27828 df-sh 28064 |
This theorem is referenced by: hhssablo 28120 hhssnv 28121 |
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