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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 31023 | . . 3 ⊢ ( +ℎ ∈ GrpOp ∧ ( +ℎ ↾ (𝐻 × 𝐻)) ∈ GrpOp ∧ ( +ℎ ↾ (𝐻 × 𝐻)) ⊆ +ℎ ) |
3 | 2 | simp2i 1137 | . 2 ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) ∈ GrpOp |
4 | 1 | shssii 30975 | . . . . 5 ⊢ 𝐻 ⊆ ℋ |
5 | xpss12 5684 | . . . . 5 ⊢ ((𝐻 ⊆ ℋ ∧ 𝐻 ⊆ ℋ) → (𝐻 × 𝐻) ⊆ ( ℋ × ℋ)) | |
6 | 4, 4, 5 | mp2an 689 | . . . 4 ⊢ (𝐻 × 𝐻) ⊆ ( ℋ × ℋ) |
7 | ax-hfvadd 30762 | . . . . 5 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | |
8 | 7 | fdmi 6723 | . . . 4 ⊢ dom +ℎ = ( ℋ × ℋ) |
9 | 6, 8 | sseqtrri 4014 | . . 3 ⊢ (𝐻 × 𝐻) ⊆ dom +ℎ |
10 | ssdmres 5998 | . . 3 ⊢ ((𝐻 × 𝐻) ⊆ dom +ℎ ↔ dom ( +ℎ ↾ (𝐻 × 𝐻)) = (𝐻 × 𝐻)) | |
11 | 9, 10 | mpbi 229 | . 2 ⊢ dom ( +ℎ ↾ (𝐻 × 𝐻)) = (𝐻 × 𝐻) |
12 | 1 | sheli 30976 | . . . 4 ⊢ (𝑥 ∈ 𝐻 → 𝑥 ∈ ℋ) |
13 | 1 | sheli 30976 | . . . 4 ⊢ (𝑦 ∈ 𝐻 → 𝑦 ∈ ℋ) |
14 | ax-hvcom 30763 | . . . 4 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 +ℎ 𝑦) = (𝑦 +ℎ 𝑥)) | |
15 | 12, 13, 14 | syl2an 595 | . . 3 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥 +ℎ 𝑦) = (𝑦 +ℎ 𝑥)) |
16 | ovres 7570 | . . 3 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥( +ℎ ↾ (𝐻 × 𝐻))𝑦) = (𝑥 +ℎ 𝑦)) | |
17 | ovres 7570 | . . . 4 ⊢ ((𝑦 ∈ 𝐻 ∧ 𝑥 ∈ 𝐻) → (𝑦( +ℎ ↾ (𝐻 × 𝐻))𝑥) = (𝑦 +ℎ 𝑥)) | |
18 | 17 | ancoms 458 | . . 3 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑦( +ℎ ↾ (𝐻 × 𝐻))𝑥) = (𝑦 +ℎ 𝑥)) |
19 | 15, 16, 18 | 3eqtr4d 2776 | . 2 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥( +ℎ ↾ (𝐻 × 𝐻))𝑦) = (𝑦( +ℎ ↾ (𝐻 × 𝐻))𝑥)) |
20 | 3, 11, 19 | isabloi 30313 | 1 ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) ∈ AbelOp |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1533 ∈ wcel 2098 ⊆ wss 3943 × cxp 5667 dom cdm 5669 ↾ cres 5671 (class class class)co 7405 GrpOpcgr 30251 AbelOpcablo 30306 ℋchba 30681 +ℎ cva 30682 Sℋ csh 30690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-hilex 30761 ax-hfvadd 30762 ax-hvcom 30763 ax-hvass 30764 ax-hv0cl 30765 ax-hvaddid 30766 ax-hfvmul 30767 ax-hvmulid 30768 ax-hvmulass 30769 ax-hvdistr1 30770 ax-hvdistr2 30771 ax-hvmul0 30772 ax-hfi 30841 ax-his1 30844 ax-his2 30845 ax-his3 30846 ax-his4 30847 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-seq 13973 df-exp 14033 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-grpo 30255 df-gid 30256 df-ginv 30257 df-ablo 30307 df-vc 30321 df-nv 30354 df-va 30357 df-ba 30358 df-sm 30359 df-0v 30360 df-nmcv 30362 df-hnorm 30730 df-hba 30731 df-hvsub 30733 df-sh 30969 |
This theorem is referenced by: hhssablo 31025 hhssnv 31026 |
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