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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 31280 | . . 3 ⊢ ( +ℎ ∈ GrpOp ∧ ( +ℎ ↾ (𝐻 × 𝐻)) ∈ GrpOp ∧ ( +ℎ ↾ (𝐻 × 𝐻)) ⊆ +ℎ ) |
| 3 | 2 | simp2i 1141 | . 2 ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) ∈ GrpOp |
| 4 | 1 | shssii 31232 | . . . . 5 ⊢ 𝐻 ⊆ ℋ |
| 5 | xpss12 5700 | . . . . 5 ⊢ ((𝐻 ⊆ ℋ ∧ 𝐻 ⊆ ℋ) → (𝐻 × 𝐻) ⊆ ( ℋ × ℋ)) | |
| 6 | 4, 4, 5 | mp2an 692 | . . . 4 ⊢ (𝐻 × 𝐻) ⊆ ( ℋ × ℋ) |
| 7 | ax-hfvadd 31019 | . . . . 5 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | |
| 8 | 7 | fdmi 6747 | . . . 4 ⊢ dom +ℎ = ( ℋ × ℋ) |
| 9 | 6, 8 | sseqtrri 4033 | . . 3 ⊢ (𝐻 × 𝐻) ⊆ dom +ℎ |
| 10 | ssdmres 6031 | . . 3 ⊢ ((𝐻 × 𝐻) ⊆ dom +ℎ ↔ dom ( +ℎ ↾ (𝐻 × 𝐻)) = (𝐻 × 𝐻)) | |
| 11 | 9, 10 | mpbi 230 | . 2 ⊢ dom ( +ℎ ↾ (𝐻 × 𝐻)) = (𝐻 × 𝐻) |
| 12 | 1 | sheli 31233 | . . . 4 ⊢ (𝑥 ∈ 𝐻 → 𝑥 ∈ ℋ) |
| 13 | 1 | sheli 31233 | . . . 4 ⊢ (𝑦 ∈ 𝐻 → 𝑦 ∈ ℋ) |
| 14 | ax-hvcom 31020 | . . . 4 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 +ℎ 𝑦) = (𝑦 +ℎ 𝑥)) | |
| 15 | 12, 13, 14 | syl2an 596 | . . 3 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥 +ℎ 𝑦) = (𝑦 +ℎ 𝑥)) |
| 16 | ovres 7599 | . . 3 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥( +ℎ ↾ (𝐻 × 𝐻))𝑦) = (𝑥 +ℎ 𝑦)) | |
| 17 | ovres 7599 | . . . 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 2787 | . 2 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥( +ℎ ↾ (𝐻 × 𝐻))𝑦) = (𝑦( +ℎ ↾ (𝐻 × 𝐻))𝑥)) |
| 20 | 3, 11, 19 | isabloi 30570 | 1 ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) ∈ AbelOp |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 × cxp 5683 dom cdm 5685 ↾ cres 5687 (class class class)co 7431 GrpOpcgr 30508 AbelOpcablo 30563 ℋchba 30938 +ℎ cva 30939 Sℋ csh 30947 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-hilex 31018 ax-hfvadd 31019 ax-hvcom 31020 ax-hvass 31021 ax-hv0cl 31022 ax-hvaddid 31023 ax-hfvmul 31024 ax-hvmulid 31025 ax-hvmulass 31026 ax-hvdistr1 31027 ax-hvdistr2 31028 ax-hvmul0 31029 ax-hfi 31098 ax-his1 31101 ax-his2 31102 ax-his3 31103 ax-his4 31104 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-grpo 30512 df-gid 30513 df-ginv 30514 df-ablo 30564 df-vc 30578 df-nv 30611 df-va 30614 df-ba 30615 df-sm 30616 df-0v 30617 df-nmcv 30619 df-hnorm 30987 df-hba 30988 df-hvsub 30990 df-sh 31226 |
| This theorem is referenced by: hhssablo 31282 hhssnv 31283 |
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