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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 30101 | . . 3 ⊢ ( +ℎ ∈ GrpOp ∧ ( +ℎ ↾ (𝐻 × 𝐻)) ∈ GrpOp ∧ ( +ℎ ↾ (𝐻 × 𝐻)) ⊆ +ℎ ) |
3 | 2 | simp2i 1140 | . 2 ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) ∈ GrpOp |
4 | 1 | shssii 30053 | . . . . 5 ⊢ 𝐻 ⊆ ℋ |
5 | xpss12 5647 | . . . . 5 ⊢ ((𝐻 ⊆ ℋ ∧ 𝐻 ⊆ ℋ) → (𝐻 × 𝐻) ⊆ ( ℋ × ℋ)) | |
6 | 4, 4, 5 | mp2an 690 | . . . 4 ⊢ (𝐻 × 𝐻) ⊆ ( ℋ × ℋ) |
7 | ax-hfvadd 29840 | . . . . 5 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | |
8 | 7 | fdmi 6678 | . . . 4 ⊢ dom +ℎ = ( ℋ × ℋ) |
9 | 6, 8 | sseqtrri 3980 | . . 3 ⊢ (𝐻 × 𝐻) ⊆ dom +ℎ |
10 | ssdmres 5959 | . . 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 30054 | . . . 4 ⊢ (𝑥 ∈ 𝐻 → 𝑥 ∈ ℋ) |
13 | 1 | sheli 30054 | . . . 4 ⊢ (𝑦 ∈ 𝐻 → 𝑦 ∈ ℋ) |
14 | ax-hvcom 29841 | . . . 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 7517 | . . 3 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥( +ℎ ↾ (𝐻 × 𝐻))𝑦) = (𝑥 +ℎ 𝑦)) | |
17 | ovres 7517 | . . . 4 ⊢ ((𝑦 ∈ 𝐻 ∧ 𝑥 ∈ 𝐻) → (𝑦( +ℎ ↾ (𝐻 × 𝐻))𝑥) = (𝑦 +ℎ 𝑥)) | |
18 | 17 | ancoms 459 | . . 3 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑦( +ℎ ↾ (𝐻 × 𝐻))𝑥) = (𝑦 +ℎ 𝑥)) |
19 | 15, 16, 18 | 3eqtr4d 2786 | . 2 ⊢ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) → (𝑥( +ℎ ↾ (𝐻 × 𝐻))𝑦) = (𝑦( +ℎ ↾ (𝐻 × 𝐻))𝑥)) |
20 | 3, 11, 19 | isabloi 29391 | 1 ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) ∈ AbelOp |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∈ wcel 2106 ⊆ wss 3909 × cxp 5630 dom cdm 5632 ↾ cres 5634 (class class class)co 7354 GrpOpcgr 29329 AbelOpcablo 29384 ℋchba 29759 +ℎ cva 29760 Sℋ csh 29768 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-cnex 11104 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 ax-pre-mulgt0 11125 ax-pre-sup 11126 ax-hilex 29839 ax-hfvadd 29840 ax-hvcom 29841 ax-hvass 29842 ax-hv0cl 29843 ax-hvaddid 29844 ax-hfvmul 29845 ax-hvmulid 29846 ax-hvmulass 29847 ax-hvdistr1 29848 ax-hvdistr2 29849 ax-hvmul0 29850 ax-hfi 29919 ax-his1 29922 ax-his2 29923 ax-his3 29924 ax-his4 29925 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-om 7800 df-1st 7918 df-2nd 7919 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-er 8645 df-en 8881 df-dom 8882 df-sdom 8883 df-sup 9375 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11384 df-neg 11385 df-div 11810 df-nn 12151 df-2 12213 df-3 12214 df-4 12215 df-n0 12411 df-z 12497 df-uz 12761 df-rp 12913 df-seq 13904 df-exp 13965 df-cj 14981 df-re 14982 df-im 14983 df-sqrt 15117 df-abs 15118 df-grpo 29333 df-gid 29334 df-ginv 29335 df-ablo 29385 df-vc 29399 df-nv 29432 df-va 29435 df-ba 29436 df-sm 29437 df-0v 29438 df-nmcv 29440 df-hnorm 29808 df-hba 29809 df-hvsub 29811 df-sh 30047 |
This theorem is referenced by: hhssablo 30103 hhssnv 30104 |
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