|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > lsmelvalx | Structured version Visualization version GIF version | ||
| Description: Subspace sum membership (for a group or vector space). Extended domain version of lsmelval 19667. (Contributed by NM, 28-Jan-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) | 
| Ref | Expression | 
|---|---|
| lsmfval.v | ⊢ 𝐵 = (Base‘𝐺) | 
| lsmfval.a | ⊢ + = (+g‘𝐺) | 
| lsmfval.s | ⊢ ⊕ = (LSSum‘𝐺) | 
| Ref | Expression | 
|---|---|
| lsmelvalx | ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑋 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 + 𝑧))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | lsmfval.v | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
| 2 | lsmfval.a | . . . 4 ⊢ + = (+g‘𝐺) | |
| 3 | lsmfval.s | . . . 4 ⊢ ⊕ = (LSSum‘𝐺) | |
| 4 | 1, 2, 3 | lsmvalx 19657 | . . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑇 ⊕ 𝑈) = ran (𝑦 ∈ 𝑇, 𝑧 ∈ 𝑈 ↦ (𝑦 + 𝑧))) | 
| 5 | 4 | eleq2d 2827 | . 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑋 ∈ (𝑇 ⊕ 𝑈) ↔ 𝑋 ∈ ran (𝑦 ∈ 𝑇, 𝑧 ∈ 𝑈 ↦ (𝑦 + 𝑧)))) | 
| 6 | eqid 2737 | . . 3 ⊢ (𝑦 ∈ 𝑇, 𝑧 ∈ 𝑈 ↦ (𝑦 + 𝑧)) = (𝑦 ∈ 𝑇, 𝑧 ∈ 𝑈 ↦ (𝑦 + 𝑧)) | |
| 7 | ovex 7464 | . . 3 ⊢ (𝑦 + 𝑧) ∈ V | |
| 8 | 6, 7 | elrnmpo 7569 | . 2 ⊢ (𝑋 ∈ ran (𝑦 ∈ 𝑇, 𝑧 ∈ 𝑈 ↦ (𝑦 + 𝑧)) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 + 𝑧)) | 
| 9 | 5, 8 | bitrdi 287 | 1 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑋 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 + 𝑧))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ⊆ wss 3951 ran crn 5686 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 Basecbs 17247 +gcplusg 17297 LSSumclsm 19652 | 
| 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 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 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-ral 3062 df-rex 3071 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-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-id 5578 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-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-lsm 19654 | 
| This theorem is referenced by: lsmelvalix 19659 lsmless1x 19662 lsmless2x 19663 lsmelval 19667 lsmsubm 19671 lsmass 19687 lsmcomx 19874 lsmcss 21710 elgrplsmsn 33418 elringlsm 33421 lsmssass 33430 grplsm0l 33431 grplsmid 33432 ssdifidlprm 33486 mxidlprm 33498 dimkerim 33678 | 
| Copyright terms: Public domain | W3C validator |