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Theorem lsmvalx 18759
 Description: Subspace sum value (for a group or vector space). Extended domain version of lsmval 18768. (Contributed by NM, 28-Jan-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
lsmfval.v 𝐵 = (Base‘𝐺)
lsmfval.a + = (+g𝐺)
lsmfval.s = (LSSum‘𝐺)
Assertion
Ref Expression
lsmvalx ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑇 𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
Distinct variable groups:   𝑥,𝑦, +   𝑥,𝐵,𝑦   𝑥,𝑇,𝑦   𝑥,𝐺,𝑦   𝑥,𝑈,𝑦
Allowed substitution hints:   (𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem lsmvalx
Dummy variables 𝑢 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lsmfval.v . . . . 5 𝐵 = (Base‘𝐺)
2 lsmfval.a . . . . 5 + = (+g𝐺)
3 lsmfval.s . . . . 5 = (LSSum‘𝐺)
41, 2, 3lsmfval 18758 . . . 4 (𝐺𝑉 = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦))))
54oveqd 7156 . . 3 (𝐺𝑉 → (𝑇 𝑈) = (𝑇(𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)))𝑈))
61fvexi 6663 . . . . 5 𝐵 ∈ V
76elpw2 5215 . . . 4 (𝑇 ∈ 𝒫 𝐵𝑇𝐵)
86elpw2 5215 . . . 4 (𝑈 ∈ 𝒫 𝐵𝑈𝐵)
9 mpoexga 7762 . . . . . 6 ((𝑇 ∈ 𝒫 𝐵𝑈 ∈ 𝒫 𝐵) → (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)) ∈ V)
10 rnexg 7599 . . . . . 6 ((𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)) ∈ V → ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)) ∈ V)
119, 10syl 17 . . . . 5 ((𝑇 ∈ 𝒫 𝐵𝑈 ∈ 𝒫 𝐵) → ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)) ∈ V)
12 mpoeq12 7210 . . . . . . 7 ((𝑡 = 𝑇𝑢 = 𝑈) → (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)) = (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
1312rneqd 5776 . . . . . 6 ((𝑡 = 𝑇𝑢 = 𝑈) → ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
14 eqid 2801 . . . . . 6 (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦))) = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)))
1513, 14ovmpoga 7287 . . . . 5 ((𝑇 ∈ 𝒫 𝐵𝑈 ∈ 𝒫 𝐵 ∧ ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)) ∈ V) → (𝑇(𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)))𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
1611, 15mpd3an3 1459 . . . 4 ((𝑇 ∈ 𝒫 𝐵𝑈 ∈ 𝒫 𝐵) → (𝑇(𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)))𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
177, 8, 16syl2anbr 601 . . 3 ((𝑇𝐵𝑈𝐵) → (𝑇(𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)))𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
185, 17sylan9eq 2856 . 2 ((𝐺𝑉 ∧ (𝑇𝐵𝑈𝐵)) → (𝑇 𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
19183impb 1112 1 ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑇 𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112  Vcvv 3444   ⊆ wss 3884  𝒫 cpw 4500  ran crn 5524  ‘cfv 6328  (class class class)co 7139   ∈ cmpo 7141  Basecbs 16478  +gcplusg 16560  LSSumclsm 18754 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-lsm 18756 This theorem is referenced by:  lsmelvalx  18760  lsmssv  18763  lsmval  18768  smndlsmidm  18776  subglsm  18794  lsmssass  31012
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