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Theorem lsmvalx 18438
Description: Subspace sum value (for a group or vector space). Extended domain version of lsmval 18447. (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 18437 . . . 4 (𝐺𝑉 = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦))))
54oveqd 6939 . . 3 (𝐺𝑉 → (𝑇 𝑈) = (𝑇(𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)))𝑈))
61fvexi 6460 . . . . 5 𝐵 ∈ V
76elpw2 5062 . . . 4 (𝑇 ∈ 𝒫 𝐵𝑇𝐵)
86elpw2 5062 . . . 4 (𝑈 ∈ 𝒫 𝐵𝑈𝐵)
9 mpt2exga 7526 . . . . . 6 ((𝑇 ∈ 𝒫 𝐵𝑈 ∈ 𝒫 𝐵) → (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)) ∈ V)
10 rnexg 7376 . . . . . 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 mpt2eq12 6992 . . . . . . 7 ((𝑡 = 𝑇𝑢 = 𝑈) → (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)) = (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
1312rneqd 5598 . . . . . 6 ((𝑡 = 𝑇𝑢 = 𝑈) → ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
14 eqid 2777 . . . . . 6 (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦))) = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)))
1513, 14ovmpt2ga 7067 . . . . 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 1535 . . . 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 592 . . 3 ((𝑇𝐵𝑈𝐵) → (𝑇(𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)))𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
185, 17sylan9eq 2833 . 2 ((𝐺𝑉 ∧ (𝑇𝐵𝑈𝐵)) → (𝑇 𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
19183impb 1104 1 ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑇 𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1071   = wceq 1601  wcel 2106  Vcvv 3397  wss 3791  𝒫 cpw 4378  ran crn 5356  cfv 6135  (class class class)co 6922  cmpt2 6924  Basecbs 16255  +gcplusg 16338  LSSumclsm 18433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-lsm 18435
This theorem is referenced by:  lsmelvalx  18439  lsmssv  18442  lsmval  18447  subglsm  18470
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