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Theorem lsmvalx 17877
 Description: Subspace sum value (for a group or vector space). Extended domain version of lsmval 17886. (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 17876 . . . 4 (𝐺𝑉 = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦))))
54oveqd 6566 . . 3 (𝐺𝑉 → (𝑇 𝑈) = (𝑇(𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)))𝑈))
6 fvex 6113 . . . . . 6 (Base‘𝐺) ∈ V
71, 6eqeltri 2684 . . . . 5 𝐵 ∈ V
87elpw2 4755 . . . 4 (𝑇 ∈ 𝒫 𝐵𝑇𝐵)
97elpw2 4755 . . . 4 (𝑈 ∈ 𝒫 𝐵𝑈𝐵)
10 mpt2exga 7135 . . . . . 6 ((𝑇 ∈ 𝒫 𝐵𝑈 ∈ 𝒫 𝐵) → (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)) ∈ V)
11 rnexg 6990 . . . . . 6 ((𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)) ∈ V → ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)) ∈ V)
1210, 11syl 17 . . . . 5 ((𝑇 ∈ 𝒫 𝐵𝑈 ∈ 𝒫 𝐵) → ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)) ∈ V)
13 mpt2eq12 6613 . . . . . . 7 ((𝑡 = 𝑇𝑢 = 𝑈) → (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)) = (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
1413rneqd 5274 . . . . . 6 ((𝑡 = 𝑇𝑢 = 𝑈) → ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
15 eqid 2610 . . . . . 6 (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦))) = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)))
1614, 15ovmpt2ga 6688 . . . . 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)))
1712, 16mpd3an3 1417 . . . 4 ((𝑇 ∈ 𝒫 𝐵𝑈 ∈ 𝒫 𝐵) → (𝑇(𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)))𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
188, 9, 17syl2anbr 496 . . 3 ((𝑇𝐵𝑈𝐵) → (𝑇(𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)))𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
195, 18sylan9eq 2664 . 2 ((𝐺𝑉 ∧ (𝑇𝐵𝑈𝐵)) → (𝑇 𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
20193impb 1252 1 ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑇 𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  ran crn 5039  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Basecbs 15695  +gcplusg 15768  LSSumclsm 17872 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-lsm 17874 This theorem is referenced by:  lsmelvalx  17878  lsmssv  17881  lsmval  17886  subglsm  17909
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