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Theorem lsmvalx 19671
Description: Subspace sum value (for a group or vector space). Extended domain version of lsmval 19680. (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 19670 . . . 4 (𝐺𝑉 = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦))))
54oveqd 7447 . . 3 (𝐺𝑉 → (𝑇 𝑈) = (𝑇(𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)))𝑈))
61fvexi 6920 . . . . 5 𝐵 ∈ V
76elpw2 5339 . . . 4 (𝑇 ∈ 𝒫 𝐵𝑇𝐵)
86elpw2 5339 . . . 4 (𝑈 ∈ 𝒫 𝐵𝑈𝐵)
9 mpoexga 8100 . . . . . 6 ((𝑇 ∈ 𝒫 𝐵𝑈 ∈ 𝒫 𝐵) → (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)) ∈ V)
10 rnexg 7924 . . . . . 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 7505 . . . . . . 7 ((𝑡 = 𝑇𝑢 = 𝑈) → (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)) = (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
1312rneqd 5951 . . . . . 6 ((𝑡 = 𝑇𝑢 = 𝑈) → ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
14 eqid 2734 . . . . . 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 7586 . . . . 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 1461 . . . 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 599 . . 3 ((𝑇𝐵𝑈𝐵) → (𝑇(𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)))𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
185, 17sylan9eq 2794 . 2 ((𝐺𝑉 ∧ (𝑇𝐵𝑈𝐵)) → (𝑇 𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
19183impb 1114 1 ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑇 𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1536  wcel 2105  Vcvv 3477  wss 3962  𝒫 cpw 4604  ran crn 5689  cfv 6562  (class class class)co 7430  cmpo 7432  Basecbs 17244  +gcplusg 17297  LSSumclsm 19666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-lsm 19668
This theorem is referenced by:  lsmelvalx  19672  lsmssv  19675  lsmval  19680  smndlsmidm  19688  subglsm  19705  lsmssass  33409
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