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Theorem lsmvalx 19429
Description: Subspace sum value (for a group or vector space). Extended domain version of lsmval 19438. (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 19428 . . . 4 (𝐺𝑉 = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦))))
54oveqd 7378 . . 3 (𝐺𝑉 → (𝑇 𝑈) = (𝑇(𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)))𝑈))
61fvexi 6860 . . . . 5 𝐵 ∈ V
76elpw2 5306 . . . 4 (𝑇 ∈ 𝒫 𝐵𝑇𝐵)
86elpw2 5306 . . . 4 (𝑈 ∈ 𝒫 𝐵𝑈𝐵)
9 mpoexga 8014 . . . . . 6 ((𝑇 ∈ 𝒫 𝐵𝑈 ∈ 𝒫 𝐵) → (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)) ∈ V)
10 rnexg 7845 . . . . . 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 7434 . . . . . . 7 ((𝑡 = 𝑇𝑢 = 𝑈) → (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)) = (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
1312rneqd 5897 . . . . . 6 ((𝑡 = 𝑇𝑢 = 𝑈) → ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
14 eqid 2733 . . . . . 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 7513 . . . . 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 1463 . . . 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 600 . . 3 ((𝑇𝐵𝑈𝐵) → (𝑇(𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦)))𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
185, 17sylan9eq 2793 . 2 ((𝐺𝑉 ∧ (𝑇𝐵𝑈𝐵)) → (𝑇 𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
19183impb 1116 1 ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑇 𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  Vcvv 3447  wss 3914  𝒫 cpw 4564  ran crn 5638  cfv 6500  (class class class)co 7361  cmpo 7363  Basecbs 17091  +gcplusg 17141  LSSumclsm 19424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-lsm 19426
This theorem is referenced by:  lsmelvalx  19430  lsmssv  19433  lsmval  19438  smndlsmidm  19446  subglsm  19463  lsmssass  32238
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