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Theorem bafval 27320
Description: Value of the function for the base set of a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
bafval.1 𝑋 = (BaseSet‘𝑈)
bafval.2 𝐺 = ( +𝑣𝑈)
Assertion
Ref Expression
bafval 𝑋 = ran 𝐺

Proof of Theorem bafval
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6150 . . . . 5 (𝑢 = 𝑈 → ( +𝑣𝑢) = ( +𝑣𝑈))
21rneqd 5315 . . . 4 (𝑢 = 𝑈 → ran ( +𝑣𝑢) = ran ( +𝑣𝑈))
3 df-ba 27312 . . . 4 BaseSet = (𝑢 ∈ V ↦ ran ( +𝑣𝑢))
4 fvex 6160 . . . . 5 ( +𝑣𝑈) ∈ V
54rnex 7050 . . . 4 ran ( +𝑣𝑈) ∈ V
62, 3, 5fvmpt 6241 . . 3 (𝑈 ∈ V → (BaseSet‘𝑈) = ran ( +𝑣𝑈))
7 rn0 5339 . . . . 5 ran ∅ = ∅
87eqcomi 2630 . . . 4 ∅ = ran ∅
9 fvprc 6144 . . . 4 𝑈 ∈ V → (BaseSet‘𝑈) = ∅)
10 fvprc 6144 . . . . 5 𝑈 ∈ V → ( +𝑣𝑈) = ∅)
1110rneqd 5315 . . . 4 𝑈 ∈ V → ran ( +𝑣𝑈) = ran ∅)
128, 9, 113eqtr4a 2681 . . 3 𝑈 ∈ V → (BaseSet‘𝑈) = ran ( +𝑣𝑈))
136, 12pm2.61i 176 . 2 (BaseSet‘𝑈) = ran ( +𝑣𝑈)
14 bafval.1 . 2 𝑋 = (BaseSet‘𝑈)
15 bafval.2 . . 3 𝐺 = ( +𝑣𝑈)
1615rneqi 5314 . 2 ran 𝐺 = ran ( +𝑣𝑈)
1713, 14, 163eqtr4i 2653 1 𝑋 = ran 𝐺
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1480  wcel 1987  Vcvv 3186  c0 3893  ran crn 5077  cfv 5849   +𝑣 cpv 27301  BaseSetcba 27302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-iota 5812  df-fun 5851  df-fv 5857  df-ba 27312
This theorem is referenced by:  nvi  27330  nvgf  27334  nvsf  27335  nvgcl  27336  nvcom  27337  nvass  27338  nvadd32  27339  nvrcan  27340  nvadd4  27341  nvscl  27342  nvsid  27343  nvsass  27344  nvdi  27346  nvdir  27347  nv2  27348  nvzcl  27350  nv0rid  27351  nv0lid  27352  nv0  27353  nvsz  27354  nvinv  27355  nvinvfval  27356  nvmval  27358  nvmfval  27360  nvnnncan1  27363  nvnegneg  27365  nvrinv  27367  nvlinv  27368  nvaddsub  27371  cnnvba  27395  sspba  27443  isph  27538  phpar  27540  ip0i  27541  ipdirilem  27545  hhba  27885  hhssabloilem  27979  hhshsslem1  27985
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