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Theorem bafval 29493
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 6842 . . . . 5 (𝑢 = 𝑈 → ( +𝑣𝑢) = ( +𝑣𝑈))
21rneqd 5893 . . . 4 (𝑢 = 𝑈 → ran ( +𝑣𝑢) = ran ( +𝑣𝑈))
3 df-ba 29485 . . . 4 BaseSet = (𝑢 ∈ V ↦ ran ( +𝑣𝑢))
4 fvex 6855 . . . . 5 ( +𝑣𝑈) ∈ V
54rnex 7848 . . . 4 ran ( +𝑣𝑈) ∈ V
62, 3, 5fvmpt 6948 . . 3 (𝑈 ∈ V → (BaseSet‘𝑈) = ran ( +𝑣𝑈))
7 rn0 5881 . . . . 5 ran ∅ = ∅
87eqcomi 2745 . . . 4 ∅ = ran ∅
9 fvprc 6834 . . . 4 𝑈 ∈ V → (BaseSet‘𝑈) = ∅)
10 fvprc 6834 . . . . 5 𝑈 ∈ V → ( +𝑣𝑈) = ∅)
1110rneqd 5893 . . . 4 𝑈 ∈ V → ran ( +𝑣𝑈) = ran ∅)
128, 9, 113eqtr4a 2802 . . 3 𝑈 ∈ V → (BaseSet‘𝑈) = ran ( +𝑣𝑈))
136, 12pm2.61i 182 . 2 (BaseSet‘𝑈) = ran ( +𝑣𝑈)
14 bafval.1 . 2 𝑋 = (BaseSet‘𝑈)
15 bafval.2 . . 3 𝐺 = ( +𝑣𝑈)
1615rneqi 5892 . 2 ran 𝐺 = ran ( +𝑣𝑈)
1713, 14, 163eqtr4i 2774 1 𝑋 = ran 𝐺
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1541  wcel 2106  Vcvv 3445  c0 4282  ran crn 5634  cfv 6496   +𝑣 cpv 29474  BaseSetcba 29475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7671
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fv 6504  df-ba 29485
This theorem is referenced by:  nvi  29503  nvgf  29507  nvsf  29508  nvgcl  29509  nvcom  29510  nvass  29511  nvadd32  29512  nvrcan  29513  nvadd4  29514  nvscl  29515  nvsid  29516  nvsass  29517  nvdi  29519  nvdir  29520  nv2  29521  nvzcl  29523  nv0rid  29524  nv0lid  29525  nv0  29526  nvsz  29527  nvinv  29528  nvinvfval  29529  nvmval  29531  nvmfval  29533  nvnnncan1  29536  nvnegneg  29538  nvrinv  29540  nvlinv  29541  nvaddsub  29544  cnnvba  29568  sspba  29616  isph  29711  phpar  29713  ip0i  29714  ipdirilem  29718  hhba  30056  hhssabloilem  30150  hhshsslem1  30156
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