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Mirrors > Home > MPE Home > Th. List > bafval | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
bafval.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
bafval.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
Ref | Expression |
---|---|
bafval | ⊢ 𝑋 = ran 𝐺 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6103 | . . . . 5 ⊢ (𝑢 = 𝑈 → ( +𝑣 ‘𝑢) = ( +𝑣 ‘𝑈)) | |
2 | 1 | rneqd 5274 | . . . 4 ⊢ (𝑢 = 𝑈 → ran ( +𝑣 ‘𝑢) = ran ( +𝑣 ‘𝑈)) |
3 | df-ba 26835 | . . . 4 ⊢ BaseSet = (𝑢 ∈ V ↦ ran ( +𝑣 ‘𝑢)) | |
4 | fvex 6113 | . . . . 5 ⊢ ( +𝑣 ‘𝑈) ∈ V | |
5 | 4 | rnex 6992 | . . . 4 ⊢ ran ( +𝑣 ‘𝑈) ∈ V |
6 | 2, 3, 5 | fvmpt 6191 | . . 3 ⊢ (𝑈 ∈ V → (BaseSet‘𝑈) = ran ( +𝑣 ‘𝑈)) |
7 | rn0 5298 | . . . . 5 ⊢ ran ∅ = ∅ | |
8 | 7 | eqcomi 2619 | . . . 4 ⊢ ∅ = ran ∅ |
9 | fvprc 6097 | . . . 4 ⊢ (¬ 𝑈 ∈ V → (BaseSet‘𝑈) = ∅) | |
10 | fvprc 6097 | . . . . 5 ⊢ (¬ 𝑈 ∈ V → ( +𝑣 ‘𝑈) = ∅) | |
11 | 10 | rneqd 5274 | . . . 4 ⊢ (¬ 𝑈 ∈ V → ran ( +𝑣 ‘𝑈) = ran ∅) |
12 | 8, 9, 11 | 3eqtr4a 2670 | . . 3 ⊢ (¬ 𝑈 ∈ V → (BaseSet‘𝑈) = ran ( +𝑣 ‘𝑈)) |
13 | 6, 12 | pm2.61i 175 | . 2 ⊢ (BaseSet‘𝑈) = ran ( +𝑣 ‘𝑈) |
14 | bafval.1 | . 2 ⊢ 𝑋 = (BaseSet‘𝑈) | |
15 | bafval.2 | . . 3 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
16 | 15 | rneqi 5273 | . 2 ⊢ ran 𝐺 = ran ( +𝑣 ‘𝑈) |
17 | 13, 14, 16 | 3eqtr4i 2642 | 1 ⊢ 𝑋 = ran 𝐺 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 ran crn 5039 ‘cfv 5804 +𝑣 cpv 26824 BaseSetcba 26825 |
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-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-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 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-iota 5768 df-fun 5806 df-fv 5812 df-ba 26835 |
This theorem is referenced by: nvi 26853 nvgf 26857 nvsf 26858 nvgcl 26859 nvcom 26860 nvass 26861 nvadd32 26862 nvrcan 26863 nvadd4 26864 nvscl 26865 nvsid 26866 nvsass 26867 nvdi 26869 nvdir 26870 nv2 26871 nvzcl 26873 nv0rid 26874 nv0lid 26875 nv0 26876 nvsz 26877 nvinv 26878 nvinvfval 26879 nvmval 26881 nvmfval 26883 nvnnncan1 26886 nvnegneg 26888 nvrinv 26890 nvlinv 26891 nvaddsub 26894 cnnvba 26918 sspba 26966 isph 27061 phpar 27063 ip0i 27064 ipdirilem 27068 hhba 27408 hhssabloilem 27502 hhshsslem1 27508 |
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