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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 6670 | . . . . 5 ⊢ (𝑢 = 𝑈 → ( +𝑣 ‘𝑢) = ( +𝑣 ‘𝑈)) | |
2 | 1 | rneqd 5808 | . . . 4 ⊢ (𝑢 = 𝑈 → ran ( +𝑣 ‘𝑢) = ran ( +𝑣 ‘𝑈)) |
3 | df-ba 28373 | . . . 4 ⊢ BaseSet = (𝑢 ∈ V ↦ ran ( +𝑣 ‘𝑢)) | |
4 | fvex 6683 | . . . . 5 ⊢ ( +𝑣 ‘𝑈) ∈ V | |
5 | 4 | rnex 7617 | . . . 4 ⊢ ran ( +𝑣 ‘𝑈) ∈ V |
6 | 2, 3, 5 | fvmpt 6768 | . . 3 ⊢ (𝑈 ∈ V → (BaseSet‘𝑈) = ran ( +𝑣 ‘𝑈)) |
7 | rn0 5796 | . . . . 5 ⊢ ran ∅ = ∅ | |
8 | 7 | eqcomi 2830 | . . . 4 ⊢ ∅ = ran ∅ |
9 | fvprc 6663 | . . . 4 ⊢ (¬ 𝑈 ∈ V → (BaseSet‘𝑈) = ∅) | |
10 | fvprc 6663 | . . . . 5 ⊢ (¬ 𝑈 ∈ V → ( +𝑣 ‘𝑈) = ∅) | |
11 | 10 | rneqd 5808 | . . . 4 ⊢ (¬ 𝑈 ∈ V → ran ( +𝑣 ‘𝑈) = ran ∅) |
12 | 8, 9, 11 | 3eqtr4a 2882 | . . 3 ⊢ (¬ 𝑈 ∈ V → (BaseSet‘𝑈) = ran ( +𝑣 ‘𝑈)) |
13 | 6, 12 | pm2.61i 184 | . 2 ⊢ (BaseSet‘𝑈) = ran ( +𝑣 ‘𝑈) |
14 | bafval.1 | . 2 ⊢ 𝑋 = (BaseSet‘𝑈) | |
15 | bafval.2 | . . 3 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
16 | 15 | rneqi 5807 | . 2 ⊢ ran 𝐺 = ran ( +𝑣 ‘𝑈) |
17 | 13, 14, 16 | 3eqtr4i 2854 | 1 ⊢ 𝑋 = ran 𝐺 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∅c0 4291 ran crn 5556 ‘cfv 6355 +𝑣 cpv 28362 BaseSetcba 28363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fv 6363 df-ba 28373 |
This theorem is referenced by: nvi 28391 nvgf 28395 nvsf 28396 nvgcl 28397 nvcom 28398 nvass 28399 nvadd32 28400 nvrcan 28401 nvadd4 28402 nvscl 28403 nvsid 28404 nvsass 28405 nvdi 28407 nvdir 28408 nv2 28409 nvzcl 28411 nv0rid 28412 nv0lid 28413 nv0 28414 nvsz 28415 nvinv 28416 nvinvfval 28417 nvmval 28419 nvmfval 28421 nvnnncan1 28424 nvnegneg 28426 nvrinv 28428 nvlinv 28429 nvaddsub 28432 cnnvba 28456 sspba 28504 isph 28599 phpar 28601 ip0i 28602 ipdirilem 28606 hhba 28944 hhssabloilem 29038 hhshsslem1 29044 |
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