![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6907 | . . . . 5 ⊢ (𝑢 = 𝑈 → ( +𝑣 ‘𝑢) = ( +𝑣 ‘𝑈)) | |
2 | 1 | rneqd 5952 | . . . 4 ⊢ (𝑢 = 𝑈 → ran ( +𝑣 ‘𝑢) = ran ( +𝑣 ‘𝑈)) |
3 | df-ba 30625 | . . . 4 ⊢ BaseSet = (𝑢 ∈ V ↦ ran ( +𝑣 ‘𝑢)) | |
4 | fvex 6920 | . . . . 5 ⊢ ( +𝑣 ‘𝑈) ∈ V | |
5 | 4 | rnex 7933 | . . . 4 ⊢ ran ( +𝑣 ‘𝑈) ∈ V |
6 | 2, 3, 5 | fvmpt 7016 | . . 3 ⊢ (𝑈 ∈ V → (BaseSet‘𝑈) = ran ( +𝑣 ‘𝑈)) |
7 | rn0 5939 | . . . . 5 ⊢ ran ∅ = ∅ | |
8 | 7 | eqcomi 2744 | . . . 4 ⊢ ∅ = ran ∅ |
9 | fvprc 6899 | . . . 4 ⊢ (¬ 𝑈 ∈ V → (BaseSet‘𝑈) = ∅) | |
10 | fvprc 6899 | . . . . 5 ⊢ (¬ 𝑈 ∈ V → ( +𝑣 ‘𝑈) = ∅) | |
11 | 10 | rneqd 5952 | . . . 4 ⊢ (¬ 𝑈 ∈ V → ran ( +𝑣 ‘𝑈) = ran ∅) |
12 | 8, 9, 11 | 3eqtr4a 2801 | . . 3 ⊢ (¬ 𝑈 ∈ V → (BaseSet‘𝑈) = ran ( +𝑣 ‘𝑈)) |
13 | 6, 12 | pm2.61i 182 | . 2 ⊢ (BaseSet‘𝑈) = ran ( +𝑣 ‘𝑈) |
14 | bafval.1 | . 2 ⊢ 𝑋 = (BaseSet‘𝑈) | |
15 | bafval.2 | . . 3 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
16 | 15 | rneqi 5951 | . 2 ⊢ ran 𝐺 = ran ( +𝑣 ‘𝑈) |
17 | 13, 14, 16 | 3eqtr4i 2773 | 1 ⊢ 𝑋 = ran 𝐺 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 ran crn 5690 ‘cfv 6563 +𝑣 cpv 30614 BaseSetcba 30615 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fv 6571 df-ba 30625 |
This theorem is referenced by: nvi 30643 nvgf 30647 nvsf 30648 nvgcl 30649 nvcom 30650 nvass 30651 nvadd32 30652 nvrcan 30653 nvadd4 30654 nvscl 30655 nvsid 30656 nvsass 30657 nvdi 30659 nvdir 30660 nv2 30661 nvzcl 30663 nv0rid 30664 nv0lid 30665 nv0 30666 nvsz 30667 nvinv 30668 nvinvfval 30669 nvmval 30671 nvmfval 30673 nvnnncan1 30676 nvnegneg 30678 nvrinv 30680 nvlinv 30681 nvaddsub 30684 cnnvba 30708 sspba 30756 isph 30851 phpar 30853 ip0i 30854 ipdirilem 30858 hhba 31196 hhssabloilem 31290 hhshsslem1 31296 |
Copyright terms: Public domain | W3C validator |