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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 6906 | . . . . 5 ⊢ (𝑢 = 𝑈 → ( +𝑣 ‘𝑢) = ( +𝑣 ‘𝑈)) | |
| 2 | 1 | rneqd 5949 | . . . 4 ⊢ (𝑢 = 𝑈 → ran ( +𝑣 ‘𝑢) = ran ( +𝑣 ‘𝑈)) |
| 3 | df-ba 30615 | . . . 4 ⊢ BaseSet = (𝑢 ∈ V ↦ ran ( +𝑣 ‘𝑢)) | |
| 4 | fvex 6919 | . . . . 5 ⊢ ( +𝑣 ‘𝑈) ∈ V | |
| 5 | 4 | rnex 7932 | . . . 4 ⊢ ran ( +𝑣 ‘𝑈) ∈ V |
| 6 | 2, 3, 5 | fvmpt 7016 | . . 3 ⊢ (𝑈 ∈ V → (BaseSet‘𝑈) = ran ( +𝑣 ‘𝑈)) |
| 7 | rn0 5936 | . . . . 5 ⊢ ran ∅ = ∅ | |
| 8 | 7 | eqcomi 2746 | . . . 4 ⊢ ∅ = ran ∅ |
| 9 | fvprc 6898 | . . . 4 ⊢ (¬ 𝑈 ∈ V → (BaseSet‘𝑈) = ∅) | |
| 10 | fvprc 6898 | . . . . 5 ⊢ (¬ 𝑈 ∈ V → ( +𝑣 ‘𝑈) = ∅) | |
| 11 | 10 | rneqd 5949 | . . . 4 ⊢ (¬ 𝑈 ∈ V → ran ( +𝑣 ‘𝑈) = ran ∅) |
| 12 | 8, 9, 11 | 3eqtr4a 2803 | . . 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 5948 | . 2 ⊢ ran 𝐺 = ran ( +𝑣 ‘𝑈) |
| 17 | 13, 14, 16 | 3eqtr4i 2775 | 1 ⊢ 𝑋 = ran 𝐺 |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 ran crn 5686 ‘cfv 6561 +𝑣 cpv 30604 BaseSetcba 30605 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-ba 30615 |
| This theorem is referenced by: nvi 30633 nvgf 30637 nvsf 30638 nvgcl 30639 nvcom 30640 nvass 30641 nvadd32 30642 nvrcan 30643 nvadd4 30644 nvscl 30645 nvsid 30646 nvsass 30647 nvdi 30649 nvdir 30650 nv2 30651 nvzcl 30653 nv0rid 30654 nv0lid 30655 nv0 30656 nvsz 30657 nvinv 30658 nvinvfval 30659 nvmval 30661 nvmfval 30663 nvnnncan1 30666 nvnegneg 30668 nvrinv 30670 nvlinv 30671 nvaddsub 30674 cnnvba 30698 sspba 30746 isph 30841 phpar 30843 ip0i 30844 ipdirilem 30848 hhba 31186 hhssabloilem 31280 hhshsslem1 31286 |
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