vafval - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > vafval		Structured version Visualization version GIF version

Theorem vafval 30585

Description: Value of the function for the vector addition (group) operation on a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)

**Hypothesis**
Ref	Expression
vafval.2	⊢ 𝐺 = ( +_𝑣 ‘𝑈)

**Assertion**
Ref	Expression
vafval	⊢ 𝐺 = (1^st ‘(1^st ‘𝑈))

**Proof of Theorem vafval**
Step	Hyp	Ref	Expression
1		vafval.2	. 2 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
2		df-va 30577	. . . . 5 ⊢ +_𝑣 = (1^st ∘ 1^st )
3	2	fveq1i 6829	. . . 4 ⊢ ( +_𝑣 ‘𝑈) = ((1^st ∘ 1^st )‘𝑈)
4		fo1st 7947	. . . . . 6 ⊢ 1^st :V–onto→V
5		fof 6740	. . . . . 6 ⊢ (1^st :V–onto→V → 1^st :V⟶V)
6	4, 5	ax-mp 5	. . . . 5 ⊢ 1^st :V⟶V
7		fvco3 6927	. . . . 5 ⊢ ((1^st :V⟶V ∧ 𝑈 ∈ V) → ((1^st ∘ 1^st )‘𝑈) = (1^st ‘(1^st ‘𝑈)))
8	6, 7	mpan 690	. . . 4 ⊢ (𝑈 ∈ V → ((1^st ∘ 1^st )‘𝑈) = (1^st ‘(1^st ‘𝑈)))
9	3, 8	eqtrid 2780	. . 3 ⊢ (𝑈 ∈ V → ( +_𝑣 ‘𝑈) = (1^st ‘(1^st ‘𝑈)))
10		fvprc 6820	. . . 4 ⊢ (¬ 𝑈 ∈ V → ( +_𝑣 ‘𝑈) = ∅)
11		fvprc 6820	. . . . . 6 ⊢ (¬ 𝑈 ∈ V → (1^st ‘𝑈) = ∅)
12	11	fveq2d 6832	. . . . 5 ⊢ (¬ 𝑈 ∈ V → (1^st ‘(1^st ‘𝑈)) = (1^st ‘∅))
13		1st0 7933	. . . . 5 ⊢ (1^st ‘∅) = ∅
14	12, 13	eqtr2di 2785	. . . 4 ⊢ (¬ 𝑈 ∈ V → ∅ = (1^st ‘(1^st ‘𝑈)))
15	10, 14	eqtrd 2768	. . 3 ⊢ (¬ 𝑈 ∈ V → ( +_𝑣 ‘𝑈) = (1^st ‘(1^st ‘𝑈)))
16	9, 15	pm2.61i 182	. 2 ⊢ ( +_𝑣 ‘𝑈) = (1^st ‘(1^st ‘𝑈))
17	1, 16	eqtri 2756	1 ⊢ 𝐺 = (1^st ‘(1^st ‘𝑈))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 = wceq 1541 ∈ wcel 2113 Vcvv 3437 ∅c0 4282 ∘ ccom 5623 ⟶wf 6482 –onto→wfo 6484 ‘cfv 6486 1^st c1st 7925 +_𝑣 cpv 30567

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5236 ax-nul 5246 ax-pr 5372 ax-un 7674

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-ral 3049 df-rex 3058 df-rab 3397 df-v 3439 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4283 df-if 4475 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-br 5094 df-opab 5156 df-mpt 5175 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-fo 6492 df-fv 6494 df-1st 7927 df-va 30577

This theorem is referenced by: nvvop 30591 nvablo 30598 nvsf 30601 nvscl 30608 nvsid 30609 nvsass 30610 nvdi 30612 nvdir 30613 nv2 30614 nv0 30619 nvsz 30620 nvinv 30621 cnnvg 30660 phop 30800 ip0i 30807 ipdirilem 30811 h2hva 30956 hhssva 31239 hhshsslem1 31249

W3C validator