Theorem hfmmvalt 9522

Description: Value of the scalar product with a Hilbert space functional.

Assertion

Ref	Expression
hfmmvalt	⊢ ((A ∈ ℂ ⋀ T: ℋ –→ℂ) → (A ·fn T) = {⟨x, y⟩∣(x ∈ ℋ ⋀ y = (A · (T ‘x)))})

Distinct variable groups:   x,y,A   x,T,y

Proof of Theorem hfmmvalt

Step	Hyp	Ref	Expression
1		ax-hilex 8876	. . . 4 ⊢ ℋ ∈ V
2	1	opabex2 3624	. . 3 ⊢ {⟨x, y⟩∣(x ∈ ℋ ⋀ y = (A · (T ‘x)))} ∈ V
3		opreq1 3982	. . . . . 6 ⊢ (f = A → (f · (g ‘x)) = (A · (g ‘x)))
4	3	eqeq2d 1493	. . . . 5 ⊢ (f = A → (y = (f · (g ‘x)) ↔ y = (A · (g ‘x))))
5	4	anbi2d 619	. . . 4 ⊢ (f = A → ((x ∈ ℋ ⋀ y = (f · (g ‘x))) ↔ (x ∈ ℋ ⋀ y = (A · (g ‘x)))))
6	5	opabbidv 2683	. . 3 ⊢ (f = A → {⟨x, y⟩∣(x ∈ ℋ ⋀ y = (f · (g ‘x)))} = {⟨x, y⟩∣(x ∈ ℋ ⋀ y = (A · (g ‘x)))})
7		fveq1 3737	. . . . . . 7 ⊢ (g = T → (g ‘x) = (T ‘x))
8	7	opreq2d 3990	. . . . . 6 ⊢ (g = T → (A · (g ‘x)) = (A · (T ‘x)))
9	8	eqeq2d 1493	. . . . 5 ⊢ (g = T → (y = (A · (g ‘x)) ↔ y = (A · (T ‘x))))
10	9	anbi2d 619	. . . 4 ⊢ (g = T → ((x ∈ ℋ ⋀ y = (A · (g ‘x))) ↔ (x ∈ ℋ ⋀ y = (A · (T ‘x)))))
11	10	opabbidv 2683	. . 3 ⊢ (g = T → {⟨x, y⟩∣(x ∈ ℋ ⋀ y = (A · (g ‘x)))} = {⟨x, y⟩∣(x ∈ ℋ ⋀ y = (A · (T ‘x)))})
12		df-hfmul 9517	. . . 4 ⊢ ·fn = {⟨⟨f, g⟩, h⟩∣((f ∈ ℂ ⋀ g: ℋ –→ℂ) ⋀ h = {⟨x, y⟩∣(x ∈ ℋ ⋀ y = (f · (g ‘x)))})}
13		axcnex 5280	. . . . . . . 8 ⊢ ℂ ∈ V
14	13, 1	elmap 4348	. . . . . . 7 ⊢ (g ∈ (ℂ ↑m ℋ ) ↔ g: ℋ –→ℂ)
15	14	anbi2i 483	. . . . . 6 ⊢ ((f ∈ ℂ ⋀ g ∈ (ℂ ↑m ℋ )) ↔ (f ∈ ℂ ⋀ g: ℋ –→ℂ))
16	15	anbi1i 484	. . . . 5 ⊢ (((f ∈ ℂ ⋀ g ∈ (ℂ ↑m ℋ )) ⋀ h = {⟨x, y⟩∣(x ∈ ℋ ⋀ y = (f · (g ‘x)))}) ↔ ((f ∈ ℂ ⋀ g: ℋ –→ℂ) ⋀ h = {⟨x, y⟩∣(x ∈ ℋ ⋀ y = (f · (g ‘x)))}))
17	16	oprabbii 4011	. . . 4 ⊢ {⟨⟨f, g⟩, h⟩∣((f ∈ ℂ ⋀ g ∈ (ℂ ↑m ℋ )) ⋀ h = {⟨x, y⟩∣(x ∈ ℋ ⋀ y = (f · (g ‘x)))})} = {⟨⟨f, g⟩, h⟩∣((f ∈ ℂ ⋀ g: ℋ –→ℂ) ⋀ h = {⟨x, y⟩∣(x ∈ ℋ ⋀ y = (f · (g ‘x)))})}
18	12, 17	eqtr4 1505	. . 3 ⊢ ·fn = {⟨⟨f, g⟩, h⟩∣((f ∈ ℂ ⋀ g ∈ (ℂ ↑m ℋ )) ⋀ h = {⟨x, y⟩∣(x ∈ ℋ ⋀ y = (f · (g ‘x)))})}
19	2, 6, 11, 18	oprabval2 4042	. 2 ⊢ ((A ∈ ℂ ⋀ T ∈ (ℂ ↑m ℋ )) → (A ·fn T) = {⟨x, y⟩∣(x ∈ ℋ ⋀ y = (A · (T ‘x)))})
20	13, 1	elmap 4348	. 2 ⊢ (T ∈ (ℂ ↑m ℋ ) ↔ T: ℋ –→ℂ)
21	19, 20	sylan2br 456	1 ⊢ ((A ∈ ℂ ⋀ T: ℋ –→ℂ) → (A ·fn T) = {⟨x, y⟩∣(x ∈ ℋ ⋀ y = (A · (T ‘x)))})

Syntax hints:   → wi 3   ⋀ wa 223   = wceq 960   ∈ wcel 962  {copab 2679  –→wf 3192   ‘cfv 3196  (class class class)co 3977  {copab2 3978   ↑_m cm 4336  ℂcc 5245   · cmul 5252   ℋ chil 8795   ·_fn chft 8818

This theorem is referenced by:  hfmvalt 9529  brafnmult 9882  kbass2t 10057

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-9 969  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1129  ax-10o 1146  ax-16 1216  ax-11o 1224  ax-ext 1466  ax-rep 2706  ax-sep 2716  ax-nul 2723  ax-pow 2756  ax-pr 2793  ax-un 2880  ax-inf2 4637  ax-hilex 8876

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 780  df-3an 781  df-ex 985  df-sb 1178  df-eu 1388  df-mo 1389  df-clab 1471  df-cleq 1476  df-clel 1479  df-ne 1594  df-ral 1656  df-rex 1657  df-v 1819  df-sbc 1949  df-csb 2010  df-dif 2058  df-un 2059  df-in 2060  df-ss 2062  df-pss 2064  df-nul 2290  df-if 2372  df-pw 2412  df-sn 2422  df-pr 2423  df-tp 2425  df-op 2426  df-uni 2516  df-br 2633  df-opab 2680  df-tr 2694  df-eprel 2846  df-id 2849  df-po 2854  df-so 2864  df-fr 2931  df-we 2948  df-ord 2965  df-on 2966  df-lim 2967  df-suc 2968  df-om 3146  df-xp 3198  df-rel 3199  df-cnv 3200  df-co 3201  df-dm 3202  df-rn 3203  df-res 3204  df-ima 3205  df-fun 3206  df-fn 3207  df-f 3208  df-fv 3212  df-opr 3979  df-oprab 3980  df-qs 4280  df-map 4338  df-ni 5013  df-nq 5051  df-np 5099  df-nr 5180  df-c 5253  df-hfmul 9517

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