HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  opsqrlem3 Structured version   Visualization version   GIF version

Theorem opsqrlem3 29526
Description: Lemma for opsqri . (Contributed by NM, 22-Aug-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
opsqrlem2.1 𝑇 ∈ HrmOp
opsqrlem2.2 𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇op (𝑥𝑥)))))
opsqrlem2.3 𝐹 = seq1(𝑆, (ℕ × { 0hop }))
Assertion
Ref Expression
opsqrlem3 ((𝐺 ∈ HrmOp ∧ 𝐻 ∈ HrmOp) → (𝐺𝑆𝐻) = (𝐺 +op ((1 / 2) ·op (𝑇op (𝐺𝐺)))))
Distinct variable group:   𝑥,𝑦,𝑇
Allowed substitution hints:   𝑆(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝐻(𝑥,𝑦)

Proof of Theorem opsqrlem3
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . . 3 (𝑧 = 𝐺𝑧 = 𝐺)
21, 1coeq12d 5490 . . . . 5 (𝑧 = 𝐺 → (𝑧𝑧) = (𝐺𝐺))
32oveq2d 6894 . . . 4 (𝑧 = 𝐺 → (𝑇op (𝑧𝑧)) = (𝑇op (𝐺𝐺)))
43oveq2d 6894 . . 3 (𝑧 = 𝐺 → ((1 / 2) ·op (𝑇op (𝑧𝑧))) = ((1 / 2) ·op (𝑇op (𝐺𝐺))))
51, 4oveq12d 6896 . 2 (𝑧 = 𝐺 → (𝑧 +op ((1 / 2) ·op (𝑇op (𝑧𝑧)))) = (𝐺 +op ((1 / 2) ·op (𝑇op (𝐺𝐺)))))
6 eqidd 2800 . 2 (𝑤 = 𝐻 → (𝐺 +op ((1 / 2) ·op (𝑇op (𝐺𝐺)))) = (𝐺 +op ((1 / 2) ·op (𝑇op (𝐺𝐺)))))
7 opsqrlem2.2 . . 3 𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇op (𝑥𝑥)))))
8 id 22 . . . . 5 (𝑥 = 𝑧𝑥 = 𝑧)
98, 8coeq12d 5490 . . . . . . 7 (𝑥 = 𝑧 → (𝑥𝑥) = (𝑧𝑧))
109oveq2d 6894 . . . . . 6 (𝑥 = 𝑧 → (𝑇op (𝑥𝑥)) = (𝑇op (𝑧𝑧)))
1110oveq2d 6894 . . . . 5 (𝑥 = 𝑧 → ((1 / 2) ·op (𝑇op (𝑥𝑥))) = ((1 / 2) ·op (𝑇op (𝑧𝑧))))
128, 11oveq12d 6896 . . . 4 (𝑥 = 𝑧 → (𝑥 +op ((1 / 2) ·op (𝑇op (𝑥𝑥)))) = (𝑧 +op ((1 / 2) ·op (𝑇op (𝑧𝑧)))))
13 eqidd 2800 . . . 4 (𝑦 = 𝑤 → (𝑧 +op ((1 / 2) ·op (𝑇op (𝑧𝑧)))) = (𝑧 +op ((1 / 2) ·op (𝑇op (𝑧𝑧)))))
1412, 13cbvmpt2v 6969 . . 3 (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇op (𝑥𝑥))))) = (𝑧 ∈ HrmOp, 𝑤 ∈ HrmOp ↦ (𝑧 +op ((1 / 2) ·op (𝑇op (𝑧𝑧)))))
157, 14eqtri 2821 . 2 𝑆 = (𝑧 ∈ HrmOp, 𝑤 ∈ HrmOp ↦ (𝑧 +op ((1 / 2) ·op (𝑇op (𝑧𝑧)))))
16 ovex 6910 . 2 (𝐺 +op ((1 / 2) ·op (𝑇op (𝐺𝐺)))) ∈ V
175, 6, 15, 16ovmpt2 7030 1 ((𝐺 ∈ HrmOp ∧ 𝐻 ∈ HrmOp) → (𝐺𝑆𝐻) = (𝐺 +op ((1 / 2) ·op (𝑇op (𝐺𝐺)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  {csn 4368   × cxp 5310  ccom 5316  (class class class)co 6878  cmpt2 6880  1c1 10225   / cdiv 10976  cn 11312  2c2 11368  seqcseq 13055   +op chos 28320   ·op chot 28321  op chod 28322   0hop ch0o 28325  HrmOpcho 28332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-iota 6064  df-fun 6103  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883
This theorem is referenced by:  opsqrlem4  29527  opsqrlem5  29528
  Copyright terms: Public domain W3C validator