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Theorem opsqrlem3 29129
 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 5319 . . . . 5 (𝑧 = 𝐺 → (𝑧𝑧) = (𝐺𝐺))
32oveq2d 6706 . . . 4 (𝑧 = 𝐺 → (𝑇op (𝑧𝑧)) = (𝑇op (𝐺𝐺)))
43oveq2d 6706 . . 3 (𝑧 = 𝐺 → ((1 / 2) ·op (𝑇op (𝑧𝑧))) = ((1 / 2) ·op (𝑇op (𝐺𝐺))))
51, 4oveq12d 6708 . 2 (𝑧 = 𝐺 → (𝑧 +op ((1 / 2) ·op (𝑇op (𝑧𝑧)))) = (𝐺 +op ((1 / 2) ·op (𝑇op (𝐺𝐺)))))
6 eqidd 2652 . 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 5319 . . . . . . 7 (𝑥 = 𝑧 → (𝑥𝑥) = (𝑧𝑧))
109oveq2d 6706 . . . . . 6 (𝑥 = 𝑧 → (𝑇op (𝑥𝑥)) = (𝑇op (𝑧𝑧)))
1110oveq2d 6706 . . . . 5 (𝑥 = 𝑧 → ((1 / 2) ·op (𝑇op (𝑥𝑥))) = ((1 / 2) ·op (𝑇op (𝑧𝑧))))
128, 11oveq12d 6708 . . . 4 (𝑥 = 𝑧 → (𝑥 +op ((1 / 2) ·op (𝑇op (𝑥𝑥)))) = (𝑧 +op ((1 / 2) ·op (𝑇op (𝑧𝑧)))))
13 eqidd 2652 . . . 4 (𝑦 = 𝑤 → (𝑧 +op ((1 / 2) ·op (𝑇op (𝑧𝑧)))) = (𝑧 +op ((1 / 2) ·op (𝑇op (𝑧𝑧)))))
1412, 13cbvmpt2v 6777 . . 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 2673 . 2 𝑆 = (𝑧 ∈ HrmOp, 𝑤 ∈ HrmOp ↦ (𝑧 +op ((1 / 2) ·op (𝑇op (𝑧𝑧)))))
16 ovex 6718 . 2 (𝐺 +op ((1 / 2) ·op (𝑇op (𝐺𝐺)))) ∈ V
175, 6, 15, 16ovmpt2 6838 1 ((𝐺 ∈ HrmOp ∧ 𝐻 ∈ HrmOp) → (𝐺𝑆𝐻) = (𝐺 +op ((1 / 2) ·op (𝑇op (𝐺𝐺)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  {csn 4210   × cxp 5141   ∘ ccom 5147  (class class class)co 6690   ↦ cmpt2 6692  1c1 9975   / cdiv 10722  ℕcn 11058  2c2 11108  seqcseq 12841   +op chos 27923   ·op chot 27924   −op chod 27925   0hop ch0o 27928  HrmOpcho 27935 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695 This theorem is referenced by:  opsqrlem4  29130  opsqrlem5  29131
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