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Theorem opsqrlem3 28841
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 5251 . . . . 5 (𝑧 = 𝐺 → (𝑧𝑧) = (𝐺𝐺))
32oveq2d 6621 . . . 4 (𝑧 = 𝐺 → (𝑇op (𝑧𝑧)) = (𝑇op (𝐺𝐺)))
43oveq2d 6621 . . 3 (𝑧 = 𝐺 → ((1 / 2) ·op (𝑇op (𝑧𝑧))) = ((1 / 2) ·op (𝑇op (𝐺𝐺))))
51, 4oveq12d 6623 . 2 (𝑧 = 𝐺 → (𝑧 +op ((1 / 2) ·op (𝑇op (𝑧𝑧)))) = (𝐺 +op ((1 / 2) ·op (𝑇op (𝐺𝐺)))))
6 eqidd 2627 . 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 5251 . . . . . . 7 (𝑥 = 𝑧 → (𝑥𝑥) = (𝑧𝑧))
109oveq2d 6621 . . . . . 6 (𝑥 = 𝑧 → (𝑇op (𝑥𝑥)) = (𝑇op (𝑧𝑧)))
1110oveq2d 6621 . . . . 5 (𝑥 = 𝑧 → ((1 / 2) ·op (𝑇op (𝑥𝑥))) = ((1 / 2) ·op (𝑇op (𝑧𝑧))))
128, 11oveq12d 6623 . . . 4 (𝑥 = 𝑧 → (𝑥 +op ((1 / 2) ·op (𝑇op (𝑥𝑥)))) = (𝑧 +op ((1 / 2) ·op (𝑇op (𝑧𝑧)))))
13 eqidd 2627 . . . 4 (𝑦 = 𝑤 → (𝑧 +op ((1 / 2) ·op (𝑇op (𝑧𝑧)))) = (𝑧 +op ((1 / 2) ·op (𝑇op (𝑧𝑧)))))
1412, 13cbvmpt2v 6689 . . 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 2648 . 2 𝑆 = (𝑧 ∈ HrmOp, 𝑤 ∈ HrmOp ↦ (𝑧 +op ((1 / 2) ·op (𝑇op (𝑧𝑧)))))
16 ovex 6633 . 2 (𝐺 +op ((1 / 2) ·op (𝑇op (𝐺𝐺)))) ∈ V
175, 6, 15, 16ovmpt2 6750 1 ((𝐺 ∈ HrmOp ∧ 𝐻 ∈ HrmOp) → (𝐺𝑆𝐻) = (𝐺 +op ((1 / 2) ·op (𝑇op (𝐺𝐺)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  {csn 4153   × cxp 5077  ccom 5083  (class class class)co 6605  cmpt2 6607  1c1 9882   / cdiv 10629  cn 10965  2c2 11015  seqcseq 12738   +op chos 27635   ·op chot 27636  op chod 27637   0hop ch0o 27640  HrmOpcho 27647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5813  df-fun 5852  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610
This theorem is referenced by:  opsqrlem4  28842  opsqrlem5  28843
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