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Theorem opsqrlem3 31904
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 5858 . . . . 5 (𝑧 = 𝐺 → (𝑧𝑧) = (𝐺𝐺))
32oveq2d 7421 . . . 4 (𝑧 = 𝐺 → (𝑇op (𝑧𝑧)) = (𝑇op (𝐺𝐺)))
43oveq2d 7421 . . 3 (𝑧 = 𝐺 → ((1 / 2) ·op (𝑇op (𝑧𝑧))) = ((1 / 2) ·op (𝑇op (𝐺𝐺))))
51, 4oveq12d 7423 . 2 (𝑧 = 𝐺 → (𝑧 +op ((1 / 2) ·op (𝑇op (𝑧𝑧)))) = (𝐺 +op ((1 / 2) ·op (𝑇op (𝐺𝐺)))))
6 eqidd 2727 . 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 5858 . . . . . . 7 (𝑥 = 𝑧 → (𝑥𝑥) = (𝑧𝑧))
109oveq2d 7421 . . . . . 6 (𝑥 = 𝑧 → (𝑇op (𝑥𝑥)) = (𝑇op (𝑧𝑧)))
1110oveq2d 7421 . . . . 5 (𝑥 = 𝑧 → ((1 / 2) ·op (𝑇op (𝑥𝑥))) = ((1 / 2) ·op (𝑇op (𝑧𝑧))))
128, 11oveq12d 7423 . . . 4 (𝑥 = 𝑧 → (𝑥 +op ((1 / 2) ·op (𝑇op (𝑥𝑥)))) = (𝑧 +op ((1 / 2) ·op (𝑇op (𝑧𝑧)))))
13 eqidd 2727 . . . 4 (𝑦 = 𝑤 → (𝑧 +op ((1 / 2) ·op (𝑇op (𝑧𝑧)))) = (𝑧 +op ((1 / 2) ·op (𝑇op (𝑧𝑧)))))
1412, 13cbvmpov 7500 . . 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 2754 . 2 𝑆 = (𝑧 ∈ HrmOp, 𝑤 ∈ HrmOp ↦ (𝑧 +op ((1 / 2) ·op (𝑇op (𝑧𝑧)))))
16 ovex 7438 . 2 (𝐺 +op ((1 / 2) ·op (𝑇op (𝐺𝐺)))) ∈ V
175, 6, 15, 16ovmpo 7564 1 ((𝐺 ∈ HrmOp ∧ 𝐻 ∈ HrmOp) → (𝐺𝑆𝐻) = (𝐺 +op ((1 / 2) ·op (𝑇op (𝐺𝐺)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  {csn 4623   × cxp 5667  ccom 5673  (class class class)co 7405  cmpo 7407  1c1 11113   / cdiv 11875  cn 12216  2c2 12271  seqcseq 13972   +op chos 30700   ·op chot 30701  op chod 30702   0hop ch0o 30705  HrmOpcho 30712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410
This theorem is referenced by:  opsqrlem4  31905  opsqrlem5  31906
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