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Theorem opsqrlem3 29934
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 5723 . . . . 5 (𝑧 = 𝐺 → (𝑧𝑧) = (𝐺𝐺))
32oveq2d 7167 . . . 4 (𝑧 = 𝐺 → (𝑇op (𝑧𝑧)) = (𝑇op (𝐺𝐺)))
43oveq2d 7167 . . 3 (𝑧 = 𝐺 → ((1 / 2) ·op (𝑇op (𝑧𝑧))) = ((1 / 2) ·op (𝑇op (𝐺𝐺))))
51, 4oveq12d 7169 . 2 (𝑧 = 𝐺 → (𝑧 +op ((1 / 2) ·op (𝑇op (𝑧𝑧)))) = (𝐺 +op ((1 / 2) ·op (𝑇op (𝐺𝐺)))))
6 eqidd 2825 . 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 5723 . . . . . . 7 (𝑥 = 𝑧 → (𝑥𝑥) = (𝑧𝑧))
109oveq2d 7167 . . . . . 6 (𝑥 = 𝑧 → (𝑇op (𝑥𝑥)) = (𝑇op (𝑧𝑧)))
1110oveq2d 7167 . . . . 5 (𝑥 = 𝑧 → ((1 / 2) ·op (𝑇op (𝑥𝑥))) = ((1 / 2) ·op (𝑇op (𝑧𝑧))))
128, 11oveq12d 7169 . . . 4 (𝑥 = 𝑧 → (𝑥 +op ((1 / 2) ·op (𝑇op (𝑥𝑥)))) = (𝑧 +op ((1 / 2) ·op (𝑇op (𝑧𝑧)))))
13 eqidd 2825 . . . 4 (𝑦 = 𝑤 → (𝑧 +op ((1 / 2) ·op (𝑇op (𝑧𝑧)))) = (𝑧 +op ((1 / 2) ·op (𝑇op (𝑧𝑧)))))
1412, 13cbvmpov 7244 . . 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 2847 . 2 𝑆 = (𝑧 ∈ HrmOp, 𝑤 ∈ HrmOp ↦ (𝑧 +op ((1 / 2) ·op (𝑇op (𝑧𝑧)))))
16 ovex 7184 . 2 (𝐺 +op ((1 / 2) ·op (𝑇op (𝐺𝐺)))) ∈ V
175, 6, 15, 16ovmpo 7305 1 ((𝐺 ∈ HrmOp ∧ 𝐻 ∈ HrmOp) → (𝐺𝑆𝐻) = (𝐺 +op ((1 / 2) ·op (𝑇op (𝐺𝐺)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  {csn 4550   × cxp 5541  ccom 5547  (class class class)co 7151  cmpo 7153  1c1 10538   / cdiv 11297  cn 11636  2c2 11691  seqcseq 13375   +op chos 28730   ·op chot 28731  op chod 28732   0hop ch0o 28735  HrmOpcho 28742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5054  df-opab 5116  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-iota 6304  df-fun 6347  df-fv 6353  df-ov 7154  df-oprab 7155  df-mpo 7156
This theorem is referenced by:  opsqrlem4  29935  opsqrlem5  29936
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