Theorem opsqrlem3 32071

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 (𝑇 −op (𝑥 ∘ 𝑥)))))
opsqrlem2.3	⊢ 𝐹 = seq1(𝑆, (ℕ × { 0hop }))

Assertion

Ref	Expression
opsqrlem3	⊢ ((𝐺 ∈ HrmOp ∧ 𝐻 ∈ HrmOp) → (𝐺𝑆𝐻) = (𝐺 +op ((1 / 2) ·op (𝑇 −op (𝐺 ∘ 𝐺)))))

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.

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ (𝑧 = 𝐺 → 𝑧 = 𝐺)
2	1, 1	coeq12d 5828	. . . . 5 ⊢ (𝑧 = 𝐺 → (𝑧 ∘ 𝑧) = (𝐺 ∘ 𝐺))
3	2	oveq2d 7403	. . . 4 ⊢ (𝑧 = 𝐺 → (𝑇 −op (𝑧 ∘ 𝑧)) = (𝑇 −op (𝐺 ∘ 𝐺)))
4	3	oveq2d 7403	. . 3 ⊢ (𝑧 = 𝐺 → ((1 / 2) ·op (𝑇 −op (𝑧 ∘ 𝑧))) = ((1 / 2) ·op (𝑇 −op (𝐺 ∘ 𝐺))))
5	1, 4	oveq12d 7405	. 2 ⊢ (𝑧 = 𝐺 → (𝑧 +op ((1 / 2) ·op (𝑇 −op (𝑧 ∘ 𝑧)))) = (𝐺 +op ((1 / 2) ·op (𝑇 −op (𝐺 ∘ 𝐺)))))
6		eqidd 2730	. 2 ⊢ (𝑤 = 𝐻 → (𝐺 +op ((1 / 2) ·op (𝑇 −op (𝐺 ∘ 𝐺)))) = (𝐺 +op ((1 / 2) ·op (𝑇 −op (𝐺 ∘ 𝐺)))))
7		opsqrlem2.2	. . 3 ⊢ 𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇 −op (𝑥 ∘ 𝑥)))))
8		id 22	. . . . 5 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
9	8, 8	coeq12d 5828	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 ∘ 𝑥) = (𝑧 ∘ 𝑧))
10	9	oveq2d 7403	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑇 −op (𝑥 ∘ 𝑥)) = (𝑇 −op (𝑧 ∘ 𝑧)))
11	10	oveq2d 7403	. . . . 5 ⊢ (𝑥 = 𝑧 → ((1 / 2) ·op (𝑇 −op (𝑥 ∘ 𝑥))) = ((1 / 2) ·op (𝑇 −op (𝑧 ∘ 𝑧))))
12	8, 11	oveq12d 7405	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 +op ((1 / 2) ·op (𝑇 −op (𝑥 ∘ 𝑥)))) = (𝑧 +op ((1 / 2) ·op (𝑇 −op (𝑧 ∘ 𝑧)))))
13		eqidd 2730	. . . 4 ⊢ (𝑦 = 𝑤 → (𝑧 +op ((1 / 2) ·op (𝑇 −op (𝑧 ∘ 𝑧)))) = (𝑧 +op ((1 / 2) ·op (𝑇 −op (𝑧 ∘ 𝑧)))))
14	12, 13	cbvmpov 7484	. . 3 ⊢ (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇 −op (𝑥 ∘ 𝑥))))) = (𝑧 ∈ HrmOp, 𝑤 ∈ HrmOp ↦ (𝑧 +op ((1 / 2) ·op (𝑇 −op (𝑧 ∘ 𝑧)))))
15	7, 14	eqtri 2752	. 2 ⊢ 𝑆 = (𝑧 ∈ HrmOp, 𝑤 ∈ HrmOp ↦ (𝑧 +op ((1 / 2) ·op (𝑇 −op (𝑧 ∘ 𝑧)))))
16		ovex 7420	. 2 ⊢ (𝐺 +op ((1 / 2) ·op (𝑇 −op (𝐺 ∘ 𝐺)))) ∈ V
17	5, 6, 15, 16	ovmpo 7549	1 ⊢ ((𝐺 ∈ HrmOp ∧ 𝐻 ∈ HrmOp) → (𝐺𝑆𝐻) = (𝐺 +op ((1 / 2) ·op (𝑇 −op (𝐺 ∘ 𝐺)))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {csn 4589   × cxp 5636   ∘ ccom 5642  (class class class)co 7387   ∈ cmpo 7389  1c1 11069   / cdiv 11835  ℕcn 12186  2c2 12241  seqcseq 13966   +_op chos 30867   ·_op chot 30868   −_op chod 30869   0_hop ch0o 30872  HrmOpcho 30879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392

This theorem is referenced by:  opsqrlem4  32072  opsqrlem5  32073