Theorem opsqrlem3 29925

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 5699	. . . . 5 ⊢ (𝑧 = 𝐺 → (𝑧 ∘ 𝑧) = (𝐺 ∘ 𝐺))
3	2	oveq2d 7151	. . . 4 ⊢ (𝑧 = 𝐺 → (𝑇 −op (𝑧 ∘ 𝑧)) = (𝑇 −op (𝐺 ∘ 𝐺)))
4	3	oveq2d 7151	. . 3 ⊢ (𝑧 = 𝐺 → ((1 / 2) ·op (𝑇 −op (𝑧 ∘ 𝑧))) = ((1 / 2) ·op (𝑇 −op (𝐺 ∘ 𝐺))))
5	1, 4	oveq12d 7153	. 2 ⊢ (𝑧 = 𝐺 → (𝑧 +op ((1 / 2) ·op (𝑇 −op (𝑧 ∘ 𝑧)))) = (𝐺 +op ((1 / 2) ·op (𝑇 −op (𝐺 ∘ 𝐺)))))
6		eqidd 2799	. 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 5699	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 ∘ 𝑥) = (𝑧 ∘ 𝑧))
10	9	oveq2d 7151	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑇 −op (𝑥 ∘ 𝑥)) = (𝑇 −op (𝑧 ∘ 𝑧)))
11	10	oveq2d 7151	. . . . 5 ⊢ (𝑥 = 𝑧 → ((1 / 2) ·op (𝑇 −op (𝑥 ∘ 𝑥))) = ((1 / 2) ·op (𝑇 −op (𝑧 ∘ 𝑧))))
12	8, 11	oveq12d 7153	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 +op ((1 / 2) ·op (𝑇 −op (𝑥 ∘ 𝑥)))) = (𝑧 +op ((1 / 2) ·op (𝑇 −op (𝑧 ∘ 𝑧)))))
13		eqidd 2799	. . . 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 7228	. . 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 2821	. 2 ⊢ 𝑆 = (𝑧 ∈ HrmOp, 𝑤 ∈ HrmOp ↦ (𝑧 +op ((1 / 2) ·op (𝑇 −op (𝑧 ∘ 𝑧)))))
16		ovex 7168	. 2 ⊢ (𝐺 +op ((1 / 2) ·op (𝑇 −op (𝐺 ∘ 𝐺)))) ∈ V
17	5, 6, 15, 16	ovmpo 7289	1 ⊢ ((𝐺 ∈ HrmOp ∧ 𝐻 ∈ HrmOp) → (𝐺𝑆𝐻) = (𝐺 +op ((1 / 2) ·op (𝑇 −op (𝐺 ∘ 𝐺)))))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {csn 4525   × cxp 5517   ∘ ccom 5523  (class class class)co 7135   ∈ cmpo 7137  1c1 10527   / cdiv 11286  ℕcn 11625  2c2 11680  seqcseq 13364   +_op chos 28721   ·_op chot 28722   −_op chod 28723   0_hop ch0o 28726  HrmOpcho 28733

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140

This theorem is referenced by:  opsqrlem4  29926  opsqrlem5  29927