Theorem opsqrlem3 32494

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 23	. . 3 ⊢ (𝑧 = 𝐺 → 𝑧 = 𝐺)
2	1, 1	coeq12d 5850	. . . . 5 ⊢ (𝑧 = 𝐺 → (𝑧 ∘ 𝑧) = (𝐺 ∘ 𝐺))
3	2	oveq2d 7426	. . . 4 ⊢ (𝑧 = 𝐺 → (𝑇 −op (𝑧 ∘ 𝑧)) = (𝑇 −op (𝐺 ∘ 𝐺)))
4	3	oveq2d 7426	. . 3 ⊢ (𝑧 = 𝐺 → ((1 / 2) ·op (𝑇 −op (𝑧 ∘ 𝑧))) = ((1 / 2) ·op (𝑇 −op (𝐺 ∘ 𝐺))))
5	1, 4	oveq12d 7428	. 2 ⊢ (𝑧 = 𝐺 → (𝑧 +op ((1 / 2) ·op (𝑇 −op (𝑧 ∘ 𝑧)))) = (𝐺 +op ((1 / 2) ·op (𝑇 −op (𝐺 ∘ 𝐺)))))
6		eqidd 2764	. 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 23	. . . . 5 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
9	8, 8	coeq12d 5850	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 ∘ 𝑥) = (𝑧 ∘ 𝑧))
10	9	oveq2d 7426	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑇 −op (𝑥 ∘ 𝑥)) = (𝑇 −op (𝑧 ∘ 𝑧)))
11	10	oveq2d 7426	. . . . 5 ⊢ (𝑥 = 𝑧 → ((1 / 2) ·op (𝑇 −op (𝑥 ∘ 𝑥))) = ((1 / 2) ·op (𝑇 −op (𝑧 ∘ 𝑧))))
12	8, 11	oveq12d 7428	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 +op ((1 / 2) ·op (𝑇 −op (𝑥 ∘ 𝑥)))) = (𝑧 +op ((1 / 2) ·op (𝑇 −op (𝑧 ∘ 𝑧)))))
13		eqidd 2764	. . . 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 7505	. . 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 2786	. 2 ⊢ 𝑆 = (𝑧 ∈ HrmOp, 𝑤 ∈ HrmOp ↦ (𝑧 +op ((1 / 2) ·op (𝑇 −op (𝑧 ∘ 𝑧)))))
16		ovex 7443	. 2 ⊢ (𝐺 +op ((1 / 2) ·op (𝑇 −op (𝐺 ∘ 𝐺)))) ∈ V
17	5, 6, 15, 16	ovmpo 7570	1 ⊢ ((𝐺 ∈ HrmOp ∧ 𝐻 ∈ HrmOp) → (𝐺𝑆𝐻) = (𝐺 +op ((1 / 2) ·op (𝑇 −op (𝐺 ∘ 𝐺)))))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {csn 4589   × cxp 5659   ∘ ccom 5665  (class class class)co 7410   ∈ cmpo 7412  1c1 11096   / cdiv 11866  ℕcn 12228  2c2 12290  seqcseq 14033   +_op chos 31290   ·_op chot 31291   −_op chod 31292   0_hop ch0o 31295  HrmOpcho 31302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415

This theorem is referenced by:  opsqrlem4  32495  opsqrlem5  32496