cnlnadjlem6 - Hilbert Space Explorer

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Theorem cnlnadjlem6 31325

Description: Lemma for cnlnadji 31329. 𝐹 is linear. (Contributed by NM, 17-Feb-2006.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
cnlnadjlem.1	⊢ 𝑇 ∈ LinOp
cnlnadjlem.2	⊢ 𝑇 ∈ ContOp
cnlnadjlem.3	⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·_ih 𝑦))
cnlnadjlem.4	⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·_ih 𝑦) = (𝑣 ·_ih 𝑤))
cnlnadjlem.5	⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵)

**Assertion**
Ref	Expression
cnlnadjlem6	⊢ 𝐹 ∈ LinOp

Distinct variable groups: 𝑣,𝑔,𝑤,𝑦 𝑤,𝐹 𝑇,𝑔,𝑣,𝑤,𝑦 𝑣,𝐺,𝑤 Allowed substitution hints: 𝐵(𝑦,𝑤,𝑣,𝑔) 𝐹(𝑦,𝑣,𝑔) 𝐺(𝑦,𝑔)

Proof of Theorem cnlnadjlem6 Dummy variables 𝑓 𝑧 𝑡 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnlnadjlem.5	. . 3 ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵)
2		cnlnadjlem.1	. . . 4 ⊢ 𝑇 ∈ LinOp
3		cnlnadjlem.2	. . . 4 ⊢ 𝑇 ∈ ContOp
4		cnlnadjlem.3	. . . 4 ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·_ih 𝑦))
5		cnlnadjlem.4	. . . 4 ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·_ih 𝑦) = (𝑣 ·_ih 𝑤))
6	2, 3, 4, 5	cnlnadjlem3 31322	. . 3 ⊢ (𝑦 ∈ ℋ → 𝐵 ∈ ℋ)
7	1, 6	fmpti 7112	. 2 ⊢ 𝐹: ℋ⟶ ℋ
8	2	lnopfi 31222	. . . . . . . . . 10 ⊢ 𝑇: ℋ⟶ ℋ
9	8	ffvelcdmi 7086	. . . . . . . . 9 ⊢ (𝑡 ∈ ℋ → (𝑇‘𝑡) ∈ ℋ)
10	9	adantl 483	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → (𝑇‘𝑡) ∈ ℋ)
11		hvmulcl 30266	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) → (𝑥 ·_ℎ 𝑓) ∈ ℋ)
12	11	ad2antrr 725	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → (𝑥 ·_ℎ 𝑓) ∈ ℋ)
13		simplr 768	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → 𝑧 ∈ ℋ)
14		his7 30343	. . . . . . . 8 ⊢ (((𝑇‘𝑡) ∈ ℋ ∧ (𝑥 ·_ℎ 𝑓) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑇‘𝑡) ·_ih ((𝑥 ·_ℎ 𝑓) +_ℎ 𝑧)) = (((𝑇‘𝑡) ·_ih (𝑥 ·_ℎ 𝑓)) + ((𝑇‘𝑡) ·_ih 𝑧)))
15	10, 12, 13, 14	syl3anc 1372	. . . . . . 7 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → ((𝑇‘𝑡) ·_ih ((𝑥 ·_ℎ 𝑓) +_ℎ 𝑧)) = (((𝑇‘𝑡) ·_ih (𝑥 ·_ℎ 𝑓)) + ((𝑇‘𝑡) ·_ih 𝑧)))
16		hvaddcl 30265	. . . . . . . . 9 ⊢ (((𝑥 ·_ℎ 𝑓) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑓) +_ℎ 𝑧) ∈ ℋ)
17	11, 16	sylan 581	. . . . . . . 8 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑓) +_ℎ 𝑧) ∈ ℋ)
18	2, 3, 4, 5, 1	cnlnadjlem5 31324	. . . . . . . 8 ⊢ ((((𝑥 ·_ℎ 𝑓) +_ℎ 𝑧) ∈ ℋ ∧ 𝑡 ∈ ℋ) → ((𝑇‘𝑡) ·_ih ((𝑥 ·_ℎ 𝑓) +_ℎ 𝑧)) = (𝑡 ·_ih (𝐹‘((𝑥 ·_ℎ 𝑓) +_ℎ 𝑧))))
19	17, 18	sylan 581	. . . . . . 7 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → ((𝑇‘𝑡) ·_ih ((𝑥 ·_ℎ 𝑓) +_ℎ 𝑧)) = (𝑡 ·_ih (𝐹‘((𝑥 ·_ℎ 𝑓) +_ℎ 𝑧))))
20		simpll 766	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → 𝑥 ∈ ℂ)
21	9	adantl 483	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → (𝑇‘𝑡) ∈ ℋ)
22		simplr 768	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → 𝑓 ∈ ℋ)
23		his5 30339	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ (𝑇‘𝑡) ∈ ℋ ∧ 𝑓 ∈ ℋ) → ((𝑇‘𝑡) ·_ih (𝑥 ·_ℎ 𝑓)) = ((∗‘𝑥) · ((𝑇‘𝑡) ·_ih 𝑓)))
24	20, 21, 22, 23	syl3anc 1372	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → ((𝑇‘𝑡) ·_ih (𝑥 ·_ℎ 𝑓)) = ((∗‘𝑥) · ((𝑇‘𝑡) ·_ih 𝑓)))
25		simpr 486	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → 𝑡 ∈ ℋ)
26	2, 3, 4, 5, 1	cnlnadjlem4 31323	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ ℋ → (𝐹‘𝑓) ∈ ℋ)
27	26	ad2antlr 726	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → (𝐹‘𝑓) ∈ ℋ)
28		his5 30339	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ 𝑡 ∈ ℋ ∧ (𝐹‘𝑓) ∈ ℋ) → (𝑡 ·_ih (𝑥 ·_ℎ (𝐹‘𝑓))) = ((∗‘𝑥) · (𝑡 ·_ih (𝐹‘𝑓))))
29	20, 25, 27, 28	syl3anc 1372	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → (𝑡 ·_ih (𝑥 ·_ℎ (𝐹‘𝑓))) = ((∗‘𝑥) · (𝑡 ·_ih (𝐹‘𝑓))))
30	2, 3, 4, 5, 1	cnlnadjlem5 31324	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ ℋ ∧ 𝑡 ∈ ℋ) → ((𝑇‘𝑡) ·_ih 𝑓) = (𝑡 ·_ih (𝐹‘𝑓)))
31	30	adantll 713	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → ((𝑇‘𝑡) ·_ih 𝑓) = (𝑡 ·_ih (𝐹‘𝑓)))
32	31	oveq2d 7425	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → ((∗‘𝑥) · ((𝑇‘𝑡) ·_ih 𝑓)) = ((∗‘𝑥) · (𝑡 ·_ih (𝐹‘𝑓))))
33	29, 32	eqtr4d 2776	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → (𝑡 ·_ih (𝑥 ·_ℎ (𝐹‘𝑓))) = ((∗‘𝑥) · ((𝑇‘𝑡) ·_ih 𝑓)))
34	24, 33	eqtr4d 2776	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → ((𝑇‘𝑡) ·_ih (𝑥 ·_ℎ 𝑓)) = (𝑡 ·_ih (𝑥 ·_ℎ (𝐹‘𝑓))))
35	34	adantlr 714	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → ((𝑇‘𝑡) ·_ih (𝑥 ·_ℎ 𝑓)) = (𝑡 ·_ih (𝑥 ·_ℎ (𝐹‘𝑓))))
36	2, 3, 4, 5, 1	cnlnadjlem5 31324	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℋ ∧ 𝑡 ∈ ℋ) → ((𝑇‘𝑡) ·_ih 𝑧) = (𝑡 ·_ih (𝐹‘𝑧)))
37	36	adantll 713	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → ((𝑇‘𝑡) ·_ih 𝑧) = (𝑡 ·_ih (𝐹‘𝑧)))
38	35, 37	oveq12d 7427	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → (((𝑇‘𝑡) ·_ih (𝑥 ·_ℎ 𝑓)) + ((𝑇‘𝑡) ·_ih 𝑧)) = ((𝑡 ·_ih (𝑥 ·_ℎ (𝐹‘𝑓))) + (𝑡 ·_ih (𝐹‘𝑧))))
39		simpr 486	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → 𝑡 ∈ ℋ)
40		hvmulcl 30266	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ (𝐹‘𝑓) ∈ ℋ) → (𝑥 ·_ℎ (𝐹‘𝑓)) ∈ ℋ)
41	26, 40	sylan2 594	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) → (𝑥 ·_ℎ (𝐹‘𝑓)) ∈ ℋ)
42	41	ad2antrr 725	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → (𝑥 ·_ℎ (𝐹‘𝑓)) ∈ ℋ)
43	2, 3, 4, 5, 1	cnlnadjlem4 31323	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℋ → (𝐹‘𝑧) ∈ ℋ)
44	43	ad2antlr 726	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → (𝐹‘𝑧) ∈ ℋ)
45		his7 30343	. . . . . . . . 9 ⊢ ((𝑡 ∈ ℋ ∧ (𝑥 ·_ℎ (𝐹‘𝑓)) ∈ ℋ ∧ (𝐹‘𝑧) ∈ ℋ) → (𝑡 ·_ih ((𝑥 ·_ℎ (𝐹‘𝑓)) +_ℎ (𝐹‘𝑧))) = ((𝑡 ·_ih (𝑥 ·_ℎ (𝐹‘𝑓))) + (𝑡 ·_ih (𝐹‘𝑧))))
46	39, 42, 44, 45	syl3anc 1372	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → (𝑡 ·_ih ((𝑥 ·_ℎ (𝐹‘𝑓)) +_ℎ (𝐹‘𝑧))) = ((𝑡 ·_ih (𝑥 ·_ℎ (𝐹‘𝑓))) + (𝑡 ·_ih (𝐹‘𝑧))))
47	38, 46	eqtr4d 2776	. . . . . . 7 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → (((𝑇‘𝑡) ·_ih (𝑥 ·_ℎ 𝑓)) + ((𝑇‘𝑡) ·_ih 𝑧)) = (𝑡 ·_ih ((𝑥 ·_ℎ (𝐹‘𝑓)) +_ℎ (𝐹‘𝑧))))
48	15, 19, 47	3eqtr3d 2781	. . . . . 6 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → (𝑡 ·_ih (𝐹‘((𝑥 ·_ℎ 𝑓) +_ℎ 𝑧))) = (𝑡 ·_ih ((𝑥 ·_ℎ (𝐹‘𝑓)) +_ℎ (𝐹‘𝑧))))
49	48	ralrimiva 3147	. . . . 5 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ∀𝑡 ∈ ℋ (𝑡 ·_ih (𝐹‘((𝑥 ·_ℎ 𝑓) +_ℎ 𝑧))) = (𝑡 ·_ih ((𝑥 ·_ℎ (𝐹‘𝑓)) +_ℎ (𝐹‘𝑧))))
50	2, 3, 4, 5, 1	cnlnadjlem4 31323	. . . . . . 7 ⊢ (((𝑥 ·_ℎ 𝑓) +_ℎ 𝑧) ∈ ℋ → (𝐹‘((𝑥 ·_ℎ 𝑓) +_ℎ 𝑧)) ∈ ℋ)
51	17, 50	syl 17	. . . . . 6 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝐹‘((𝑥 ·_ℎ 𝑓) +_ℎ 𝑧)) ∈ ℋ)
52		hvaddcl 30265	. . . . . . 7 ⊢ (((𝑥 ·_ℎ (𝐹‘𝑓)) ∈ ℋ ∧ (𝐹‘𝑧) ∈ ℋ) → ((𝑥 ·_ℎ (𝐹‘𝑓)) +_ℎ (𝐹‘𝑧)) ∈ ℋ)
53	41, 43, 52	syl2an 597	. . . . . 6 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ (𝐹‘𝑓)) +_ℎ (𝐹‘𝑧)) ∈ ℋ)
54		hial2eq2 30360	. . . . . 6 ⊢ (((𝐹‘((𝑥 ·_ℎ 𝑓) +_ℎ 𝑧)) ∈ ℋ ∧ ((𝑥 ·_ℎ (𝐹‘𝑓)) +_ℎ (𝐹‘𝑧)) ∈ ℋ) → (∀𝑡 ∈ ℋ (𝑡 ·_ih (𝐹‘((𝑥 ·_ℎ 𝑓) +_ℎ 𝑧))) = (𝑡 ·_ih ((𝑥 ·_ℎ (𝐹‘𝑓)) +_ℎ (𝐹‘𝑧))) ↔ (𝐹‘((𝑥 ·_ℎ 𝑓) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝐹‘𝑓)) +_ℎ (𝐹‘𝑧))))
55	51, 53, 54	syl2anc 585	. . . . 5 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (∀𝑡 ∈ ℋ (𝑡 ·_ih (𝐹‘((𝑥 ·_ℎ 𝑓) +_ℎ 𝑧))) = (𝑡 ·_ih ((𝑥 ·_ℎ (𝐹‘𝑓)) +_ℎ (𝐹‘𝑧))) ↔ (𝐹‘((𝑥 ·_ℎ 𝑓) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝐹‘𝑓)) +_ℎ (𝐹‘𝑧))))
56	49, 55	mpbid 231	. . . 4 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝐹‘((𝑥 ·_ℎ 𝑓) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝐹‘𝑓)) +_ℎ (𝐹‘𝑧)))
57	56	ralrimiva 3147	. . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) → ∀𝑧 ∈ ℋ (𝐹‘((𝑥 ·_ℎ 𝑓) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝐹‘𝑓)) +_ℎ (𝐹‘𝑧)))
58	57	rgen2 3198	. 2 ⊢ ∀𝑥 ∈ ℂ ∀𝑓 ∈ ℋ ∀𝑧 ∈ ℋ (𝐹‘((𝑥 ·_ℎ 𝑓) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝐹‘𝑓)) +_ℎ (𝐹‘𝑧))
59		ellnop 31111	. 2 ⊢ (𝐹 ∈ LinOp ↔ (𝐹: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑓 ∈ ℋ ∀𝑧 ∈ ℋ (𝐹‘((𝑥 ·_ℎ 𝑓) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝐹‘𝑓)) +_ℎ (𝐹‘𝑧))))
60	7, 58, 59	mpbir2an 710	1 ⊢ 𝐹 ∈ LinOp

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ↦ cmpt 5232 ⟶wf 6540 ‘cfv 6544 ℩crio 7364 (class class class)co 7409 ℂcc 11108 + caddc 11113 · cmul 11115 ∗ccj 15043 ℋchba 30172 +_ℎ cva 30173 ·_ℎ csm 30174 ·_ih csp 30175 ContOpccop 30199 LinOpclo 30200

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cc 10430 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190 ax-hilex 30252 ax-hfvadd 30253 ax-hvcom 30254 ax-hvass 30255 ax-hv0cl 30256 ax-hvaddid 30257 ax-hfvmul 30258 ax-hvmulid 30259 ax-hvmulass 30260 ax-hvdistr1 30261 ax-hvdistr2 30262 ax-hvmul0 30263 ax-hfi 30332 ax-his1 30335 ax-his2 30336 ax-his3 30337 ax-his4 30338 ax-hcompl 30455

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-omul 8471 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-acn 9937 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-rlim 15433 df-sum 15633 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-mulg 18951 df-cntz 19181 df-cmn 19650 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-cn 22731 df-cnp 22732 df-lm 22733 df-t1 22818 df-haus 22819 df-tx 23066 df-hmeo 23259 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-xms 23826 df-ms 23827 df-tms 23828 df-cfil 24772 df-cau 24773 df-cmet 24774 df-grpo 29746 df-gid 29747 df-ginv 29748 df-gdiv 29749 df-ablo 29798 df-vc 29812 df-nv 29845 df-va 29848 df-ba 29849 df-sm 29850 df-0v 29851 df-vs 29852 df-nmcv 29853 df-ims 29854 df-dip 29954 df-ssp 29975 df-ph 30066 df-cbn 30116 df-hnorm 30221 df-hba 30222 df-hvsub 30224 df-hlim 30225 df-hcau 30226 df-sh 30460 df-ch 30474 df-oc 30505 df-ch0 30506 df-nmop 31092 df-cnop 31093 df-lnop 31094 df-nmfn 31098 df-nlfn 31099 df-cnfn 31100 df-lnfn 31101

This theorem is referenced by: cnlnadjlem8 31327 cnlnadjlem9 31328

W3C validator