cnlnadjlem6 - Hilbert Space Explorer

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Theorem cnlnadjlem6 31320

Description: Lemma for cnlnadji 31324. 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 31317	. . 3 ⊢ (𝑦 ∈ ℋ → 𝐵 ∈ ℋ)
7	1, 6	fmpti 7111	. 2 ⊢ 𝐹: ℋ⟶ ℋ
8	2	lnopfi 31217	. . . . . . . . . 10 ⊢ 𝑇: ℋ⟶ ℋ
9	8	ffvelcdmi 7085	. . . . . . . . 9 ⊢ (𝑡 ∈ ℋ → (𝑇‘𝑡) ∈ ℋ)
10	9	adantl 482	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → (𝑇‘𝑡) ∈ ℋ)
11		hvmulcl 30261	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) → (𝑥 ·_ℎ 𝑓) ∈ ℋ)
12	11	ad2antrr 724	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → (𝑥 ·_ℎ 𝑓) ∈ ℋ)
13		simplr 767	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → 𝑧 ∈ ℋ)
14		his7 30338	. . . . . . . 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 1371	. . . . . . 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 30260	. . . . . . . . 9 ⊢ (((𝑥 ·_ℎ 𝑓) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑓) +_ℎ 𝑧) ∈ ℋ)
17	11, 16	sylan 580	. . . . . . . 8 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑓) +_ℎ 𝑧) ∈ ℋ)
18	2, 3, 4, 5, 1	cnlnadjlem5 31319	. . . . . . . 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 580	. . . . . . 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 765	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → 𝑥 ∈ ℂ)
21	9	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → (𝑇‘𝑡) ∈ ℋ)
22		simplr 767	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → 𝑓 ∈ ℋ)
23		his5 30334	. . . . . . . . . . . 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 1371	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → ((𝑇‘𝑡) ·_ih (𝑥 ·_ℎ 𝑓)) = ((∗‘𝑥) · ((𝑇‘𝑡) ·_ih 𝑓)))
25		simpr 485	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → 𝑡 ∈ ℋ)
26	2, 3, 4, 5, 1	cnlnadjlem4 31318	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ ℋ → (𝐹‘𝑓) ∈ ℋ)
27	26	ad2antlr 725	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → (𝐹‘𝑓) ∈ ℋ)
28		his5 30334	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 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 31319	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ ℋ ∧ 𝑡 ∈ ℋ) → ((𝑇‘𝑡) ·_ih 𝑓) = (𝑡 ·_ih (𝐹‘𝑓)))
31	30	adantll 712	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → ((𝑇‘𝑡) ·_ih 𝑓) = (𝑡 ·_ih (𝐹‘𝑓)))
32	31	oveq2d 7424	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → ((∗‘𝑥) · ((𝑇‘𝑡) ·_ih 𝑓)) = ((∗‘𝑥) · (𝑡 ·_ih (𝐹‘𝑓))))
33	29, 32	eqtr4d 2775	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → (𝑡 ·_ih (𝑥 ·_ℎ (𝐹‘𝑓))) = ((∗‘𝑥) · ((𝑇‘𝑡) ·_ih 𝑓)))
34	24, 33	eqtr4d 2775	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → ((𝑇‘𝑡) ·_ih (𝑥 ·_ℎ 𝑓)) = (𝑡 ·_ih (𝑥 ·_ℎ (𝐹‘𝑓))))
35	34	adantlr 713	. . . . . . . . 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 31319	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℋ ∧ 𝑡 ∈ ℋ) → ((𝑇‘𝑡) ·_ih 𝑧) = (𝑡 ·_ih (𝐹‘𝑧)))
37	36	adantll 712	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → ((𝑇‘𝑡) ·_ih 𝑧) = (𝑡 ·_ih (𝐹‘𝑧)))
38	35, 37	oveq12d 7426	. . . . . . . 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 485	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → 𝑡 ∈ ℋ)
40		hvmulcl 30261	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ (𝐹‘𝑓) ∈ ℋ) → (𝑥 ·_ℎ (𝐹‘𝑓)) ∈ ℋ)
41	26, 40	sylan2 593	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) → (𝑥 ·_ℎ (𝐹‘𝑓)) ∈ ℋ)
42	41	ad2antrr 724	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → (𝑥 ·_ℎ (𝐹‘𝑓)) ∈ ℋ)
43	2, 3, 4, 5, 1	cnlnadjlem4 31318	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℋ → (𝐹‘𝑧) ∈ ℋ)
44	43	ad2antlr 725	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → (𝐹‘𝑧) ∈ ℋ)
45		his7 30338	. . . . . . . . 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 1371	. . . . . . . 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 2775	. . . . . . 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 2780	. . . . . 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 3146	. . . . 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 31318	. . . . . . 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 30260	. . . . . . 7 ⊢ (((𝑥 ·_ℎ (𝐹‘𝑓)) ∈ ℋ ∧ (𝐹‘𝑧) ∈ ℋ) → ((𝑥 ·_ℎ (𝐹‘𝑓)) +_ℎ (𝐹‘𝑧)) ∈ ℋ)
53	41, 43, 52	syl2an 596	. . . . . 6 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ (𝐹‘𝑓)) +_ℎ (𝐹‘𝑧)) ∈ ℋ)
54		hial2eq2 30355	. . . . . 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 584	. . . . 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 3146	. . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) → ∀𝑧 ∈ ℋ (𝐹‘((𝑥 ·_ℎ 𝑓) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝐹‘𝑓)) +_ℎ (𝐹‘𝑧)))
58	57	rgen2 3197	. 2 ⊢ ∀𝑥 ∈ ℂ ∀𝑓 ∈ ℋ ∀𝑧 ∈ ℋ (𝐹‘((𝑥 ·_ℎ 𝑓) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝐹‘𝑓)) +_ℎ (𝐹‘𝑧))
59		ellnop 31106	. 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 709	1 ⊢ 𝐹 ∈ LinOp

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ↦ cmpt 5231 ⟶wf 6539 ‘cfv 6543 ℩crio 7363 (class class class)co 7408 ℂcc 11107 + caddc 11112 · cmul 11114 ∗ccj 15042 ℋchba 30167 +_ℎ cva 30168 ·_ℎ csm 30169 ·_ih csp 30170 ContOpccop 30194 LinOpclo 30195

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-inf2 9635 ax-cc 10429 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189 ax-hilex 30247 ax-hfvadd 30248 ax-hvcom 30249 ax-hvass 30250 ax-hv0cl 30251 ax-hvaddid 30252 ax-hfvmul 30253 ax-hvmulid 30254 ax-hvmulass 30255 ax-hvdistr1 30256 ax-hvdistr2 30257 ax-hvmul0 30258 ax-hfi 30327 ax-his1 30330 ax-his2 30331 ax-his3 30332 ax-his4 30333 ax-hcompl 30450

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-om 7855 df-1st 7974 df-2nd 7975 df-supp 8146 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-2o 8466 df-oadd 8469 df-omul 8470 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-acn 9936 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-q 12932 df-rp 12974 df-xneg 13091 df-xadd 13092 df-xmul 13093 df-ioo 13327 df-ico 13329 df-icc 13330 df-fz 13484 df-fzo 13627 df-fl 13756 df-seq 13966 df-exp 14027 df-hash 14290 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-clim 15431 df-rlim 15432 df-sum 15632 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17367 df-topn 17368 df-0g 17386 df-gsum 17387 df-topgen 17388 df-pt 17389 df-prds 17392 df-xrs 17447 df-qtop 17452 df-imas 17453 df-xps 17455 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-submnd 18671 df-mulg 18950 df-cntz 19180 df-cmn 19649 df-psmet 20935 df-xmet 20936 df-met 20937 df-bl 20938 df-mopn 20939 df-fbas 20940 df-fg 20941 df-cnfld 20944 df-top 22395 df-topon 22412 df-topsp 22434 df-bases 22448 df-cld 22522 df-ntr 22523 df-cls 22524 df-nei 22601 df-cn 22730 df-cnp 22731 df-lm 22732 df-t1 22817 df-haus 22818 df-tx 23065 df-hmeo 23258 df-fil 23349 df-fm 23441 df-flim 23442 df-flf 23443 df-xms 23825 df-ms 23826 df-tms 23827 df-cfil 24771 df-cau 24772 df-cmet 24773 df-grpo 29741 df-gid 29742 df-ginv 29743 df-gdiv 29744 df-ablo 29793 df-vc 29807 df-nv 29840 df-va 29843 df-ba 29844 df-sm 29845 df-0v 29846 df-vs 29847 df-nmcv 29848 df-ims 29849 df-dip 29949 df-ssp 29970 df-ph 30061 df-cbn 30111 df-hnorm 30216 df-hba 30217 df-hvsub 30219 df-hlim 30220 df-hcau 30221 df-sh 30455 df-ch 30469 df-oc 30500 df-ch0 30501 df-nmop 31087 df-cnop 31088 df-lnop 31089 df-nmfn 31093 df-nlfn 31094 df-cnfn 31095 df-lnfn 31096

This theorem is referenced by: cnlnadjlem8 31322 cnlnadjlem9 31323

W3C validator