cnlnadjlem6 - Hilbert Space Explorer

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Theorem cnlnadjlem6 31592

Description: Lemma for cnlnadji 31596. 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 31589	. . 3 ⊢ (𝑦 ∈ ℋ → 𝐵 ∈ ℋ)
7	1, 6	fmpti 7112	. 2 ⊢ 𝐹: ℋ⟶ ℋ
8	2	lnopfi 31489	. . . . . . . . . 10 ⊢ 𝑇: ℋ⟶ ℋ
9	8	ffvelcdmi 7084	. . . . . . . . 9 ⊢ (𝑡 ∈ ℋ → (𝑇‘𝑡) ∈ ℋ)
10	9	adantl 480	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → (𝑇‘𝑡) ∈ ℋ)
11		hvmulcl 30533	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) → (𝑥 ·_ℎ 𝑓) ∈ ℋ)
12	11	ad2antrr 722	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → (𝑥 ·_ℎ 𝑓) ∈ ℋ)
13		simplr 765	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → 𝑧 ∈ ℋ)
14		his7 30610	. . . . . . . 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 1369	. . . . . . 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 30532	. . . . . . . . 9 ⊢ (((𝑥 ·_ℎ 𝑓) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑓) +_ℎ 𝑧) ∈ ℋ)
17	11, 16	sylan 578	. . . . . . . 8 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑓) +_ℎ 𝑧) ∈ ℋ)
18	2, 3, 4, 5, 1	cnlnadjlem5 31591	. . . . . . . 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 578	. . . . . . 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 763	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → 𝑥 ∈ ℂ)
21	9	adantl 480	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → (𝑇‘𝑡) ∈ ℋ)
22		simplr 765	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → 𝑓 ∈ ℋ)
23		his5 30606	. . . . . . . . . . . 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 1369	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → ((𝑇‘𝑡) ·_ih (𝑥 ·_ℎ 𝑓)) = ((∗‘𝑥) · ((𝑇‘𝑡) ·_ih 𝑓)))
25		simpr 483	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → 𝑡 ∈ ℋ)
26	2, 3, 4, 5, 1	cnlnadjlem4 31590	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ ℋ → (𝐹‘𝑓) ∈ ℋ)
27	26	ad2antlr 723	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → (𝐹‘𝑓) ∈ ℋ)
28		his5 30606	. . . . . . . . . . . . 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 1369	. . . . . . . . . . . 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 31591	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ ℋ ∧ 𝑡 ∈ ℋ) → ((𝑇‘𝑡) ·_ih 𝑓) = (𝑡 ·_ih (𝐹‘𝑓)))
31	30	adantll 710	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → ((𝑇‘𝑡) ·_ih 𝑓) = (𝑡 ·_ih (𝐹‘𝑓)))
32	31	oveq2d 7427	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → ((∗‘𝑥) · ((𝑇‘𝑡) ·_ih 𝑓)) = ((∗‘𝑥) · (𝑡 ·_ih (𝐹‘𝑓))))
33	29, 32	eqtr4d 2773	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → (𝑡 ·_ih (𝑥 ·_ℎ (𝐹‘𝑓))) = ((∗‘𝑥) · ((𝑇‘𝑡) ·_ih 𝑓)))
34	24, 33	eqtr4d 2773	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → ((𝑇‘𝑡) ·_ih (𝑥 ·_ℎ 𝑓)) = (𝑡 ·_ih (𝑥 ·_ℎ (𝐹‘𝑓))))
35	34	adantlr 711	. . . . . . . . 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 31591	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℋ ∧ 𝑡 ∈ ℋ) → ((𝑇‘𝑡) ·_ih 𝑧) = (𝑡 ·_ih (𝐹‘𝑧)))
37	36	adantll 710	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → ((𝑇‘𝑡) ·_ih 𝑧) = (𝑡 ·_ih (𝐹‘𝑧)))
38	35, 37	oveq12d 7429	. . . . . . . 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 483	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → 𝑡 ∈ ℋ)
40		hvmulcl 30533	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ (𝐹‘𝑓) ∈ ℋ) → (𝑥 ·_ℎ (𝐹‘𝑓)) ∈ ℋ)
41	26, 40	sylan2 591	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) → (𝑥 ·_ℎ (𝐹‘𝑓)) ∈ ℋ)
42	41	ad2antrr 722	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → (𝑥 ·_ℎ (𝐹‘𝑓)) ∈ ℋ)
43	2, 3, 4, 5, 1	cnlnadjlem4 31590	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℋ → (𝐹‘𝑧) ∈ ℋ)
44	43	ad2antlr 723	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ 𝑡 ∈ ℋ) → (𝐹‘𝑧) ∈ ℋ)
45		his7 30610	. . . . . . . . 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 1369	. . . . . . . 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 2773	. . . . . . 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 2778	. . . . . 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 3144	. . . . 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 31590	. . . . . . 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 30532	. . . . . . 7 ⊢ (((𝑥 ·_ℎ (𝐹‘𝑓)) ∈ ℋ ∧ (𝐹‘𝑧) ∈ ℋ) → ((𝑥 ·_ℎ (𝐹‘𝑓)) +_ℎ (𝐹‘𝑧)) ∈ ℋ)
53	41, 43, 52	syl2an 594	. . . . . 6 ⊢ (((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ (𝐹‘𝑓)) +_ℎ (𝐹‘𝑧)) ∈ ℋ)
54		hial2eq2 30627	. . . . . 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 582	. . . . 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 3144	. . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑓 ∈ ℋ) → ∀𝑧 ∈ ℋ (𝐹‘((𝑥 ·_ℎ 𝑓) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝐹‘𝑓)) +_ℎ (𝐹‘𝑧)))
58	57	rgen2 3195	. 2 ⊢ ∀𝑥 ∈ ℂ ∀𝑓 ∈ ℋ ∀𝑧 ∈ ℋ (𝐹‘((𝑥 ·_ℎ 𝑓) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝐹‘𝑓)) +_ℎ (𝐹‘𝑧))
59		ellnop 31378	. 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 707	1 ⊢ 𝐹 ∈ LinOp

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ∀wral 3059 ↦ cmpt 5230 ⟶wf 6538 ‘cfv 6542 ℩crio 7366 (class class class)co 7411 ℂcc 11110 + caddc 11115 · cmul 11117 ∗ccj 15047 ℋchba 30439 +_ℎ cva 30440 ·_ℎ csm 30441 ·_ih csp 30442 ContOpccop 30466 LinOpclo 30467

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 ax-cc 10432 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192 ax-hilex 30519 ax-hfvadd 30520 ax-hvcom 30521 ax-hvass 30522 ax-hv0cl 30523 ax-hvaddid 30524 ax-hfvmul 30525 ax-hvmulid 30526 ax-hvmulass 30527 ax-hvdistr1 30528 ax-hvdistr2 30529 ax-hvmul0 30530 ax-hfi 30599 ax-his1 30602 ax-his2 30603 ax-his3 30604 ax-his4 30605 ax-hcompl 30722

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-oadd 8472 df-omul 8473 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-acn 9939 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-ioo 13332 df-ico 13334 df-icc 13335 df-fz 13489 df-fzo 13632 df-fl 13761 df-seq 13971 df-exp 14032 df-hash 14295 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15436 df-rlim 15437 df-sum 15637 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-hom 17225 df-cco 17226 df-rest 17372 df-topn 17373 df-0g 17391 df-gsum 17392 df-topgen 17393 df-pt 17394 df-prds 17397 df-xrs 17452 df-qtop 17457 df-imas 17458 df-xps 17460 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-mulg 18987 df-cntz 19222 df-cmn 19691 df-psmet 21136 df-xmet 21137 df-met 21138 df-bl 21139 df-mopn 21140 df-fbas 21141 df-fg 21142 df-cnfld 21145 df-top 22616 df-topon 22633 df-topsp 22655 df-bases 22669 df-cld 22743 df-ntr 22744 df-cls 22745 df-nei 22822 df-cn 22951 df-cnp 22952 df-lm 22953 df-t1 23038 df-haus 23039 df-tx 23286 df-hmeo 23479 df-fil 23570 df-fm 23662 df-flim 23663 df-flf 23664 df-xms 24046 df-ms 24047 df-tms 24048 df-cfil 25003 df-cau 25004 df-cmet 25005 df-grpo 30013 df-gid 30014 df-ginv 30015 df-gdiv 30016 df-ablo 30065 df-vc 30079 df-nv 30112 df-va 30115 df-ba 30116 df-sm 30117 df-0v 30118 df-vs 30119 df-nmcv 30120 df-ims 30121 df-dip 30221 df-ssp 30242 df-ph 30333 df-cbn 30383 df-hnorm 30488 df-hba 30489 df-hvsub 30491 df-hlim 30492 df-hcau 30493 df-sh 30727 df-ch 30741 df-oc 30772 df-ch0 30773 df-nmop 31359 df-cnop 31360 df-lnop 31361 df-nmfn 31365 df-nlfn 31366 df-cnfn 31367 df-lnfn 31368

This theorem is referenced by: cnlnadjlem8 31594 cnlnadjlem9 31595

W3C validator