Theorem nmophmi 29802

Description: The norm of the scalar product of a bounded linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)

Hypothesis

Ref	Expression
nmophm.1	⊢ 𝑇 ∈ BndLinOp

Assertion

Ref	Expression
nmophmi	⊢ (𝐴 ∈ ℂ → (normop‘(𝐴 ·op 𝑇)) = ((abs‘𝐴) · (normop‘𝑇)))

Proof of Theorem nmophmi

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nmophm.1	. . . . . . . . . . 11 ⊢ 𝑇 ∈ BndLinOp
2		bdopf 29633	. . . . . . . . . . 11 ⊢ (𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ)
3	1, 2	ax-mp 5	. . . . . . . . . 10 ⊢ 𝑇: ℋ⟶ ℋ
4		homval 29512	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 ·ℎ (𝑇‘𝑥)))
5	3, 4	mp3an2 1445	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 ·ℎ (𝑇‘𝑥)))
6	5	fveq2d 6669	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) → (normℎ‘((𝐴 ·op 𝑇)‘𝑥)) = (normℎ‘(𝐴 ·ℎ (𝑇‘𝑥))))
7	3	ffvelrni 6845	. . . . . . . . 9 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ)
8		norm-iii 28911	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇‘𝑥) ∈ ℋ) → (normℎ‘(𝐴 ·ℎ (𝑇‘𝑥))) = ((abs‘𝐴) · (normℎ‘(𝑇‘𝑥))))
9	7, 8	sylan2 594	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) → (normℎ‘(𝐴 ·ℎ (𝑇‘𝑥))) = ((abs‘𝐴) · (normℎ‘(𝑇‘𝑥))))
10	6, 9	eqtrd 2856	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) → (normℎ‘((𝐴 ·op 𝑇)‘𝑥)) = ((abs‘𝐴) · (normℎ‘(𝑇‘𝑥))))
11	10	adantr 483	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘((𝐴 ·op 𝑇)‘𝑥)) = ((abs‘𝐴) · (normℎ‘(𝑇‘𝑥))))
12		normcl 28896	. . . . . . . . 9 ⊢ ((𝑇‘𝑥) ∈ ℋ → (normℎ‘(𝑇‘𝑥)) ∈ ℝ)
13	7, 12	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ ℋ → (normℎ‘(𝑇‘𝑥)) ∈ ℝ)
14	13	ad2antlr 725	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘(𝑇‘𝑥)) ∈ ℝ)
15		abscl 14632	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ)
16		absge0 14641	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → 0 ≤ (abs‘𝐴))
17	15, 16	jca 514	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)))
18	17	ad2antrr 724	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (normℎ‘𝑥) ≤ 1) → ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)))
19		nmoplb 29678	. . . . . . . . 9 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘(𝑇‘𝑥)) ≤ (normop‘𝑇))
20	3, 19	mp3an1 1444	. . . . . . . 8 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘(𝑇‘𝑥)) ≤ (normop‘𝑇))
21	20	adantll 712	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘(𝑇‘𝑥)) ≤ (normop‘𝑇))
22		nmopre 29641	. . . . . . . . 9 ⊢ (𝑇 ∈ BndLinOp → (normop‘𝑇) ∈ ℝ)
23	1, 22	ax-mp 5	. . . . . . . 8 ⊢ (normop‘𝑇) ∈ ℝ
24		lemul2a 11489	. . . . . . . 8 ⊢ ((((normℎ‘(𝑇‘𝑥)) ∈ ℝ ∧ (normop‘𝑇) ∈ ℝ ∧ ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴))) ∧ (normℎ‘(𝑇‘𝑥)) ≤ (normop‘𝑇)) → ((abs‘𝐴) · (normℎ‘(𝑇‘𝑥))) ≤ ((abs‘𝐴) · (normop‘𝑇)))
25	23, 24	mp3anl2 1452	. . . . . . 7 ⊢ ((((normℎ‘(𝑇‘𝑥)) ∈ ℝ ∧ ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴))) ∧ (normℎ‘(𝑇‘𝑥)) ≤ (normop‘𝑇)) → ((abs‘𝐴) · (normℎ‘(𝑇‘𝑥))) ≤ ((abs‘𝐴) · (normop‘𝑇)))
26	14, 18, 21, 25	syl21anc 835	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (normℎ‘𝑥) ≤ 1) → ((abs‘𝐴) · (normℎ‘(𝑇‘𝑥))) ≤ ((abs‘𝐴) · (normop‘𝑇)))
27	11, 26	eqbrtrd 5081	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘((𝐴 ·op 𝑇)‘𝑥)) ≤ ((abs‘𝐴) · (normop‘𝑇)))
28	27	ex 415	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((normℎ‘𝑥) ≤ 1 → (normℎ‘((𝐴 ·op 𝑇)‘𝑥)) ≤ ((abs‘𝐴) · (normop‘𝑇))))
29	28	ralrimiva 3182	. . 3 ⊢ (𝐴 ∈ ℂ → ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (normℎ‘((𝐴 ·op 𝑇)‘𝑥)) ≤ ((abs‘𝐴) · (normop‘𝑇))))
30		homulcl 29530	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
31	3, 30	mpan2 689	. . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
32		remulcl 10616	. . . . . 6 ⊢ (((abs‘𝐴) ∈ ℝ ∧ (normop‘𝑇) ∈ ℝ) → ((abs‘𝐴) · (normop‘𝑇)) ∈ ℝ)
33	15, 23, 32	sylancl 588	. . . . 5 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) · (normop‘𝑇)) ∈ ℝ)
34	33	rexrd 10685	. . . 4 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) · (normop‘𝑇)) ∈ ℝ*)
35		nmopub 29679	. . . 4 ⊢ (((𝐴 ·op 𝑇): ℋ⟶ ℋ ∧ ((abs‘𝐴) · (normop‘𝑇)) ∈ ℝ*) → ((normop‘(𝐴 ·op 𝑇)) ≤ ((abs‘𝐴) · (normop‘𝑇)) ↔ ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (normℎ‘((𝐴 ·op 𝑇)‘𝑥)) ≤ ((abs‘𝐴) · (normop‘𝑇)))))
36	31, 34, 35	syl2anc 586	. . 3 ⊢ (𝐴 ∈ ℂ → ((normop‘(𝐴 ·op 𝑇)) ≤ ((abs‘𝐴) · (normop‘𝑇)) ↔ ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (normℎ‘((𝐴 ·op 𝑇)‘𝑥)) ≤ ((abs‘𝐴) · (normop‘𝑇)))))
37	29, 36	mpbird 259	. 2 ⊢ (𝐴 ∈ ℂ → (normop‘(𝐴 ·op 𝑇)) ≤ ((abs‘𝐴) · (normop‘𝑇)))
38		fveq2 6665	. . . . . . . 8 ⊢ (𝐴 = 0 → (abs‘𝐴) = (abs‘0))
39		abs0 14639	. . . . . . . 8 ⊢ (abs‘0) = 0
40	38, 39	syl6eq 2872	. . . . . . 7 ⊢ (𝐴 = 0 → (abs‘𝐴) = 0)
41	40	oveq1d 7165	. . . . . 6 ⊢ (𝐴 = 0 → ((abs‘𝐴) · (normop‘𝑇)) = (0 · (normop‘𝑇)))
42	23	recni 10649	. . . . . . 7 ⊢ (normop‘𝑇) ∈ ℂ
43	42	mul02i 10823	. . . . . 6 ⊢ (0 · (normop‘𝑇)) = 0
44	41, 43	syl6eq 2872	. . . . 5 ⊢ (𝐴 = 0 → ((abs‘𝐴) · (normop‘𝑇)) = 0)
45	44	adantl 484	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 = 0) → ((abs‘𝐴) · (normop‘𝑇)) = 0)
46		nmopge0 29682	. . . . . 6 ⊢ ((𝐴 ·op 𝑇): ℋ⟶ ℋ → 0 ≤ (normop‘(𝐴 ·op 𝑇)))
47	31, 46	syl 17	. . . . 5 ⊢ (𝐴 ∈ ℂ → 0 ≤ (normop‘(𝐴 ·op 𝑇)))
48	47	adantr 483	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 = 0) → 0 ≤ (normop‘(𝐴 ·op 𝑇)))
49	45, 48	eqbrtrd 5081	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 = 0) → ((abs‘𝐴) · (normop‘𝑇)) ≤ (normop‘(𝐴 ·op 𝑇)))
50		nmoplb 29678	. . . . . . . . . . . 12 ⊢ (((𝐴 ·op 𝑇): ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘((𝐴 ·op 𝑇)‘𝑥)) ≤ (normop‘(𝐴 ·op 𝑇)))
51	31, 50	syl3an1 1159	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘((𝐴 ·op 𝑇)‘𝑥)) ≤ (normop‘(𝐴 ·op 𝑇)))
52	51	3expa 1114	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘((𝐴 ·op 𝑇)‘𝑥)) ≤ (normop‘(𝐴 ·op 𝑇)))
53	11, 52	eqbrtrrd 5083	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (normℎ‘𝑥) ≤ 1) → ((abs‘𝐴) · (normℎ‘(𝑇‘𝑥))) ≤ (normop‘(𝐴 ·op 𝑇)))
54	53	adantllr 717	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) ∧ (normℎ‘𝑥) ≤ 1) → ((abs‘𝐴) · (normℎ‘(𝑇‘𝑥))) ≤ (normop‘(𝐴 ·op 𝑇)))
55	13	adantl 484	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) → (normℎ‘(𝑇‘𝑥)) ∈ ℝ)
56		nmopxr 29637	. . . . . . . . . . . . 13 ⊢ ((𝐴 ·op 𝑇): ℋ⟶ ℋ → (normop‘(𝐴 ·op 𝑇)) ∈ ℝ*)
57	31, 56	syl 17	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → (normop‘(𝐴 ·op 𝑇)) ∈ ℝ*)
58		nmopgtmnf 29639	. . . . . . . . . . . . 13 ⊢ ((𝐴 ·op 𝑇): ℋ⟶ ℋ → -∞ < (normop‘(𝐴 ·op 𝑇)))
59	31, 58	syl 17	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → -∞ < (normop‘(𝐴 ·op 𝑇)))
60		xrre 12556	. . . . . . . . . . . 12 ⊢ ((((normop‘(𝐴 ·op 𝑇)) ∈ ℝ* ∧ ((abs‘𝐴) · (normop‘𝑇)) ∈ ℝ) ∧ (-∞ < (normop‘(𝐴 ·op 𝑇)) ∧ (normop‘(𝐴 ·op 𝑇)) ≤ ((abs‘𝐴) · (normop‘𝑇)))) → (normop‘(𝐴 ·op 𝑇)) ∈ ℝ)
61	57, 33, 59, 37, 60	syl22anc 836	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → (normop‘(𝐴 ·op 𝑇)) ∈ ℝ)
62	61	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) → (normop‘(𝐴 ·op 𝑇)) ∈ ℝ)
63	15	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) → (abs‘𝐴) ∈ ℝ)
64		absgt0 14678	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → (𝐴 ≠ 0 ↔ 0 < (abs‘𝐴)))
65	64	biimpa 479	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 0 < (abs‘𝐴))
66	65	adantr 483	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) → 0 < (abs‘𝐴))
67		lemuldiv2 11515	. . . . . . . . . 10 ⊢ (((normℎ‘(𝑇‘𝑥)) ∈ ℝ ∧ (normop‘(𝐴 ·op 𝑇)) ∈ ℝ ∧ ((abs‘𝐴) ∈ ℝ ∧ 0 < (abs‘𝐴))) → (((abs‘𝐴) · (normℎ‘(𝑇‘𝑥))) ≤ (normop‘(𝐴 ·op 𝑇)) ↔ (normℎ‘(𝑇‘𝑥)) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴))))
68	55, 62, 63, 66, 67	syl112anc 1370	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) → (((abs‘𝐴) · (normℎ‘(𝑇‘𝑥))) ≤ (normop‘(𝐴 ·op 𝑇)) ↔ (normℎ‘(𝑇‘𝑥)) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴))))
69	68	adantr 483	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) ∧ (normℎ‘𝑥) ≤ 1) → (((abs‘𝐴) · (normℎ‘(𝑇‘𝑥))) ≤ (normop‘(𝐴 ·op 𝑇)) ↔ (normℎ‘(𝑇‘𝑥)) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴))))
70	54, 69	mpbid 234	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘(𝑇‘𝑥)) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴)))
71	70	ex 415	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) → ((normℎ‘𝑥) ≤ 1 → (normℎ‘(𝑇‘𝑥)) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴))))
72	71	ralrimiva 3182	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (normℎ‘(𝑇‘𝑥)) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴))))
73	61	adantr 483	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (normop‘(𝐴 ·op 𝑇)) ∈ ℝ)
74	15	adantr 483	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘𝐴) ∈ ℝ)
75		abs00 14643	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) = 0 ↔ 𝐴 = 0))
76	75	necon3bid 3060	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0))
77	76	biimpar 480	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘𝐴) ≠ 0)
78	73, 74, 77	redivcld 11462	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴)) ∈ ℝ)
79	78	rexrd 10685	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴)) ∈ ℝ*)
80		nmopub 29679	. . . . . 6 ⊢ ((𝑇: ℋ⟶ ℋ ∧ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴)) ∈ ℝ*) → ((normop‘𝑇) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴)) ↔ ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (normℎ‘(𝑇‘𝑥)) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴)))))
81	3, 79, 80	sylancr 589	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((normop‘𝑇) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴)) ↔ ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (normℎ‘(𝑇‘𝑥)) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴)))))
82	72, 81	mpbird 259	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (normop‘𝑇) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴)))
83	23	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (normop‘𝑇) ∈ ℝ)
84		lemuldiv2 11515	. . . . 5 ⊢ (((normop‘𝑇) ∈ ℝ ∧ (normop‘(𝐴 ·op 𝑇)) ∈ ℝ ∧ ((abs‘𝐴) ∈ ℝ ∧ 0 < (abs‘𝐴))) → (((abs‘𝐴) · (normop‘𝑇)) ≤ (normop‘(𝐴 ·op 𝑇)) ↔ (normop‘𝑇) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴))))
85	83, 73, 74, 65, 84	syl112anc 1370	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (((abs‘𝐴) · (normop‘𝑇)) ≤ (normop‘(𝐴 ·op 𝑇)) ↔ (normop‘𝑇) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴))))
86	82, 85	mpbird 259	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((abs‘𝐴) · (normop‘𝑇)) ≤ (normop‘(𝐴 ·op 𝑇)))
87	49, 86	pm2.61dane 3104	. 2 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) · (normop‘𝑇)) ≤ (normop‘(𝐴 ·op 𝑇)))
88	61, 33	letri3d 10776	. 2 ⊢ (𝐴 ∈ ℂ → ((normop‘(𝐴 ·op 𝑇)) = ((abs‘𝐴) · (normop‘𝑇)) ↔ ((normop‘(𝐴 ·op 𝑇)) ≤ ((abs‘𝐴) · (normop‘𝑇)) ∧ ((abs‘𝐴) · (normop‘𝑇)) ≤ (normop‘(𝐴 ·op 𝑇)))))
89	37, 87, 88	mpbir2and 711	1 ⊢ (𝐴 ∈ ℂ → (normop‘(𝐴 ·op 𝑇)) = ((abs‘𝐴) · (normop‘𝑇)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138   class class class wbr 5059  ⟶wf 6346  ‘cfv 6350  (class class class)co 7150  ℂcc 10529  ℝcr 10530  0cc0 10531  1c1 10532   · cmul 10536  -∞cmnf 10667  ℝ^*cxr 10668   < clt 10669   ≤ cle 10670   / cdiv 11291  abscabs 14587   ℋchba 28690   ·_ℎ csm 28692  norm_ℎcno 28694   ·_op chot 28710  norm_opcnop 28716  BndLinOpcbo 28719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609  ax-hilex 28770  ax-hfvadd 28771  ax-hvcom 28772  ax-hvass 28773  ax-hv0cl 28774  ax-hvaddid 28775  ax-hfvmul 28776  ax-hvmulid 28777  ax-hvmulass 28778  ax-hvdistr1 28779  ax-hvdistr2 28780  ax-hvmul0 28781  ax-hfi 28850  ax-his1 28853  ax-his2 28854  ax-his3 28855  ax-his4 28856

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8283  df-map 8402  df-en 8504  df-dom 8505  df-sdom 8506  df-sup 8900  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-n0 11892  df-z 11976  df-uz 12238  df-rp 12384  df-seq 13364  df-exp 13424  df-cj 14452  df-re 14453  df-im 14454  df-sqrt 14588  df-abs 14589  df-grpo 28264  df-gid 28265  df-ablo 28316  df-vc 28330  df-nv 28363  df-va 28366  df-ba 28367  df-sm 28368  df-0v 28369  df-nmcv 28371  df-hnorm 28739  df-hba 28740  df-hvsub 28742  df-homul 29502  df-nmop 29610  df-lnop 29612  df-bdop 29613

This theorem is referenced by:  bdophmi  29803