HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  nmophmi Structured version   Visualization version   GIF version

Theorem nmophmi 29966
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) · (normop𝑇)))

Proof of Theorem nmophmi
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 nmophm.1 . . . . . . . . . . 11 𝑇 ∈ BndLinOp
2 bdopf 29797 . . . . . . . . . . 11 (𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ)
31, 2ax-mp 5 . . . . . . . . . 10 𝑇: ℋ⟶ ℋ
4 homval 29676 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 · (𝑇𝑥)))
53, 4mp3an2 1450 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 · (𝑇𝑥)))
65fveq2d 6679 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) → (norm‘((𝐴 ·op 𝑇)‘𝑥)) = (norm‘(𝐴 · (𝑇𝑥))))
73ffvelrni 6861 . . . . . . . . 9 (𝑥 ∈ ℋ → (𝑇𝑥) ∈ ℋ)
8 norm-iii 29075 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ (𝑇𝑥) ∈ ℋ) → (norm‘(𝐴 · (𝑇𝑥))) = ((abs‘𝐴) · (norm‘(𝑇𝑥))))
97, 8sylan2 596 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) → (norm‘(𝐴 · (𝑇𝑥))) = ((abs‘𝐴) · (norm‘(𝑇𝑥))))
106, 9eqtrd 2773 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) → (norm‘((𝐴 ·op 𝑇)‘𝑥)) = ((abs‘𝐴) · (norm‘(𝑇𝑥))))
1110adantr 484 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → (norm‘((𝐴 ·op 𝑇)‘𝑥)) = ((abs‘𝐴) · (norm‘(𝑇𝑥))))
12 normcl 29060 . . . . . . . . 9 ((𝑇𝑥) ∈ ℋ → (norm‘(𝑇𝑥)) ∈ ℝ)
137, 12syl 17 . . . . . . . 8 (𝑥 ∈ ℋ → (norm‘(𝑇𝑥)) ∈ ℝ)
1413ad2antlr 727 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → (norm‘(𝑇𝑥)) ∈ ℝ)
15 abscl 14729 . . . . . . . . 9 (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ)
16 absge0 14738 . . . . . . . . 9 (𝐴 ∈ ℂ → 0 ≤ (abs‘𝐴))
1715, 16jca 515 . . . . . . . 8 (𝐴 ∈ ℂ → ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)))
1817ad2antrr 726 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)))
19 nmoplb 29842 . . . . . . . . 9 ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1) → (norm‘(𝑇𝑥)) ≤ (normop𝑇))
203, 19mp3an1 1449 . . . . . . . 8 ((𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1) → (norm‘(𝑇𝑥)) ≤ (normop𝑇))
2120adantll 714 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → (norm‘(𝑇𝑥)) ≤ (normop𝑇))
22 nmopre 29805 . . . . . . . . 9 (𝑇 ∈ BndLinOp → (normop𝑇) ∈ ℝ)
231, 22ax-mp 5 . . . . . . . 8 (normop𝑇) ∈ ℝ
24 lemul2a 11574 . . . . . . . 8 ((((norm‘(𝑇𝑥)) ∈ ℝ ∧ (normop𝑇) ∈ ℝ ∧ ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴))) ∧ (norm‘(𝑇𝑥)) ≤ (normop𝑇)) → ((abs‘𝐴) · (norm‘(𝑇𝑥))) ≤ ((abs‘𝐴) · (normop𝑇)))
2523, 24mp3anl2 1457 . . . . . . 7 ((((norm‘(𝑇𝑥)) ∈ ℝ ∧ ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴))) ∧ (norm‘(𝑇𝑥)) ≤ (normop𝑇)) → ((abs‘𝐴) · (norm‘(𝑇𝑥))) ≤ ((abs‘𝐴) · (normop𝑇)))
2614, 18, 21, 25syl21anc 837 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → ((abs‘𝐴) · (norm‘(𝑇𝑥))) ≤ ((abs‘𝐴) · (normop𝑇)))
2711, 26eqbrtrd 5053 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → (norm‘((𝐴 ·op 𝑇)‘𝑥)) ≤ ((abs‘𝐴) · (normop𝑇)))
2827ex 416 . . . 4 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((norm𝑥) ≤ 1 → (norm‘((𝐴 ·op 𝑇)‘𝑥)) ≤ ((abs‘𝐴) · (normop𝑇))))
2928ralrimiva 3096 . . 3 (𝐴 ∈ ℂ → ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘((𝐴 ·op 𝑇)‘𝑥)) ≤ ((abs‘𝐴) · (normop𝑇))))
30 homulcl 29694 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
313, 30mpan2 691 . . . 4 (𝐴 ∈ ℂ → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
32 remulcl 10701 . . . . . 6 (((abs‘𝐴) ∈ ℝ ∧ (normop𝑇) ∈ ℝ) → ((abs‘𝐴) · (normop𝑇)) ∈ ℝ)
3315, 23, 32sylancl 589 . . . . 5 (𝐴 ∈ ℂ → ((abs‘𝐴) · (normop𝑇)) ∈ ℝ)
3433rexrd 10770 . . . 4 (𝐴 ∈ ℂ → ((abs‘𝐴) · (normop𝑇)) ∈ ℝ*)
35 nmopub 29843 . . . 4 (((𝐴 ·op 𝑇): ℋ⟶ ℋ ∧ ((abs‘𝐴) · (normop𝑇)) ∈ ℝ*) → ((normop‘(𝐴 ·op 𝑇)) ≤ ((abs‘𝐴) · (normop𝑇)) ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘((𝐴 ·op 𝑇)‘𝑥)) ≤ ((abs‘𝐴) · (normop𝑇)))))
3631, 34, 35syl2anc 587 . . 3 (𝐴 ∈ ℂ → ((normop‘(𝐴 ·op 𝑇)) ≤ ((abs‘𝐴) · (normop𝑇)) ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘((𝐴 ·op 𝑇)‘𝑥)) ≤ ((abs‘𝐴) · (normop𝑇)))))
3729, 36mpbird 260 . 2 (𝐴 ∈ ℂ → (normop‘(𝐴 ·op 𝑇)) ≤ ((abs‘𝐴) · (normop𝑇)))
38 fveq2 6675 . . . . . . . 8 (𝐴 = 0 → (abs‘𝐴) = (abs‘0))
39 abs0 14736 . . . . . . . 8 (abs‘0) = 0
4038, 39eqtrdi 2789 . . . . . . 7 (𝐴 = 0 → (abs‘𝐴) = 0)
4140oveq1d 7186 . . . . . 6 (𝐴 = 0 → ((abs‘𝐴) · (normop𝑇)) = (0 · (normop𝑇)))
4223recni 10734 . . . . . . 7 (normop𝑇) ∈ ℂ
4342mul02i 10908 . . . . . 6 (0 · (normop𝑇)) = 0
4441, 43eqtrdi 2789 . . . . 5 (𝐴 = 0 → ((abs‘𝐴) · (normop𝑇)) = 0)
4544adantl 485 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐴 = 0) → ((abs‘𝐴) · (normop𝑇)) = 0)
46 nmopge0 29846 . . . . . 6 ((𝐴 ·op 𝑇): ℋ⟶ ℋ → 0 ≤ (normop‘(𝐴 ·op 𝑇)))
4731, 46syl 17 . . . . 5 (𝐴 ∈ ℂ → 0 ≤ (normop‘(𝐴 ·op 𝑇)))
4847adantr 484 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐴 = 0) → 0 ≤ (normop‘(𝐴 ·op 𝑇)))
4945, 48eqbrtrd 5053 . . 3 ((𝐴 ∈ ℂ ∧ 𝐴 = 0) → ((abs‘𝐴) · (normop𝑇)) ≤ (normop‘(𝐴 ·op 𝑇)))
50 nmoplb 29842 . . . . . . . . . . . 12 (((𝐴 ·op 𝑇): ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1) → (norm‘((𝐴 ·op 𝑇)‘𝑥)) ≤ (normop‘(𝐴 ·op 𝑇)))
5131, 50syl3an1 1164 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1) → (norm‘((𝐴 ·op 𝑇)‘𝑥)) ≤ (normop‘(𝐴 ·op 𝑇)))
52513expa 1119 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → (norm‘((𝐴 ·op 𝑇)‘𝑥)) ≤ (normop‘(𝐴 ·op 𝑇)))
5311, 52eqbrtrrd 5055 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → ((abs‘𝐴) · (norm‘(𝑇𝑥))) ≤ (normop‘(𝐴 ·op 𝑇)))
5453adantllr 719 . . . . . . . 8 ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → ((abs‘𝐴) · (norm‘(𝑇𝑥))) ≤ (normop‘(𝐴 ·op 𝑇)))
5513adantl 485 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) → (norm‘(𝑇𝑥)) ∈ ℝ)
56 nmopxr 29801 . . . . . . . . . . . . 13 ((𝐴 ·op 𝑇): ℋ⟶ ℋ → (normop‘(𝐴 ·op 𝑇)) ∈ ℝ*)
5731, 56syl 17 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → (normop‘(𝐴 ·op 𝑇)) ∈ ℝ*)
58 nmopgtmnf 29803 . . . . . . . . . . . . 13 ((𝐴 ·op 𝑇): ℋ⟶ ℋ → -∞ < (normop‘(𝐴 ·op 𝑇)))
5931, 58syl 17 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → -∞ < (normop‘(𝐴 ·op 𝑇)))
60 xrre 12646 . . . . . . . . . . . 12 ((((normop‘(𝐴 ·op 𝑇)) ∈ ℝ* ∧ ((abs‘𝐴) · (normop𝑇)) ∈ ℝ) ∧ (-∞ < (normop‘(𝐴 ·op 𝑇)) ∧ (normop‘(𝐴 ·op 𝑇)) ≤ ((abs‘𝐴) · (normop𝑇)))) → (normop‘(𝐴 ·op 𝑇)) ∈ ℝ)
6157, 33, 59, 37, 60syl22anc 838 . . . . . . . . . . 11 (𝐴 ∈ ℂ → (normop‘(𝐴 ·op 𝑇)) ∈ ℝ)
6261ad2antrr 726 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) → (normop‘(𝐴 ·op 𝑇)) ∈ ℝ)
6315ad2antrr 726 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) → (abs‘𝐴) ∈ ℝ)
64 absgt0 14775 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → (𝐴 ≠ 0 ↔ 0 < (abs‘𝐴)))
6564biimpa 480 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 0 < (abs‘𝐴))
6665adantr 484 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) → 0 < (abs‘𝐴))
67 lemuldiv2 11600 . . . . . . . . . 10 (((norm‘(𝑇𝑥)) ∈ ℝ ∧ (normop‘(𝐴 ·op 𝑇)) ∈ ℝ ∧ ((abs‘𝐴) ∈ ℝ ∧ 0 < (abs‘𝐴))) → (((abs‘𝐴) · (norm‘(𝑇𝑥))) ≤ (normop‘(𝐴 ·op 𝑇)) ↔ (norm‘(𝑇𝑥)) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴))))
6855, 62, 63, 66, 67syl112anc 1375 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) → (((abs‘𝐴) · (norm‘(𝑇𝑥))) ≤ (normop‘(𝐴 ·op 𝑇)) ↔ (norm‘(𝑇𝑥)) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴))))
6968adantr 484 . . . . . . . 8 ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → (((abs‘𝐴) · (norm‘(𝑇𝑥))) ≤ (normop‘(𝐴 ·op 𝑇)) ↔ (norm‘(𝑇𝑥)) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴))))
7054, 69mpbid 235 . . . . . . 7 ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → (norm‘(𝑇𝑥)) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴)))
7170ex 416 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) → ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴))))
7271ralrimiva 3096 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴))))
7361adantr 484 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (normop‘(𝐴 ·op 𝑇)) ∈ ℝ)
7415adantr 484 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘𝐴) ∈ ℝ)
75 abs00 14740 . . . . . . . . . 10 (𝐴 ∈ ℂ → ((abs‘𝐴) = 0 ↔ 𝐴 = 0))
7675necon3bid 2978 . . . . . . . . 9 (𝐴 ∈ ℂ → ((abs‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0))
7776biimpar 481 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘𝐴) ≠ 0)
7873, 74, 77redivcld 11547 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴)) ∈ ℝ)
7978rexrd 10770 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴)) ∈ ℝ*)
80 nmopub 29843 . . . . . 6 ((𝑇: ℋ⟶ ℋ ∧ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴)) ∈ ℝ*) → ((normop𝑇) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴)) ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴)))))
813, 79, 80sylancr 590 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((normop𝑇) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴)) ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴)))))
8272, 81mpbird 260 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (normop𝑇) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴)))
8323a1i 11 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (normop𝑇) ∈ ℝ)
84 lemuldiv2 11600 . . . . 5 (((normop𝑇) ∈ ℝ ∧ (normop‘(𝐴 ·op 𝑇)) ∈ ℝ ∧ ((abs‘𝐴) ∈ ℝ ∧ 0 < (abs‘𝐴))) → (((abs‘𝐴) · (normop𝑇)) ≤ (normop‘(𝐴 ·op 𝑇)) ↔ (normop𝑇) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴))))
8583, 73, 74, 65, 84syl112anc 1375 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (((abs‘𝐴) · (normop𝑇)) ≤ (normop‘(𝐴 ·op 𝑇)) ↔ (normop𝑇) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴))))
8682, 85mpbird 260 . . 3 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((abs‘𝐴) · (normop𝑇)) ≤ (normop‘(𝐴 ·op 𝑇)))
8749, 86pm2.61dane 3021 . 2 (𝐴 ∈ ℂ → ((abs‘𝐴) · (normop𝑇)) ≤ (normop‘(𝐴 ·op 𝑇)))
8861, 33letri3d 10861 . 2 (𝐴 ∈ ℂ → ((normop‘(𝐴 ·op 𝑇)) = ((abs‘𝐴) · (normop𝑇)) ↔ ((normop‘(𝐴 ·op 𝑇)) ≤ ((abs‘𝐴) · (normop𝑇)) ∧ ((abs‘𝐴) · (normop𝑇)) ≤ (normop‘(𝐴 ·op 𝑇)))))
8937, 87, 88mpbir2and 713 1 (𝐴 ∈ ℂ → (normop‘(𝐴 ·op 𝑇)) = ((abs‘𝐴) · (normop𝑇)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2113  wne 2934  wral 3053   class class class wbr 5031  wf 6336  cfv 6340  (class class class)co 7171  cc 10614  cr 10615  0cc0 10616  1c1 10617   · cmul 10621  -∞cmnf 10752  *cxr 10753   < clt 10754  cle 10755   / cdiv 11376  abscabs 14684  chba 28854   · csm 28856  normcno 28858   ·op chot 28874  normopcnop 28880  BndLinOpcbo 28883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7480  ax-cnex 10672  ax-resscn 10673  ax-1cn 10674  ax-icn 10675  ax-addcl 10676  ax-addrcl 10677  ax-mulcl 10678  ax-mulrcl 10679  ax-mulcom 10680  ax-addass 10681  ax-mulass 10682  ax-distr 10683  ax-i2m1 10684  ax-1ne0 10685  ax-1rid 10686  ax-rnegex 10687  ax-rrecex 10688  ax-cnre 10689  ax-pre-lttri 10690  ax-pre-lttrn 10691  ax-pre-ltadd 10692  ax-pre-mulgt0 10693  ax-pre-sup 10694  ax-hilex 28934  ax-hfvadd 28935  ax-hvcom 28936  ax-hvass 28937  ax-hv0cl 28938  ax-hvaddid 28939  ax-hfvmul 28940  ax-hvmulid 28941  ax-hvmulass 28942  ax-hvdistr1 28943  ax-hvdistr2 28944  ax-hvmul0 28945  ax-hfi 29014  ax-his1 29017  ax-his2 29018  ax-his3 29019  ax-his4 29020
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3683  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-pss 3863  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-tp 4522  df-op 4524  df-uni 4798  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5484  df-we 5486  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7128  df-ov 7174  df-oprab 7175  df-mpo 7176  df-om 7601  df-1st 7715  df-2nd 7716  df-wrecs 7977  df-recs 8038  df-rdg 8076  df-er 8321  df-map 8440  df-en 8557  df-dom 8558  df-sdom 8559  df-sup 8980  df-pnf 10756  df-mnf 10757  df-xr 10758  df-ltxr 10759  df-le 10760  df-sub 10951  df-neg 10952  df-div 11377  df-nn 11718  df-2 11780  df-3 11781  df-4 11782  df-n0 11978  df-z 12064  df-uz 12326  df-rp 12474  df-seq 13462  df-exp 13523  df-cj 14549  df-re 14550  df-im 14551  df-sqrt 14685  df-abs 14686  df-grpo 28428  df-gid 28429  df-ablo 28480  df-vc 28494  df-nv 28527  df-va 28530  df-ba 28531  df-sm 28532  df-0v 28533  df-nmcv 28535  df-hnorm 28903  df-hba 28904  df-hvsub 28906  df-homul 29666  df-nmop 29774  df-lnop 29776  df-bdop 29777
This theorem is referenced by:  bdophmi  29967
  Copyright terms: Public domain W3C validator