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Theorem nmophmi 28274
 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 28105 . . . . . . . . . . 11 (𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ)
31, 2ax-mp 5 . . . . . . . . . 10 𝑇: ℋ⟶ ℋ
4 homval 27984 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 · (𝑇𝑥)))
53, 4mp3an2 1404 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 · (𝑇𝑥)))
65fveq2d 6107 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) → (norm‘((𝐴 ·op 𝑇)‘𝑥)) = (norm‘(𝐴 · (𝑇𝑥))))
73ffvelrni 6266 . . . . . . . . 9 (𝑥 ∈ ℋ → (𝑇𝑥) ∈ ℋ)
8 norm-iii 27381 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ (𝑇𝑥) ∈ ℋ) → (norm‘(𝐴 · (𝑇𝑥))) = ((abs‘𝐴) · (norm‘(𝑇𝑥))))
97, 8sylan2 490 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) → (norm‘(𝐴 · (𝑇𝑥))) = ((abs‘𝐴) · (norm‘(𝑇𝑥))))
106, 9eqtrd 2644 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) → (norm‘((𝐴 ·op 𝑇)‘𝑥)) = ((abs‘𝐴) · (norm‘(𝑇𝑥))))
1110adantr 480 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → (norm‘((𝐴 ·op 𝑇)‘𝑥)) = ((abs‘𝐴) · (norm‘(𝑇𝑥))))
12 normcl 27366 . . . . . . . . 9 ((𝑇𝑥) ∈ ℋ → (norm‘(𝑇𝑥)) ∈ ℝ)
137, 12syl 17 . . . . . . . 8 (𝑥 ∈ ℋ → (norm‘(𝑇𝑥)) ∈ ℝ)
1413ad2antlr 759 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → (norm‘(𝑇𝑥)) ∈ ℝ)
15 abscl 13866 . . . . . . . . 9 (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ)
16 absge0 13875 . . . . . . . . 9 (𝐴 ∈ ℂ → 0 ≤ (abs‘𝐴))
1715, 16jca 553 . . . . . . . 8 (𝐴 ∈ ℂ → ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)))
1817ad2antrr 758 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)))
19 nmoplb 28150 . . . . . . . . 9 ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1) → (norm‘(𝑇𝑥)) ≤ (normop𝑇))
203, 19mp3an1 1403 . . . . . . . 8 ((𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1) → (norm‘(𝑇𝑥)) ≤ (normop𝑇))
2120adantll 746 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → (norm‘(𝑇𝑥)) ≤ (normop𝑇))
22 nmopre 28113 . . . . . . . . 9 (𝑇 ∈ BndLinOp → (normop𝑇) ∈ ℝ)
231, 22ax-mp 5 . . . . . . . 8 (normop𝑇) ∈ ℝ
24 lemul2a 10757 . . . . . . . 8 ((((norm‘(𝑇𝑥)) ∈ ℝ ∧ (normop𝑇) ∈ ℝ ∧ ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴))) ∧ (norm‘(𝑇𝑥)) ≤ (normop𝑇)) → ((abs‘𝐴) · (norm‘(𝑇𝑥))) ≤ ((abs‘𝐴) · (normop𝑇)))
2523, 24mp3anl2 1411 . . . . . . 7 ((((norm‘(𝑇𝑥)) ∈ ℝ ∧ ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴))) ∧ (norm‘(𝑇𝑥)) ≤ (normop𝑇)) → ((abs‘𝐴) · (norm‘(𝑇𝑥))) ≤ ((abs‘𝐴) · (normop𝑇)))
2614, 18, 21, 25syl21anc 1317 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → ((abs‘𝐴) · (norm‘(𝑇𝑥))) ≤ ((abs‘𝐴) · (normop𝑇)))
2711, 26eqbrtrd 4605 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → (norm‘((𝐴 ·op 𝑇)‘𝑥)) ≤ ((abs‘𝐴) · (normop𝑇)))
2827ex 449 . . . 4 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((norm𝑥) ≤ 1 → (norm‘((𝐴 ·op 𝑇)‘𝑥)) ≤ ((abs‘𝐴) · (normop𝑇))))
2928ralrimiva 2949 . . 3 (𝐴 ∈ ℂ → ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘((𝐴 ·op 𝑇)‘𝑥)) ≤ ((abs‘𝐴) · (normop𝑇))))
30 homulcl 28002 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
313, 30mpan2 703 . . . 4 (𝐴 ∈ ℂ → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
32 remulcl 9900 . . . . . 6 (((abs‘𝐴) ∈ ℝ ∧ (normop𝑇) ∈ ℝ) → ((abs‘𝐴) · (normop𝑇)) ∈ ℝ)
3315, 23, 32sylancl 693 . . . . 5 (𝐴 ∈ ℂ → ((abs‘𝐴) · (normop𝑇)) ∈ ℝ)
3433rexrd 9968 . . . 4 (𝐴 ∈ ℂ → ((abs‘𝐴) · (normop𝑇)) ∈ ℝ*)
35 nmopub 28151 . . . 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 691 . . 3 (𝐴 ∈ ℂ → ((normop‘(𝐴 ·op 𝑇)) ≤ ((abs‘𝐴) · (normop𝑇)) ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘((𝐴 ·op 𝑇)‘𝑥)) ≤ ((abs‘𝐴) · (normop𝑇)))))
3729, 36mpbird 246 . 2 (𝐴 ∈ ℂ → (normop‘(𝐴 ·op 𝑇)) ≤ ((abs‘𝐴) · (normop𝑇)))
38 fveq2 6103 . . . . . . . 8 (𝐴 = 0 → (abs‘𝐴) = (abs‘0))
39 abs0 13873 . . . . . . . 8 (abs‘0) = 0
4038, 39syl6eq 2660 . . . . . . 7 (𝐴 = 0 → (abs‘𝐴) = 0)
4140oveq1d 6564 . . . . . 6 (𝐴 = 0 → ((abs‘𝐴) · (normop𝑇)) = (0 · (normop𝑇)))
4223recni 9931 . . . . . . 7 (normop𝑇) ∈ ℂ
4342mul02i 10104 . . . . . 6 (0 · (normop𝑇)) = 0
4441, 43syl6eq 2660 . . . . 5 (𝐴 = 0 → ((abs‘𝐴) · (normop𝑇)) = 0)
4544adantl 481 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐴 = 0) → ((abs‘𝐴) · (normop𝑇)) = 0)
46 nmopge0 28154 . . . . . 6 ((𝐴 ·op 𝑇): ℋ⟶ ℋ → 0 ≤ (normop‘(𝐴 ·op 𝑇)))
4731, 46syl 17 . . . . 5 (𝐴 ∈ ℂ → 0 ≤ (normop‘(𝐴 ·op 𝑇)))
4847adantr 480 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐴 = 0) → 0 ≤ (normop‘(𝐴 ·op 𝑇)))
4945, 48eqbrtrd 4605 . . 3 ((𝐴 ∈ ℂ ∧ 𝐴 = 0) → ((abs‘𝐴) · (normop𝑇)) ≤ (normop‘(𝐴 ·op 𝑇)))
50 nmoplb 28150 . . . . . . . . . . . 12 (((𝐴 ·op 𝑇): ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1) → (norm‘((𝐴 ·op 𝑇)‘𝑥)) ≤ (normop‘(𝐴 ·op 𝑇)))
5131, 50syl3an1 1351 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1) → (norm‘((𝐴 ·op 𝑇)‘𝑥)) ≤ (normop‘(𝐴 ·op 𝑇)))
52513expa 1257 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → (norm‘((𝐴 ·op 𝑇)‘𝑥)) ≤ (normop‘(𝐴 ·op 𝑇)))
5311, 52eqbrtrrd 4607 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → ((abs‘𝐴) · (norm‘(𝑇𝑥))) ≤ (normop‘(𝐴 ·op 𝑇)))
5453adantllr 751 . . . . . . . 8 ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → ((abs‘𝐴) · (norm‘(𝑇𝑥))) ≤ (normop‘(𝐴 ·op 𝑇)))
5513adantl 481 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) → (norm‘(𝑇𝑥)) ∈ ℝ)
56 nmopxr 28109 . . . . . . . . . . . . 13 ((𝐴 ·op 𝑇): ℋ⟶ ℋ → (normop‘(𝐴 ·op 𝑇)) ∈ ℝ*)
5731, 56syl 17 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → (normop‘(𝐴 ·op 𝑇)) ∈ ℝ*)
58 nmopgtmnf 28111 . . . . . . . . . . . . 13 ((𝐴 ·op 𝑇): ℋ⟶ ℋ → -∞ < (normop‘(𝐴 ·op 𝑇)))
5931, 58syl 17 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → -∞ < (normop‘(𝐴 ·op 𝑇)))
60 xrre 11874 . . . . . . . . . . . 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 1319 . . . . . . . . . . 11 (𝐴 ∈ ℂ → (normop‘(𝐴 ·op 𝑇)) ∈ ℝ)
6261ad2antrr 758 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) → (normop‘(𝐴 ·op 𝑇)) ∈ ℝ)
6315ad2antrr 758 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) → (abs‘𝐴) ∈ ℝ)
64 absgt0 13912 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → (𝐴 ≠ 0 ↔ 0 < (abs‘𝐴)))
6564biimpa 500 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 0 < (abs‘𝐴))
6665adantr 480 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) → 0 < (abs‘𝐴))
67 lemuldiv2 10783 . . . . . . . . . 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 1322 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) → (((abs‘𝐴) · (norm‘(𝑇𝑥))) ≤ (normop‘(𝐴 ·op 𝑇)) ↔ (norm‘(𝑇𝑥)) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴))))
6968adantr 480 . . . . . . . 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 221 . . . . . . 7 ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → (norm‘(𝑇𝑥)) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴)))
7170ex 449 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) → ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴))))
7271ralrimiva 2949 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴))))
7361adantr 480 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (normop‘(𝐴 ·op 𝑇)) ∈ ℝ)
7415adantr 480 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘𝐴) ∈ ℝ)
75 abs00 13877 . . . . . . . . . 10 (𝐴 ∈ ℂ → ((abs‘𝐴) = 0 ↔ 𝐴 = 0))
7675necon3bid 2826 . . . . . . . . 9 (𝐴 ∈ ℂ → ((abs‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0))
7776biimpar 501 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘𝐴) ≠ 0)
7873, 74, 77redivcld 10732 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴)) ∈ ℝ)
7978rexrd 9968 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴)) ∈ ℝ*)
80 nmopub 28151 . . . . . 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 694 . . . . 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 246 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (normop𝑇) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴)))
8323a1i 11 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (normop𝑇) ∈ ℝ)
84 lemuldiv2 10783 . . . . 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 1322 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (((abs‘𝐴) · (normop𝑇)) ≤ (normop‘(𝐴 ·op 𝑇)) ↔ (normop𝑇) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴))))
8682, 85mpbird 246 . . 3 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((abs‘𝐴) · (normop𝑇)) ≤ (normop‘(𝐴 ·op 𝑇)))
8749, 86pm2.61dane 2869 . 2 (𝐴 ∈ ℂ → ((abs‘𝐴) · (normop𝑇)) ≤ (normop‘(𝐴 ·op 𝑇)))
8861, 33letri3d 10058 . 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 959 1 (𝐴 ∈ ℂ → (normop‘(𝐴 ·op 𝑇)) = ((abs‘𝐴) · (normop𝑇)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896   class class class wbr 4583  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   · cmul 9820  -∞cmnf 9951  ℝ*cxr 9952   < clt 9953   ≤ cle 9954   / cdiv 10563  abscabs 13822   ℋchil 27160   ·ℎ csm 27162  normℎcno 27164   ·op chot 27180  normopcnop 27186  BndLinOpcbo 27189 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-hilex 27240  ax-hfvadd 27241  ax-hvcom 27242  ax-hvass 27243  ax-hv0cl 27244  ax-hvaddid 27245  ax-hfvmul 27246  ax-hvmulid 27247  ax-hvmulass 27248  ax-hvdistr1 27249  ax-hvdistr2 27250  ax-hvmul0 27251  ax-hfi 27320  ax-his1 27323  ax-his2 27324  ax-his3 27325  ax-his4 27326 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-grpo 26731  df-gid 26732  df-ablo 26783  df-vc 26798  df-nv 26831  df-va 26834  df-ba 26835  df-sm 26836  df-0v 26837  df-nmcv 26839  df-hnorm 27209  df-hba 27210  df-hvsub 27212  df-homul 27974  df-nmop 28082  df-lnop 28084  df-bdop 28085 This theorem is referenced by:  bdophmi  28275
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