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Theorem nmophmi 28730
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 28561 . . . . . . . . . . 11 (𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ)
31, 2ax-mp 5 . . . . . . . . . 10 𝑇: ℋ⟶ ℋ
4 homval 28440 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 · (𝑇𝑥)))
53, 4mp3an2 1409 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 · (𝑇𝑥)))
65fveq2d 6154 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) → (norm‘((𝐴 ·op 𝑇)‘𝑥)) = (norm‘(𝐴 · (𝑇𝑥))))
73ffvelrni 6315 . . . . . . . . 9 (𝑥 ∈ ℋ → (𝑇𝑥) ∈ ℋ)
8 norm-iii 27837 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ (𝑇𝑥) ∈ ℋ) → (norm‘(𝐴 · (𝑇𝑥))) = ((abs‘𝐴) · (norm‘(𝑇𝑥))))
97, 8sylan2 491 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) → (norm‘(𝐴 · (𝑇𝑥))) = ((abs‘𝐴) · (norm‘(𝑇𝑥))))
106, 9eqtrd 2660 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) → (norm‘((𝐴 ·op 𝑇)‘𝑥)) = ((abs‘𝐴) · (norm‘(𝑇𝑥))))
1110adantr 481 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → (norm‘((𝐴 ·op 𝑇)‘𝑥)) = ((abs‘𝐴) · (norm‘(𝑇𝑥))))
12 normcl 27822 . . . . . . . . 9 ((𝑇𝑥) ∈ ℋ → (norm‘(𝑇𝑥)) ∈ ℝ)
137, 12syl 17 . . . . . . . 8 (𝑥 ∈ ℋ → (norm‘(𝑇𝑥)) ∈ ℝ)
1413ad2antlr 762 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → (norm‘(𝑇𝑥)) ∈ ℝ)
15 abscl 13947 . . . . . . . . 9 (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ)
16 absge0 13956 . . . . . . . . 9 (𝐴 ∈ ℂ → 0 ≤ (abs‘𝐴))
1715, 16jca 554 . . . . . . . 8 (𝐴 ∈ ℂ → ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)))
1817ad2antrr 761 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)))
19 nmoplb 28606 . . . . . . . . 9 ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1) → (norm‘(𝑇𝑥)) ≤ (normop𝑇))
203, 19mp3an1 1408 . . . . . . . 8 ((𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1) → (norm‘(𝑇𝑥)) ≤ (normop𝑇))
2120adantll 749 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → (norm‘(𝑇𝑥)) ≤ (normop𝑇))
22 nmopre 28569 . . . . . . . . 9 (𝑇 ∈ BndLinOp → (normop𝑇) ∈ ℝ)
231, 22ax-mp 5 . . . . . . . 8 (normop𝑇) ∈ ℝ
24 lemul2a 10823 . . . . . . . 8 ((((norm‘(𝑇𝑥)) ∈ ℝ ∧ (normop𝑇) ∈ ℝ ∧ ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴))) ∧ (norm‘(𝑇𝑥)) ≤ (normop𝑇)) → ((abs‘𝐴) · (norm‘(𝑇𝑥))) ≤ ((abs‘𝐴) · (normop𝑇)))
2523, 24mp3anl2 1416 . . . . . . 7 ((((norm‘(𝑇𝑥)) ∈ ℝ ∧ ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴))) ∧ (norm‘(𝑇𝑥)) ≤ (normop𝑇)) → ((abs‘𝐴) · (norm‘(𝑇𝑥))) ≤ ((abs‘𝐴) · (normop𝑇)))
2614, 18, 21, 25syl21anc 1322 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → ((abs‘𝐴) · (norm‘(𝑇𝑥))) ≤ ((abs‘𝐴) · (normop𝑇)))
2711, 26eqbrtrd 4640 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → (norm‘((𝐴 ·op 𝑇)‘𝑥)) ≤ ((abs‘𝐴) · (normop𝑇)))
2827ex 450 . . . 4 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((norm𝑥) ≤ 1 → (norm‘((𝐴 ·op 𝑇)‘𝑥)) ≤ ((abs‘𝐴) · (normop𝑇))))
2928ralrimiva 2965 . . 3 (𝐴 ∈ ℂ → ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘((𝐴 ·op 𝑇)‘𝑥)) ≤ ((abs‘𝐴) · (normop𝑇))))
30 homulcl 28458 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
313, 30mpan2 706 . . . 4 (𝐴 ∈ ℂ → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
32 remulcl 9966 . . . . . 6 (((abs‘𝐴) ∈ ℝ ∧ (normop𝑇) ∈ ℝ) → ((abs‘𝐴) · (normop𝑇)) ∈ ℝ)
3315, 23, 32sylancl 693 . . . . 5 (𝐴 ∈ ℂ → ((abs‘𝐴) · (normop𝑇)) ∈ ℝ)
3433rexrd 10034 . . . 4 (𝐴 ∈ ℂ → ((abs‘𝐴) · (normop𝑇)) ∈ ℝ*)
35 nmopub 28607 . . . 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 692 . . 3 (𝐴 ∈ ℂ → ((normop‘(𝐴 ·op 𝑇)) ≤ ((abs‘𝐴) · (normop𝑇)) ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘((𝐴 ·op 𝑇)‘𝑥)) ≤ ((abs‘𝐴) · (normop𝑇)))))
3729, 36mpbird 247 . 2 (𝐴 ∈ ℂ → (normop‘(𝐴 ·op 𝑇)) ≤ ((abs‘𝐴) · (normop𝑇)))
38 fveq2 6150 . . . . . . . 8 (𝐴 = 0 → (abs‘𝐴) = (abs‘0))
39 abs0 13954 . . . . . . . 8 (abs‘0) = 0
4038, 39syl6eq 2676 . . . . . . 7 (𝐴 = 0 → (abs‘𝐴) = 0)
4140oveq1d 6620 . . . . . 6 (𝐴 = 0 → ((abs‘𝐴) · (normop𝑇)) = (0 · (normop𝑇)))
4223recni 9997 . . . . . . 7 (normop𝑇) ∈ ℂ
4342mul02i 10170 . . . . . 6 (0 · (normop𝑇)) = 0
4441, 43syl6eq 2676 . . . . 5 (𝐴 = 0 → ((abs‘𝐴) · (normop𝑇)) = 0)
4544adantl 482 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐴 = 0) → ((abs‘𝐴) · (normop𝑇)) = 0)
46 nmopge0 28610 . . . . . 6 ((𝐴 ·op 𝑇): ℋ⟶ ℋ → 0 ≤ (normop‘(𝐴 ·op 𝑇)))
4731, 46syl 17 . . . . 5 (𝐴 ∈ ℂ → 0 ≤ (normop‘(𝐴 ·op 𝑇)))
4847adantr 481 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐴 = 0) → 0 ≤ (normop‘(𝐴 ·op 𝑇)))
4945, 48eqbrtrd 4640 . . 3 ((𝐴 ∈ ℂ ∧ 𝐴 = 0) → ((abs‘𝐴) · (normop𝑇)) ≤ (normop‘(𝐴 ·op 𝑇)))
50 nmoplb 28606 . . . . . . . . . . . 12 (((𝐴 ·op 𝑇): ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1) → (norm‘((𝐴 ·op 𝑇)‘𝑥)) ≤ (normop‘(𝐴 ·op 𝑇)))
5131, 50syl3an1 1356 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1) → (norm‘((𝐴 ·op 𝑇)‘𝑥)) ≤ (normop‘(𝐴 ·op 𝑇)))
52513expa 1262 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → (norm‘((𝐴 ·op 𝑇)‘𝑥)) ≤ (normop‘(𝐴 ·op 𝑇)))
5311, 52eqbrtrrd 4642 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → ((abs‘𝐴) · (norm‘(𝑇𝑥))) ≤ (normop‘(𝐴 ·op 𝑇)))
5453adantllr 754 . . . . . . . 8 ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → ((abs‘𝐴) · (norm‘(𝑇𝑥))) ≤ (normop‘(𝐴 ·op 𝑇)))
5513adantl 482 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) → (norm‘(𝑇𝑥)) ∈ ℝ)
56 nmopxr 28565 . . . . . . . . . . . . 13 ((𝐴 ·op 𝑇): ℋ⟶ ℋ → (normop‘(𝐴 ·op 𝑇)) ∈ ℝ*)
5731, 56syl 17 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → (normop‘(𝐴 ·op 𝑇)) ∈ ℝ*)
58 nmopgtmnf 28567 . . . . . . . . . . . . 13 ((𝐴 ·op 𝑇): ℋ⟶ ℋ → -∞ < (normop‘(𝐴 ·op 𝑇)))
5931, 58syl 17 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → -∞ < (normop‘(𝐴 ·op 𝑇)))
60 xrre 11942 . . . . . . . . . . . 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 1324 . . . . . . . . . . 11 (𝐴 ∈ ℂ → (normop‘(𝐴 ·op 𝑇)) ∈ ℝ)
6261ad2antrr 761 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) → (normop‘(𝐴 ·op 𝑇)) ∈ ℝ)
6315ad2antrr 761 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) → (abs‘𝐴) ∈ ℝ)
64 absgt0 13993 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → (𝐴 ≠ 0 ↔ 0 < (abs‘𝐴)))
6564biimpa 501 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 0 < (abs‘𝐴))
6665adantr 481 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) → 0 < (abs‘𝐴))
67 lemuldiv2 10849 . . . . . . . . . 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 1327 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) → (((abs‘𝐴) · (norm‘(𝑇𝑥))) ≤ (normop‘(𝐴 ·op 𝑇)) ↔ (norm‘(𝑇𝑥)) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴))))
6968adantr 481 . . . . . . . 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 222 . . . . . . 7 ((((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → (norm‘(𝑇𝑥)) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴)))
7170ex 450 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑥 ∈ ℋ) → ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴))))
7271ralrimiva 2965 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴))))
7361adantr 481 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (normop‘(𝐴 ·op 𝑇)) ∈ ℝ)
7415adantr 481 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘𝐴) ∈ ℝ)
75 abs00 13958 . . . . . . . . . 10 (𝐴 ∈ ℂ → ((abs‘𝐴) = 0 ↔ 𝐴 = 0))
7675necon3bid 2840 . . . . . . . . 9 (𝐴 ∈ ℂ → ((abs‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0))
7776biimpar 502 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘𝐴) ≠ 0)
7873, 74, 77redivcld 10798 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴)) ∈ ℝ)
7978rexrd 10034 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴)) ∈ ℝ*)
80 nmopub 28607 . . . . . 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 247 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (normop𝑇) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴)))
8323a1i 11 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (normop𝑇) ∈ ℝ)
84 lemuldiv2 10849 . . . . 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 1327 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (((abs‘𝐴) · (normop𝑇)) ≤ (normop‘(𝐴 ·op 𝑇)) ↔ (normop𝑇) ≤ ((normop‘(𝐴 ·op 𝑇)) / (abs‘𝐴))))
8682, 85mpbird 247 . . 3 ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((abs‘𝐴) · (normop𝑇)) ≤ (normop‘(𝐴 ·op 𝑇)))
8749, 86pm2.61dane 2883 . 2 (𝐴 ∈ ℂ → ((abs‘𝐴) · (normop𝑇)) ≤ (normop‘(𝐴 ·op 𝑇)))
8861, 33letri3d 10124 . 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 956 1 (𝐴 ∈ ℂ → (normop‘(𝐴 ·op 𝑇)) = ((abs‘𝐴) · (normop𝑇)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  wne 2796  wral 2912   class class class wbr 4618  wf 5846  cfv 5850  (class class class)co 6605  cc 9879  cr 9880  0cc0 9881  1c1 9882   · cmul 9886  -∞cmnf 10017  *cxr 10018   < clt 10019  cle 10020   / cdiv 10629  abscabs 13903  chil 27616   · csm 27618  normcno 27620   ·op chot 27636  normopcnop 27642  BndLinOpcbo 27645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958  ax-pre-sup 9959  ax-hilex 27696  ax-hfvadd 27697  ax-hvcom 27698  ax-hvass 27699  ax-hv0cl 27700  ax-hvaddid 27701  ax-hfvmul 27702  ax-hvmulid 27703  ax-hvmulass 27704  ax-hvdistr1 27705  ax-hvdistr2 27706  ax-hvmul0 27707  ax-hfi 27776  ax-his1 27779  ax-his2 27780  ax-his3 27781  ax-his4 27782
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-map 7805  df-en 7901  df-dom 7902  df-sdom 7903  df-sup 8293  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-div 10630  df-nn 10966  df-2 11024  df-3 11025  df-4 11026  df-n0 11238  df-z 11323  df-uz 11632  df-rp 11777  df-seq 12739  df-exp 12798  df-cj 13768  df-re 13769  df-im 13770  df-sqrt 13904  df-abs 13905  df-grpo 27187  df-gid 27188  df-ablo 27239  df-vc 27254  df-nv 27287  df-va 27290  df-ba 27291  df-sm 27292  df-0v 27293  df-nmcv 27295  df-hnorm 27665  df-hba 27666  df-hvsub 27668  df-homul 28430  df-nmop 28538  df-lnop 28540  df-bdop 28541
This theorem is referenced by:  bdophmi  28731
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