Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  nmoco Structured version   Visualization version   GIF version

Theorem nmoco 22451
 Description: An upper bound on the operator norm of a composition. (Contributed by Mario Carneiro, 20-Oct-2015.)
Hypotheses
Ref Expression
nmoco.1 𝑁 = (𝑆 normOp 𝑈)
nmoco.2 𝐿 = (𝑇 normOp 𝑈)
nmoco.3 𝑀 = (𝑆 normOp 𝑇)
Assertion
Ref Expression
nmoco ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘(𝐹𝐺)) ≤ ((𝐿𝐹) · (𝑀𝐺)))

Proof of Theorem nmoco
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 nmoco.1 . 2 𝑁 = (𝑆 normOp 𝑈)
2 eqid 2621 . 2 (Base‘𝑆) = (Base‘𝑆)
3 eqid 2621 . 2 (norm‘𝑆) = (norm‘𝑆)
4 eqid 2621 . 2 (norm‘𝑈) = (norm‘𝑈)
5 eqid 2621 . 2 (0g𝑆) = (0g𝑆)
6 nghmrcl1 22446 . . 3 (𝐺 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp)
76adantl 482 . 2 ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝑆 ∈ NrmGrp)
8 nghmrcl2 22447 . . 3 (𝐹 ∈ (𝑇 NGHom 𝑈) → 𝑈 ∈ NrmGrp)
98adantr 481 . 2 ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝑈 ∈ NrmGrp)
10 nghmghm 22448 . . 3 (𝐹 ∈ (𝑇 NGHom 𝑈) → 𝐹 ∈ (𝑇 GrpHom 𝑈))
11 nghmghm 22448 . . 3 (𝐺 ∈ (𝑆 NGHom 𝑇) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
12 ghmco 17601 . . 3 ((𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝐹𝐺) ∈ (𝑆 GrpHom 𝑈))
1310, 11, 12syl2an 494 . 2 ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝐹𝐺) ∈ (𝑆 GrpHom 𝑈))
14 nmoco.2 . . . 4 𝐿 = (𝑇 normOp 𝑈)
1514nghmcl 22441 . . 3 (𝐹 ∈ (𝑇 NGHom 𝑈) → (𝐿𝐹) ∈ ℝ)
16 nmoco.3 . . . 4 𝑀 = (𝑆 normOp 𝑇)
1716nghmcl 22441 . . 3 (𝐺 ∈ (𝑆 NGHom 𝑇) → (𝑀𝐺) ∈ ℝ)
18 remulcl 9965 . . 3 (((𝐿𝐹) ∈ ℝ ∧ (𝑀𝐺) ∈ ℝ) → ((𝐿𝐹) · (𝑀𝐺)) ∈ ℝ)
1915, 17, 18syl2an 494 . 2 ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → ((𝐿𝐹) · (𝑀𝐺)) ∈ ℝ)
20 nghmrcl1 22446 . . . . 5 (𝐹 ∈ (𝑇 NGHom 𝑈) → 𝑇 ∈ NrmGrp)
2114nmoge0 22435 . . . . 5 ((𝑇 ∈ NrmGrp ∧ 𝑈 ∈ NrmGrp ∧ 𝐹 ∈ (𝑇 GrpHom 𝑈)) → 0 ≤ (𝐿𝐹))
2220, 8, 10, 21syl3anc 1323 . . . 4 (𝐹 ∈ (𝑇 NGHom 𝑈) → 0 ≤ (𝐿𝐹))
2315, 22jca 554 . . 3 (𝐹 ∈ (𝑇 NGHom 𝑈) → ((𝐿𝐹) ∈ ℝ ∧ 0 ≤ (𝐿𝐹)))
24 nghmrcl2 22447 . . . . 5 (𝐺 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ NrmGrp)
2516nmoge0 22435 . . . . 5 ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑀𝐺))
266, 24, 11, 25syl3anc 1323 . . . 4 (𝐺 ∈ (𝑆 NGHom 𝑇) → 0 ≤ (𝑀𝐺))
2717, 26jca 554 . . 3 (𝐺 ∈ (𝑆 NGHom 𝑇) → ((𝑀𝐺) ∈ ℝ ∧ 0 ≤ (𝑀𝐺)))
28 mulge0 10490 . . 3 ((((𝐿𝐹) ∈ ℝ ∧ 0 ≤ (𝐿𝐹)) ∧ ((𝑀𝐺) ∈ ℝ ∧ 0 ≤ (𝑀𝐺))) → 0 ≤ ((𝐿𝐹) · (𝑀𝐺)))
2923, 27, 28syl2an 494 . 2 ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 0 ≤ ((𝐿𝐹) · (𝑀𝐺)))
308ad2antrr 761 . . . . 5 (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝑈 ∈ NrmGrp)
3110ad2antrr 761 . . . . . . 7 (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝐹 ∈ (𝑇 GrpHom 𝑈))
32 eqid 2621 . . . . . . . 8 (Base‘𝑇) = (Base‘𝑇)
33 eqid 2621 . . . . . . . 8 (Base‘𝑈) = (Base‘𝑈)
3432, 33ghmf 17585 . . . . . . 7 (𝐹 ∈ (𝑇 GrpHom 𝑈) → 𝐹:(Base‘𝑇)⟶(Base‘𝑈))
3531, 34syl 17 . . . . . 6 (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝐹:(Base‘𝑇)⟶(Base‘𝑈))
3611ad2antlr 762 . . . . . . . 8 (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
372, 32ghmf 17585 . . . . . . . 8 (𝐺 ∈ (𝑆 GrpHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
3836, 37syl 17 . . . . . . 7 (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
39 simprl 793 . . . . . . 7 (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝑥 ∈ (Base‘𝑆))
4038, 39ffvelrnd 6316 . . . . . 6 (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (𝐺𝑥) ∈ (Base‘𝑇))
4135, 40ffvelrnd 6316 . . . . 5 (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (𝐹‘(𝐺𝑥)) ∈ (Base‘𝑈))
4233, 4nmcl 22330 . . . . 5 ((𝑈 ∈ NrmGrp ∧ (𝐹‘(𝐺𝑥)) ∈ (Base‘𝑈)) → ((norm‘𝑈)‘(𝐹‘(𝐺𝑥))) ∈ ℝ)
4330, 41, 42syl2anc 692 . . . 4 (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((norm‘𝑈)‘(𝐹‘(𝐺𝑥))) ∈ ℝ)
4415ad2antrr 761 . . . . 5 (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (𝐿𝐹) ∈ ℝ)
4520ad2antrr 761 . . . . . 6 (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝑇 ∈ NrmGrp)
46 eqid 2621 . . . . . . 7 (norm‘𝑇) = (norm‘𝑇)
4732, 46nmcl 22330 . . . . . 6 ((𝑇 ∈ NrmGrp ∧ (𝐺𝑥) ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝐺𝑥)) ∈ ℝ)
4845, 40, 47syl2anc 692 . . . . 5 (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((norm‘𝑇)‘(𝐺𝑥)) ∈ ℝ)
4944, 48remulcld 10014 . . . 4 (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((𝐿𝐹) · ((norm‘𝑇)‘(𝐺𝑥))) ∈ ℝ)
5017ad2antlr 762 . . . . . 6 (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (𝑀𝐺) ∈ ℝ)
512, 3nmcl 22330 . . . . . . . 8 ((𝑆 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑆)‘𝑥) ∈ ℝ)
526, 51sylan 488 . . . . . . 7 ((𝐺 ∈ (𝑆 NGHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑆)‘𝑥) ∈ ℝ)
5352ad2ant2lr 783 . . . . . 6 (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((norm‘𝑆)‘𝑥) ∈ ℝ)
5450, 53remulcld 10014 . . . . 5 (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((𝑀𝐺) · ((norm‘𝑆)‘𝑥)) ∈ ℝ)
5544, 54remulcld 10014 . . . 4 (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((𝐿𝐹) · ((𝑀𝐺) · ((norm‘𝑆)‘𝑥))) ∈ ℝ)
56 simpll 789 . . . . 5 (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → 𝐹 ∈ (𝑇 NGHom 𝑈))
5714, 32, 46, 4nmoi 22442 . . . . 5 ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ (𝐺𝑥) ∈ (Base‘𝑇)) → ((norm‘𝑈)‘(𝐹‘(𝐺𝑥))) ≤ ((𝐿𝐹) · ((norm‘𝑇)‘(𝐺𝑥))))
5856, 40, 57syl2anc 692 . . . 4 (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((norm‘𝑈)‘(𝐹‘(𝐺𝑥))) ≤ ((𝐿𝐹) · ((norm‘𝑇)‘(𝐺𝑥))))
5923ad2antrr 761 . . . . 5 (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((𝐿𝐹) ∈ ℝ ∧ 0 ≤ (𝐿𝐹)))
6016, 2, 3, 46nmoi 22442 . . . . . 6 ((𝐺 ∈ (𝑆 NGHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((norm‘𝑇)‘(𝐺𝑥)) ≤ ((𝑀𝐺) · ((norm‘𝑆)‘𝑥)))
6160ad2ant2lr 783 . . . . 5 (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((norm‘𝑇)‘(𝐺𝑥)) ≤ ((𝑀𝐺) · ((norm‘𝑆)‘𝑥)))
62 lemul2a 10822 . . . . 5 (((((norm‘𝑇)‘(𝐺𝑥)) ∈ ℝ ∧ ((𝑀𝐺) · ((norm‘𝑆)‘𝑥)) ∈ ℝ ∧ ((𝐿𝐹) ∈ ℝ ∧ 0 ≤ (𝐿𝐹))) ∧ ((norm‘𝑇)‘(𝐺𝑥)) ≤ ((𝑀𝐺) · ((norm‘𝑆)‘𝑥))) → ((𝐿𝐹) · ((norm‘𝑇)‘(𝐺𝑥))) ≤ ((𝐿𝐹) · ((𝑀𝐺) · ((norm‘𝑆)‘𝑥))))
6348, 54, 59, 61, 62syl31anc 1326 . . . 4 (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((𝐿𝐹) · ((norm‘𝑇)‘(𝐺𝑥))) ≤ ((𝐿𝐹) · ((𝑀𝐺) · ((norm‘𝑆)‘𝑥))))
6443, 49, 55, 58, 63letrd 10138 . . 3 (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((norm‘𝑈)‘(𝐹‘(𝐺𝑥))) ≤ ((𝐿𝐹) · ((𝑀𝐺) · ((norm‘𝑆)‘𝑥))))
65 fvco3 6232 . . . . 5 ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
6638, 39, 65syl2anc 692 . . . 4 (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
6766fveq2d 6152 . . 3 (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((norm‘𝑈)‘((𝐹𝐺)‘𝑥)) = ((norm‘𝑈)‘(𝐹‘(𝐺𝑥))))
6844recnd 10012 . . . 4 (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (𝐿𝐹) ∈ ℂ)
6950recnd 10012 . . . 4 (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (𝑀𝐺) ∈ ℂ)
7053recnd 10012 . . . 4 (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((norm‘𝑆)‘𝑥) ∈ ℂ)
7168, 69, 70mulassd 10007 . . 3 (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → (((𝐿𝐹) · (𝑀𝐺)) · ((norm‘𝑆)‘𝑥)) = ((𝐿𝐹) · ((𝑀𝐺) · ((norm‘𝑆)‘𝑥))))
7264, 67, 713brtr4d 4645 . 2 (((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))) → ((norm‘𝑈)‘((𝐹𝐺)‘𝑥)) ≤ (((𝐿𝐹) · (𝑀𝐺)) · ((norm‘𝑆)‘𝑥)))
731, 2, 3, 4, 5, 7, 9, 13, 19, 29, 72nmolb2d 22432 1 ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘(𝐹𝐺)) ≤ ((𝐿𝐹) · (𝑀𝐺)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987   ≠ wne 2790   class class class wbr 4613   ∘ ccom 5078  ⟶wf 5843  ‘cfv 5847  (class class class)co 6604  ℝcr 9879  0cc0 9880   · cmul 9885   ≤ cle 10019  Basecbs 15781  0gc0g 16021   GrpHom cghm 17578  normcnm 22291  NrmGrpcngp 22292   normOp cnmo 22419   NGHom cnghm 22420 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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958 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 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-sup 8292  df-inf 8293  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-n0 11237  df-z 11322  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-ico 12123  df-0g 16023  df-topgen 16025  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-mhm 17256  df-grp 17346  df-ghm 17579  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-mopn 19661  df-top 20621  df-bases 20622  df-topon 20623  df-topsp 20624  df-xms 22035  df-ms 22036  df-nm 22297  df-ngp 22298  df-nmo 22422  df-nghm 22423 This theorem is referenced by:  nghmco  22452
 Copyright terms: Public domain W3C validator