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Theorem reghmph 22405
Description: Regularity is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
Assertion
Ref Expression
reghmph (𝐽𝐾 → (𝐽 ∈ Reg → 𝐾 ∈ Reg))

Proof of Theorem reghmph
Dummy variables 𝑤 𝑓 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hmph 22388 . 2 (𝐽𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2 n0 4293 . . 3 ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
3 hmeocn 22372 . . . . . . . 8 (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓 ∈ (𝐽 Cn 𝐾))
43adantl 485 . . . . . . 7 ((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝑓 ∈ (𝐽 Cn 𝐾))
5 cntop2 21853 . . . . . . 7 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
64, 5syl 17 . . . . . 6 ((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝐾 ∈ Top)
7 simpll 766 . . . . . . . . 9 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝐽 ∈ Reg)
84adantr 484 . . . . . . . . . 10 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝑓 ∈ (𝐽 Cn 𝐾))
9 simprl 770 . . . . . . . . . 10 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝑥𝐾)
10 cnima 21877 . . . . . . . . . 10 ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑥𝐾) → (𝑓𝑥) ∈ 𝐽)
118, 9, 10syl2anc 587 . . . . . . . . 9 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → (𝑓𝑥) ∈ 𝐽)
12 eqid 2824 . . . . . . . . . . . . 13 𝐽 = 𝐽
13 eqid 2824 . . . . . . . . . . . . 13 𝐾 = 𝐾
1412, 13hmeof1o 22376 . . . . . . . . . . . 12 (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓: 𝐽1-1-onto 𝐾)
1514ad2antlr 726 . . . . . . . . . . 11 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝑓: 𝐽1-1-onto 𝐾)
16 f1ocnv 6618 . . . . . . . . . . 11 (𝑓: 𝐽1-1-onto 𝐾𝑓: 𝐾1-1-onto 𝐽)
17 f1ofn 6607 . . . . . . . . . . 11 (𝑓: 𝐾1-1-onto 𝐽𝑓 Fn 𝐾)
1815, 16, 173syl 18 . . . . . . . . . 10 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝑓 Fn 𝐾)
19 elssuni 4854 . . . . . . . . . . 11 (𝑥𝐾𝑥 𝐾)
2019ad2antrl 727 . . . . . . . . . 10 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝑥 𝐾)
21 simprr 772 . . . . . . . . . 10 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝑦𝑥)
22 fnfvima 6987 . . . . . . . . . 10 ((𝑓 Fn 𝐾𝑥 𝐾𝑦𝑥) → (𝑓𝑦) ∈ (𝑓𝑥))
2318, 20, 21, 22syl3anc 1368 . . . . . . . . 9 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → (𝑓𝑦) ∈ (𝑓𝑥))
24 regsep 21946 . . . . . . . . 9 ((𝐽 ∈ Reg ∧ (𝑓𝑥) ∈ 𝐽 ∧ (𝑓𝑦) ∈ (𝑓𝑥)) → ∃𝑤𝐽 ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))
257, 11, 23, 24syl3anc 1368 . . . . . . . 8 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → ∃𝑤𝐽 ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))
26 simpllr 775 . . . . . . . . . 10 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑓 ∈ (𝐽Homeo𝐾))
27 simprl 770 . . . . . . . . . 10 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑤𝐽)
28 hmeoima 22377 . . . . . . . . . 10 ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑤𝐽) → (𝑓𝑤) ∈ 𝐾)
2926, 27, 28syl2anc 587 . . . . . . . . 9 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → (𝑓𝑤) ∈ 𝐾)
3020, 21sseldd 3954 . . . . . . . . . . . 12 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝑦 𝐾)
3130adantr 484 . . . . . . . . . . 11 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑦 𝐾)
32 simprrl 780 . . . . . . . . . . 11 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → (𝑓𝑦) ∈ 𝑤)
3318adantr 484 . . . . . . . . . . . 12 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑓 Fn 𝐾)
34 elpreima 6819 . . . . . . . . . . . 12 (𝑓 Fn 𝐾 → (𝑦 ∈ (𝑓𝑤) ↔ (𝑦 𝐾 ∧ (𝑓𝑦) ∈ 𝑤)))
3533, 34syl 17 . . . . . . . . . . 11 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → (𝑦 ∈ (𝑓𝑤) ↔ (𝑦 𝐾 ∧ (𝑓𝑦) ∈ 𝑤)))
3631, 32, 35mpbir2and 712 . . . . . . . . . 10 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑦 ∈ (𝑓𝑤))
37 imacnvcnv 6050 . . . . . . . . . 10 (𝑓𝑤) = (𝑓𝑤)
3836, 37eleqtrdi 2926 . . . . . . . . 9 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑦 ∈ (𝑓𝑤))
39 elssuni 4854 . . . . . . . . . . . 12 (𝑤𝐽𝑤 𝐽)
4039ad2antrl 727 . . . . . . . . . . 11 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑤 𝐽)
4112hmeocls 22380 . . . . . . . . . . 11 ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑤 𝐽) → ((cls‘𝐾)‘(𝑓𝑤)) = (𝑓 “ ((cls‘𝐽)‘𝑤)))
4226, 40, 41syl2anc 587 . . . . . . . . . 10 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ((cls‘𝐾)‘(𝑓𝑤)) = (𝑓 “ ((cls‘𝐽)‘𝑤)))
43 simprrr 781 . . . . . . . . . . 11 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥))
4415adantr 484 . . . . . . . . . . . . 13 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑓: 𝐽1-1-onto 𝐾)
45 f1ofun 6608 . . . . . . . . . . . . 13 (𝑓: 𝐽1-1-onto 𝐾 → Fun 𝑓)
4644, 45syl 17 . . . . . . . . . . . 12 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → Fun 𝑓)
477adantr 484 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝐽 ∈ Reg)
48 regtop 21945 . . . . . . . . . . . . . . 15 (𝐽 ∈ Reg → 𝐽 ∈ Top)
4947, 48syl 17 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝐽 ∈ Top)
5012clsss3 21671 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑤 𝐽) → ((cls‘𝐽)‘𝑤) ⊆ 𝐽)
5149, 40, 50syl2anc 587 . . . . . . . . . . . . 13 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ((cls‘𝐽)‘𝑤) ⊆ 𝐽)
52 f1odm 6610 . . . . . . . . . . . . . 14 (𝑓: 𝐽1-1-onto 𝐾 → dom 𝑓 = 𝐽)
5344, 52syl 17 . . . . . . . . . . . . 13 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → dom 𝑓 = 𝐽)
5451, 53sseqtrrd 3994 . . . . . . . . . . . 12 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ((cls‘𝐽)‘𝑤) ⊆ dom 𝑓)
55 funimass3 6815 . . . . . . . . . . . 12 ((Fun 𝑓 ∧ ((cls‘𝐽)‘𝑤) ⊆ dom 𝑓) → ((𝑓 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))
5646, 54, 55syl2anc 587 . . . . . . . . . . 11 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ((𝑓 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))
5743, 56mpbird 260 . . . . . . . . . 10 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → (𝑓 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑥)
5842, 57eqsstrd 3991 . . . . . . . . 9 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ((cls‘𝐾)‘(𝑓𝑤)) ⊆ 𝑥)
59 eleq2 2904 . . . . . . . . . . 11 (𝑧 = (𝑓𝑤) → (𝑦𝑧𝑦 ∈ (𝑓𝑤)))
60 fveq2 6661 . . . . . . . . . . . 12 (𝑧 = (𝑓𝑤) → ((cls‘𝐾)‘𝑧) = ((cls‘𝐾)‘(𝑓𝑤)))
6160sseq1d 3984 . . . . . . . . . . 11 (𝑧 = (𝑓𝑤) → (((cls‘𝐾)‘𝑧) ⊆ 𝑥 ↔ ((cls‘𝐾)‘(𝑓𝑤)) ⊆ 𝑥))
6259, 61anbi12d 633 . . . . . . . . . 10 (𝑧 = (𝑓𝑤) → ((𝑦𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥) ↔ (𝑦 ∈ (𝑓𝑤) ∧ ((cls‘𝐾)‘(𝑓𝑤)) ⊆ 𝑥)))
6362rspcev 3609 . . . . . . . . 9 (((𝑓𝑤) ∈ 𝐾 ∧ (𝑦 ∈ (𝑓𝑤) ∧ ((cls‘𝐾)‘(𝑓𝑤)) ⊆ 𝑥)) → ∃𝑧𝐾 (𝑦𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
6429, 38, 58, 63syl12anc 835 . . . . . . . 8 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ∃𝑧𝐾 (𝑦𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
6525, 64rexlimddv 3283 . . . . . . 7 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → ∃𝑧𝐾 (𝑦𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
6665ralrimivva 3186 . . . . . 6 ((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → ∀𝑥𝐾𝑦𝑥𝑧𝐾 (𝑦𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
67 isreg 21944 . . . . . 6 (𝐾 ∈ Reg ↔ (𝐾 ∈ Top ∧ ∀𝑥𝐾𝑦𝑥𝑧𝐾 (𝑦𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥)))
686, 66, 67sylanbrc 586 . . . . 5 ((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝐾 ∈ Reg)
6968expcom 417 . . . 4 (𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Reg → 𝐾 ∈ Reg))
7069exlimiv 1932 . . 3 (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Reg → 𝐾 ∈ Reg))
712, 70sylbi 220 . 2 ((𝐽Homeo𝐾) ≠ ∅ → (𝐽 ∈ Reg → 𝐾 ∈ Reg))
721, 71sylbi 220 1 (𝐽𝐾 → (𝐽 ∈ Reg → 𝐾 ∈ Reg))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wex 1781  wcel 2115  wne 3014  wral 3133  wrex 3134  wss 3919  c0 4276   cuni 4824   class class class wbr 5052  ccnv 5541  dom cdm 5542  cima 5545  Fun wfun 6337   Fn wfn 6338  1-1-ontowf1o 6342  cfv 6343  (class class class)co 7149  Topctop 21505  clsccl 21630   Cn ccn 21836  Regcreg 21921  Homeochmeo 22365  chmph 22366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-1o 8098  df-map 8404  df-top 21506  df-topon 21523  df-cld 21631  df-cls 21633  df-cn 21839  df-reg 21928  df-hmeo 22367  df-hmph 22368
This theorem is referenced by: (None)
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