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Theorem reghmph 22329
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 22312 . 2 (𝐽𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2 n0 4307 . . 3 ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
3 hmeocn 22296 . . . . . . . 8 (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓 ∈ (𝐽 Cn 𝐾))
43adantl 482 . . . . . . 7 ((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝑓 ∈ (𝐽 Cn 𝐾))
5 cntop2 21777 . . . . . . 7 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
64, 5syl 17 . . . . . 6 ((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝐾 ∈ Top)
7 simpll 763 . . . . . . . . 9 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝐽 ∈ Reg)
84adantr 481 . . . . . . . . . 10 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝑓 ∈ (𝐽 Cn 𝐾))
9 simprl 767 . . . . . . . . . 10 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝑥𝐾)
10 cnima 21801 . . . . . . . . . 10 ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑥𝐾) → (𝑓𝑥) ∈ 𝐽)
118, 9, 10syl2anc 584 . . . . . . . . 9 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → (𝑓𝑥) ∈ 𝐽)
12 eqid 2818 . . . . . . . . . . . . 13 𝐽 = 𝐽
13 eqid 2818 . . . . . . . . . . . . 13 𝐾 = 𝐾
1412, 13hmeof1o 22300 . . . . . . . . . . . 12 (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓: 𝐽1-1-onto 𝐾)
1514ad2antlr 723 . . . . . . . . . . 11 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝑓: 𝐽1-1-onto 𝐾)
16 f1ocnv 6620 . . . . . . . . . . 11 (𝑓: 𝐽1-1-onto 𝐾𝑓: 𝐾1-1-onto 𝐽)
17 f1ofn 6609 . . . . . . . . . . 11 (𝑓: 𝐾1-1-onto 𝐽𝑓 Fn 𝐾)
1815, 16, 173syl 18 . . . . . . . . . 10 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝑓 Fn 𝐾)
19 elssuni 4859 . . . . . . . . . . 11 (𝑥𝐾𝑥 𝐾)
2019ad2antrl 724 . . . . . . . . . 10 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝑥 𝐾)
21 simprr 769 . . . . . . . . . 10 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝑦𝑥)
22 fnfvima 6986 . . . . . . . . . 10 ((𝑓 Fn 𝐾𝑥 𝐾𝑦𝑥) → (𝑓𝑦) ∈ (𝑓𝑥))
2318, 20, 21, 22syl3anc 1363 . . . . . . . . 9 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → (𝑓𝑦) ∈ (𝑓𝑥))
24 regsep 21870 . . . . . . . . 9 ((𝐽 ∈ Reg ∧ (𝑓𝑥) ∈ 𝐽 ∧ (𝑓𝑦) ∈ (𝑓𝑥)) → ∃𝑤𝐽 ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))
257, 11, 23, 24syl3anc 1363 . . . . . . . 8 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → ∃𝑤𝐽 ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))
26 simpllr 772 . . . . . . . . . 10 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑓 ∈ (𝐽Homeo𝐾))
27 simprl 767 . . . . . . . . . 10 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑤𝐽)
28 hmeoima 22301 . . . . . . . . . 10 ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑤𝐽) → (𝑓𝑤) ∈ 𝐾)
2926, 27, 28syl2anc 584 . . . . . . . . 9 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → (𝑓𝑤) ∈ 𝐾)
3020, 21sseldd 3965 . . . . . . . . . . . 12 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → 𝑦 𝐾)
3130adantr 481 . . . . . . . . . . 11 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑦 𝐾)
32 simprrl 777 . . . . . . . . . . 11 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → (𝑓𝑦) ∈ 𝑤)
3318adantr 481 . . . . . . . . . . . 12 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑓 Fn 𝐾)
34 elpreima 6820 . . . . . . . . . . . 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 709 . . . . . . . . . 10 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑦 ∈ (𝑓𝑤))
37 imacnvcnv 6056 . . . . . . . . . 10 (𝑓𝑤) = (𝑓𝑤)
3836, 37eleqtrdi 2920 . . . . . . . . 9 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑦 ∈ (𝑓𝑤))
39 elssuni 4859 . . . . . . . . . . . 12 (𝑤𝐽𝑤 𝐽)
4039ad2antrl 724 . . . . . . . . . . 11 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑤 𝐽)
4112hmeocls 22304 . . . . . . . . . . 11 ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑤 𝐽) → ((cls‘𝐾)‘(𝑓𝑤)) = (𝑓 “ ((cls‘𝐽)‘𝑤)))
4226, 40, 41syl2anc 584 . . . . . . . . . 10 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ((cls‘𝐾)‘(𝑓𝑤)) = (𝑓 “ ((cls‘𝐽)‘𝑤)))
43 simprrr 778 . . . . . . . . . . 11 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥))
4415adantr 481 . . . . . . . . . . . . 13 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑓: 𝐽1-1-onto 𝐾)
45 f1ofun 6610 . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝐽 ∈ Reg)
48 regtop 21869 . . . . . . . . . . . . . . 15 (𝐽 ∈ Reg → 𝐽 ∈ Top)
4947, 48syl 17 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝐽 ∈ Top)
5012clsss3 21595 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑤 𝐽) → ((cls‘𝐽)‘𝑤) ⊆ 𝐽)
5149, 40, 50syl2anc 584 . . . . . . . . . . . . 13 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ((cls‘𝐽)‘𝑤) ⊆ 𝐽)
52 f1odm 6612 . . . . . . . . . . . . . 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 4005 . . . . . . . . . . . 12 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ((cls‘𝐽)‘𝑤) ⊆ dom 𝑓)
55 funimass3 6816 . . . . . . . . . . . 12 ((Fun 𝑓 ∧ ((cls‘𝐽)‘𝑤) ⊆ dom 𝑓) → ((𝑓 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))
5646, 54, 55syl2anc 584 . . . . . . . . . . 11 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ((𝑓 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))
5743, 56mpbird 258 . . . . . . . . . 10 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → (𝑓 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑥)
5842, 57eqsstrd 4002 . . . . . . . . 9 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ((cls‘𝐾)‘(𝑓𝑤)) ⊆ 𝑥)
59 eleq2 2898 . . . . . . . . . . 11 (𝑧 = (𝑓𝑤) → (𝑦𝑧𝑦 ∈ (𝑓𝑤)))
60 fveq2 6663 . . . . . . . . . . . 12 (𝑧 = (𝑓𝑤) → ((cls‘𝐾)‘𝑧) = ((cls‘𝐾)‘(𝑓𝑤)))
6160sseq1d 3995 . . . . . . . . . . 11 (𝑧 = (𝑓𝑤) → (((cls‘𝐾)‘𝑧) ⊆ 𝑥 ↔ ((cls‘𝐾)‘(𝑓𝑤)) ⊆ 𝑥))
6259, 61anbi12d 630 . . . . . . . . . 10 (𝑧 = (𝑓𝑤) → ((𝑦𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥) ↔ (𝑦 ∈ (𝑓𝑤) ∧ ((cls‘𝐾)‘(𝑓𝑤)) ⊆ 𝑥)))
6362rspcev 3620 . . . . . . . . 9 (((𝑓𝑤) ∈ 𝐾 ∧ (𝑦 ∈ (𝑓𝑤) ∧ ((cls‘𝐾)‘(𝑓𝑤)) ⊆ 𝑥)) → ∃𝑧𝐾 (𝑦𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
6429, 38, 58, 63syl12anc 832 . . . . . . . 8 ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ∃𝑧𝐾 (𝑦𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
6525, 64rexlimddv 3288 . . . . . . 7 (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦𝑥)) → ∃𝑧𝐾 (𝑦𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
6665ralrimivva 3188 . . . . . 6 ((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → ∀𝑥𝐾𝑦𝑥𝑧𝐾 (𝑦𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
67 isreg 21868 . . . . . 6 (𝐾 ∈ Reg ↔ (𝐾 ∈ Top ∧ ∀𝑥𝐾𝑦𝑥𝑧𝐾 (𝑦𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥)))
686, 66, 67sylanbrc 583 . . . . 5 ((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝐾 ∈ Reg)
6968expcom 414 . . . 4 (𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Reg → 𝐾 ∈ Reg))
7069exlimiv 1922 . . 3 (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Reg → 𝐾 ∈ Reg))
712, 70sylbi 218 . 2 ((𝐽Homeo𝐾) ≠ ∅ → (𝐽 ∈ Reg → 𝐾 ∈ Reg))
721, 71sylbi 218 1 (𝐽𝐾 → (𝐽 ∈ Reg → 𝐾 ∈ Reg))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wex 1771  wcel 2105  wne 3013  wral 3135  wrex 3136  wss 3933  c0 4288   cuni 4830   class class class wbr 5057  ccnv 5547  dom cdm 5548  cima 5551  Fun wfun 6342   Fn wfn 6343  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  Topctop 21429  clsccl 21554   Cn ccn 21760  Regcreg 21845  Homeochmeo 22289  chmph 22290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-1o 8091  df-map 8397  df-top 21430  df-topon 21447  df-cld 21555  df-cls 21557  df-cn 21763  df-reg 21852  df-hmeo 22291  df-hmph 22292
This theorem is referenced by: (None)
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