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

Theorem ghmcnp 24110
Description: A group homomorphism on topological groups is continuous everywhere if it is continuous at any point. (Contributed by Mario Carneiro, 21-Oct-2015.)
Hypotheses
Ref Expression
ghmcnp.x 𝑋 = (Base‘𝐺)
ghmcnp.j 𝐽 = (TopOpen‘𝐺)
ghmcnp.k 𝐾 = (TopOpen‘𝐻)
Assertion
Ref Expression
ghmcnp ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐴𝑋𝐹 ∈ (𝐽 Cn 𝐾))))

Proof of Theorem ghmcnp
Dummy variables 𝑣 𝑢 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2726 . . . . . 6 𝐽 = 𝐽
21cnprcl 23240 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐴 𝐽)
32a1i 11 . . . 4 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐴 𝐽))
4 ghmcnp.j . . . . . . . . . 10 𝐽 = (TopOpen‘𝐺)
5 ghmcnp.x . . . . . . . . . 10 𝑋 = (Base‘𝐺)
64, 5tmdtopon 24076 . . . . . . . . 9 (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝑋))
763ad2ant1 1130 . . . . . . . 8 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → 𝐽 ∈ (TopOn‘𝑋))
87adantr 479 . . . . . . 7 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐽 ∈ (TopOn‘𝑋))
9 simpl2 1189 . . . . . . . 8 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐻 ∈ TopMnd)
10 ghmcnp.k . . . . . . . . 9 𝐾 = (TopOpen‘𝐻)
11 eqid 2726 . . . . . . . . 9 (Base‘𝐻) = (Base‘𝐻)
1210, 11tmdtopon 24076 . . . . . . . 8 (𝐻 ∈ TopMnd → 𝐾 ∈ (TopOn‘(Base‘𝐻)))
139, 12syl 17 . . . . . . 7 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐾 ∈ (TopOn‘(Base‘𝐻)))
14 simpr 483 . . . . . . 7 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
15 cnpf2 23245 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘(Base‘𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶(Base‘𝐻))
168, 13, 14, 15syl3anc 1368 . . . . . 6 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶(Base‘𝐻))
1716adantr 479 . . . . . . . 8 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → 𝐹:𝑋⟶(Base‘𝐻))
1814adantr 479 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
19 eqid 2726 . . . . . . . . . . . . . . 15 (𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤)) = (𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤))
2019mptpreima 6249 . . . . . . . . . . . . . 14 ((𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤)) “ 𝑦) = {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦}
219adantr 479 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝐻 ∈ TopMnd)
2216adantr 479 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝐹:𝑋⟶(Base‘𝐻))
23 simpll3 1211 . . . . . . . . . . . . . . . . . . 19 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
24 ghmgrp1 19212 . . . . . . . . . . . . . . . . . . 19 (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp)
2523, 24syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝐺 ∈ Grp)
26 simprl 769 . . . . . . . . . . . . . . . . . 18 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝑥𝑋)
272adantl 480 . . . . . . . . . . . . . . . . . . . 20 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 𝐽)
28 toponuni 22907 . . . . . . . . . . . . . . . . . . . . 21 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
298, 28syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝑋 = 𝐽)
3027, 29eleqtrrd 2829 . . . . . . . . . . . . . . . . . . 19 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴𝑋)
3130adantr 479 . . . . . . . . . . . . . . . . . 18 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝐴𝑋)
32 eqid 2726 . . . . . . . . . . . . . . . . . . 19 (-g𝐺) = (-g𝐺)
335, 32grpsubcl 19014 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ Grp ∧ 𝑥𝑋𝐴𝑋) → (𝑥(-g𝐺)𝐴) ∈ 𝑋)
3425, 26, 31, 33syl3anc 1368 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (𝑥(-g𝐺)𝐴) ∈ 𝑋)
3522, 34ffvelcdmd 7099 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (𝐹‘(𝑥(-g𝐺)𝐴)) ∈ (Base‘𝐻))
36 eqid 2726 . . . . . . . . . . . . . . . . 17 (+g𝐻) = (+g𝐻)
3719, 11, 36, 10tmdlactcn 24097 . . . . . . . . . . . . . . . 16 ((𝐻 ∈ TopMnd ∧ (𝐹‘(𝑥(-g𝐺)𝐴)) ∈ (Base‘𝐻)) → (𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤)) ∈ (𝐾 Cn 𝐾))
3821, 35, 37syl2anc 582 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤)) ∈ (𝐾 Cn 𝐾))
39 simprrl 779 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝑦𝐾)
40 cnima 23260 . . . . . . . . . . . . . . 15 (((𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤)) ∈ (𝐾 Cn 𝐾) ∧ 𝑦𝐾) → ((𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤)) “ 𝑦) ∈ 𝐾)
4138, 39, 40syl2anc 582 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → ((𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤)) “ 𝑦) ∈ 𝐾)
4220, 41eqeltrrid 2831 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦} ∈ 𝐾)
43 oveq2 7432 . . . . . . . . . . . . . . 15 (𝑤 = (𝐹𝐴) → ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) = ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝐴)))
4443eleq1d 2811 . . . . . . . . . . . . . 14 (𝑤 = (𝐹𝐴) → (((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦 ↔ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝐴)) ∈ 𝑦))
4522, 31ffvelcdmd 7099 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (𝐹𝐴) ∈ (Base‘𝐻))
46 eqid 2726 . . . . . . . . . . . . . . . . . . 19 (-g𝐻) = (-g𝐻)
475, 32, 46ghmsub 19218 . . . . . . . . . . . . . . . . . 18 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥𝑋𝐴𝑋) → (𝐹‘(𝑥(-g𝐺)𝐴)) = ((𝐹𝑥)(-g𝐻)(𝐹𝐴)))
4823, 26, 31, 47syl3anc 1368 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (𝐹‘(𝑥(-g𝐺)𝐴)) = ((𝐹𝑥)(-g𝐻)(𝐹𝐴)))
4948oveq1d 7439 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝐴)) = (((𝐹𝑥)(-g𝐻)(𝐹𝐴))(+g𝐻)(𝐹𝐴)))
50 ghmgrp2 19213 . . . . . . . . . . . . . . . . . 18 (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐻 ∈ Grp)
5123, 50syl 17 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝐻 ∈ Grp)
5222, 26ffvelcdmd 7099 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (𝐹𝑥) ∈ (Base‘𝐻))
5311, 36, 46grpnpcan 19026 . . . . . . . . . . . . . . . . 17 ((𝐻 ∈ Grp ∧ (𝐹𝑥) ∈ (Base‘𝐻) ∧ (𝐹𝐴) ∈ (Base‘𝐻)) → (((𝐹𝑥)(-g𝐻)(𝐹𝐴))(+g𝐻)(𝐹𝐴)) = (𝐹𝑥))
5451, 52, 45, 53syl3anc 1368 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (((𝐹𝑥)(-g𝐻)(𝐹𝐴))(+g𝐻)(𝐹𝐴)) = (𝐹𝑥))
5549, 54eqtrd 2766 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝐴)) = (𝐹𝑥))
56 simprrr 780 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (𝐹𝑥) ∈ 𝑦)
5755, 56eqeltrd 2826 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝐴)) ∈ 𝑦)
5844, 45, 57elrabd 3683 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (𝐹𝐴) ∈ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦})
59 cnpimaex 23251 . . . . . . . . . . . . 13 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦} ∈ 𝐾 ∧ (𝐹𝐴) ∈ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦}) → ∃𝑧𝐽 (𝐴𝑧 ∧ (𝐹𝑧) ⊆ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦}))
6018, 42, 58, 59syl3anc 1368 . . . . . . . . . . . 12 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → ∃𝑧𝐽 (𝐴𝑧 ∧ (𝐹𝑧) ⊆ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦}))
61 ssrab 4069 . . . . . . . . . . . . . . . 16 ((𝐹𝑧) ⊆ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦} ↔ ((𝐹𝑧) ⊆ (Base‘𝐻) ∧ ∀𝑤 ∈ (𝐹𝑧)((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦))
6261simprbi 495 . . . . . . . . . . . . . . 15 ((𝐹𝑧) ⊆ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦} → ∀𝑤 ∈ (𝐹𝑧)((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦)
6322adantr 479 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑧𝐽) → 𝐹:𝑋⟶(Base‘𝐻))
6463ffnd 6729 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑧𝐽) → 𝐹 Fn 𝑋)
658adantr 479 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝐽 ∈ (TopOn‘𝑋))
66 toponss 22920 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → 𝑧𝑋)
6765, 66sylan 578 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑧𝐽) → 𝑧𝑋)
68 oveq2 7432 . . . . . . . . . . . . . . . . . 18 (𝑤 = (𝐹𝑣) → ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) = ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)))
6968eleq1d 2811 . . . . . . . . . . . . . . . . 17 (𝑤 = (𝐹𝑣) → (((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦 ↔ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))
7069ralima 7255 . . . . . . . . . . . . . . . 16 ((𝐹 Fn 𝑋𝑧𝑋) → (∀𝑤 ∈ (𝐹𝑧)((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦 ↔ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))
7164, 67, 70syl2anc 582 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑧𝐽) → (∀𝑤 ∈ (𝐹𝑧)((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦 ↔ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))
7262, 71imbitrid 243 . . . . . . . . . . . . . 14 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑧𝐽) → ((𝐹𝑧) ⊆ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦} → ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))
73 eqid 2726 . . . . . . . . . . . . . . . . . 18 (𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)) = (𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))
7473mptpreima 6249 . . . . . . . . . . . . . . . . 17 ((𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)) “ 𝑧) = {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧}
75 simpl1 1188 . . . . . . . . . . . . . . . . . . . 20 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐺 ∈ TopMnd)
7675ad2antrr 724 . . . . . . . . . . . . . . . . . . 19 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → 𝐺 ∈ TopMnd)
7725adantr 479 . . . . . . . . . . . . . . . . . . . 20 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → 𝐺 ∈ Grp)
7831adantr 479 . . . . . . . . . . . . . . . . . . . 20 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → 𝐴𝑋)
7926adantr 479 . . . . . . . . . . . . . . . . . . . 20 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → 𝑥𝑋)
805, 32grpsubcl 19014 . . . . . . . . . . . . . . . . . . . 20 ((𝐺 ∈ Grp ∧ 𝐴𝑋𝑥𝑋) → (𝐴(-g𝐺)𝑥) ∈ 𝑋)
8177, 78, 79, 80syl3anc 1368 . . . . . . . . . . . . . . . . . . 19 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → (𝐴(-g𝐺)𝑥) ∈ 𝑋)
82 eqid 2726 . . . . . . . . . . . . . . . . . . . 20 (+g𝐺) = (+g𝐺)
8373, 5, 82, 4tmdlactcn 24097 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ TopMnd ∧ (𝐴(-g𝐺)𝑥) ∈ 𝑋) → (𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)) ∈ (𝐽 Cn 𝐽))
8476, 81, 83syl2anc 582 . . . . . . . . . . . . . . . . . 18 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → (𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)) ∈ (𝐽 Cn 𝐽))
85 simprl 769 . . . . . . . . . . . . . . . . . 18 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → 𝑧𝐽)
86 cnima 23260 . . . . . . . . . . . . . . . . . 18 (((𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)) ∈ (𝐽 Cn 𝐽) ∧ 𝑧𝐽) → ((𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)) “ 𝑧) ∈ 𝐽)
8784, 85, 86syl2anc 582 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → ((𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)) “ 𝑧) ∈ 𝐽)
8874, 87eqeltrrid 2831 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} ∈ 𝐽)
89 oveq2 7432 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑥 → ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) = ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑥))
9089eleq1d 2811 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑥 → (((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧 ↔ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑥) ∈ 𝑧))
915, 82, 32grpnpcan 19026 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ Grp ∧ 𝐴𝑋𝑥𝑋) → ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑥) = 𝐴)
9277, 78, 79, 91syl3anc 1368 . . . . . . . . . . . . . . . . . 18 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑥) = 𝐴)
93 simprrl 779 . . . . . . . . . . . . . . . . . 18 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → 𝐴𝑧)
9492, 93eqeltrd 2826 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑥) ∈ 𝑧)
9590, 79, 94elrabd 3683 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → 𝑥 ∈ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧})
96 simprrr 780 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦)
97 fveq2 6901 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑣 = ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) → (𝐹𝑣) = (𝐹‘((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)))
9897oveq2d 7440 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 = ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) → ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) = ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹‘((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))))
9998eleq1d 2811 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 = ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) → (((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦 ↔ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹‘((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))) ∈ 𝑦))
10099rspccv 3605 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦 → (((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧 → ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹‘((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))) ∈ 𝑦))
10196, 100syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → (((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧 → ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹‘((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))) ∈ 𝑦))
102101adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) ∧ 𝑤𝑋) → (((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧 → ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹‘((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))) ∈ 𝑦))
10323adantr 479 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
10434adantr 479 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → (𝑥(-g𝐺)𝐴) ∈ 𝑋)
105103, 24syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → 𝐺 ∈ Grp)
10631adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → 𝐴𝑋)
10726adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → 𝑥𝑋)
108105, 106, 107, 80syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → (𝐴(-g𝐺)𝑥) ∈ 𝑋)
109 simpr 483 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → 𝑤𝑋)
1105, 82grpcl 18936 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐺 ∈ Grp ∧ (𝐴(-g𝐺)𝑥) ∈ 𝑋𝑤𝑋) → ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑋)
111105, 108, 109, 110syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑋)
1125, 82, 36ghmlin 19215 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ (𝑥(-g𝐺)𝐴) ∈ 𝑋 ∧ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑋) → (𝐹‘((𝑥(-g𝐺)𝐴)(+g𝐺)((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))) = ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹‘((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))))
113103, 104, 111, 112syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → (𝐹‘((𝑥(-g𝐺)𝐴)(+g𝐺)((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))) = ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹‘((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))))
114 eqid 2726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (invg𝐺) = (invg𝐺)
1155, 32, 114grpinvsub 19016 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐺 ∈ Grp ∧ 𝑥𝑋𝐴𝑋) → ((invg𝐺)‘(𝑥(-g𝐺)𝐴)) = (𝐴(-g𝐺)𝑥))
116105, 107, 106, 115syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → ((invg𝐺)‘(𝑥(-g𝐺)𝐴)) = (𝐴(-g𝐺)𝑥))
117116oveq2d 7440 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → ((𝑥(-g𝐺)𝐴)(+g𝐺)((invg𝐺)‘(𝑥(-g𝐺)𝐴))) = ((𝑥(-g𝐺)𝐴)(+g𝐺)(𝐴(-g𝐺)𝑥)))
118 eqid 2726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (0g𝐺) = (0g𝐺)
1195, 82, 118, 114grprinv 18985 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐺 ∈ Grp ∧ (𝑥(-g𝐺)𝐴) ∈ 𝑋) → ((𝑥(-g𝐺)𝐴)(+g𝐺)((invg𝐺)‘(𝑥(-g𝐺)𝐴))) = (0g𝐺))
120105, 104, 119syl2anc 582 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → ((𝑥(-g𝐺)𝐴)(+g𝐺)((invg𝐺)‘(𝑥(-g𝐺)𝐴))) = (0g𝐺))
121117, 120eqtr3d 2768 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → ((𝑥(-g𝐺)𝐴)(+g𝐺)(𝐴(-g𝐺)𝑥)) = (0g𝐺))
122121oveq1d 7439 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → (((𝑥(-g𝐺)𝐴)(+g𝐺)(𝐴(-g𝐺)𝑥))(+g𝐺)𝑤) = ((0g𝐺)(+g𝐺)𝑤))
1235, 82grpass 18937 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐺 ∈ Grp ∧ ((𝑥(-g𝐺)𝐴) ∈ 𝑋 ∧ (𝐴(-g𝐺)𝑥) ∈ 𝑋𝑤𝑋)) → (((𝑥(-g𝐺)𝐴)(+g𝐺)(𝐴(-g𝐺)𝑥))(+g𝐺)𝑤) = ((𝑥(-g𝐺)𝐴)(+g𝐺)((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)))
124105, 104, 108, 109, 123syl13anc 1369 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → (((𝑥(-g𝐺)𝐴)(+g𝐺)(𝐴(-g𝐺)𝑥))(+g𝐺)𝑤) = ((𝑥(-g𝐺)𝐴)(+g𝐺)((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)))
1255, 82, 118grplid 18962 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐺 ∈ Grp ∧ 𝑤𝑋) → ((0g𝐺)(+g𝐺)𝑤) = 𝑤)
126105, 109, 125syl2anc 582 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → ((0g𝐺)(+g𝐺)𝑤) = 𝑤)
127122, 124, 1263eqtr3d 2774 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → ((𝑥(-g𝐺)𝐴)(+g𝐺)((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)) = 𝑤)
128127fveq2d 6905 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → (𝐹‘((𝑥(-g𝐺)𝐴)(+g𝐺)((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))) = (𝐹𝑤))
129113, 128eqtr3d 2768 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹‘((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))) = (𝐹𝑤))
130129adantlr 713 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) ∧ 𝑤𝑋) → ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹‘((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))) = (𝐹𝑤))
131130eleq1d 2811 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) ∧ 𝑤𝑋) → (((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹‘((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))) ∈ 𝑦 ↔ (𝐹𝑤) ∈ 𝑦))
132102, 131sylibd 238 . . . . . . . . . . . . . . . . . . 19 ((((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) ∧ 𝑤𝑋) → (((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧 → (𝐹𝑤) ∈ 𝑦))
133132ralrimiva 3136 . . . . . . . . . . . . . . . . . 18 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → ∀𝑤𝑋 (((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧 → (𝐹𝑤) ∈ 𝑦))
134 fveq2 6901 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝑤 → (𝐹𝑣) = (𝐹𝑤))
135134eleq1d 2811 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝑤 → ((𝐹𝑣) ∈ 𝑦 ↔ (𝐹𝑤) ∈ 𝑦))
136135ralrab2 3692 . . . . . . . . . . . . . . . . . 18 (∀𝑣 ∈ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} (𝐹𝑣) ∈ 𝑦 ↔ ∀𝑤𝑋 (((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧 → (𝐹𝑤) ∈ 𝑦))
137133, 136sylibr 233 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → ∀𝑣 ∈ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} (𝐹𝑣) ∈ 𝑦)
13822adantr 479 . . . . . . . . . . . . . . . . . . 19 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → 𝐹:𝑋⟶(Base‘𝐻))
139138ffund 6732 . . . . . . . . . . . . . . . . . 18 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → Fun 𝐹)
140 ssrab2 4076 . . . . . . . . . . . . . . . . . . 19 {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} ⊆ 𝑋
141138fdmd 6738 . . . . . . . . . . . . . . . . . . 19 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → dom 𝐹 = 𝑋)
142140, 141sseqtrrid 4033 . . . . . . . . . . . . . . . . . 18 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} ⊆ dom 𝐹)
143 funimass4 6967 . . . . . . . . . . . . . . . . . 18 ((Fun 𝐹 ∧ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} ⊆ dom 𝐹) → ((𝐹 “ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧}) ⊆ 𝑦 ↔ ∀𝑣 ∈ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} (𝐹𝑣) ∈ 𝑦))
144139, 142, 143syl2anc 582 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → ((𝐹 “ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧}) ⊆ 𝑦 ↔ ∀𝑣 ∈ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} (𝐹𝑣) ∈ 𝑦))
145137, 144mpbird 256 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → (𝐹 “ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧}) ⊆ 𝑦)
146 eleq2 2815 . . . . . . . . . . . . . . . . . 18 (𝑢 = {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} → (𝑥𝑢𝑥 ∈ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧}))
147 imaeq2 6065 . . . . . . . . . . . . . . . . . . 19 (𝑢 = {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} → (𝐹𝑢) = (𝐹 “ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧}))
148147sseq1d 4011 . . . . . . . . . . . . . . . . . 18 (𝑢 = {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} → ((𝐹𝑢) ⊆ 𝑦 ↔ (𝐹 “ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧}) ⊆ 𝑦))
149146, 148anbi12d 630 . . . . . . . . . . . . . . . . 17 (𝑢 = {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} → ((𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦) ↔ (𝑥 ∈ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} ∧ (𝐹 “ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧}) ⊆ 𝑦)))
150149rspcev 3608 . . . . . . . . . . . . . . . 16 (({𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} ∈ 𝐽 ∧ (𝑥 ∈ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} ∧ (𝐹 “ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧}) ⊆ 𝑦)) → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦))
15188, 95, 145, 150syl12anc 835 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦))
152151expr 455 . . . . . . . . . . . . . 14 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑧𝐽) → ((𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦) → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦)))
15372, 152sylan2d 603 . . . . . . . . . . . . 13 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑧𝐽) → ((𝐴𝑧 ∧ (𝐹𝑧) ⊆ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦}) → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦)))
154153rexlimdva 3145 . . . . . . . . . . . 12 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (∃𝑧𝐽 (𝐴𝑧 ∧ (𝐹𝑧) ⊆ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦}) → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦)))
15560, 154mpd 15 . . . . . . . . . . 11 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦))
156155anassrs 466 . . . . . . . . . 10 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦)) → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦))
157156expr 455 . . . . . . . . 9 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) ∧ 𝑦𝐾) → ((𝐹𝑥) ∈ 𝑦 → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦)))
158157ralrimiva 3136 . . . . . . . 8 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → ∀𝑦𝐾 ((𝐹𝑥) ∈ 𝑦 → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦)))
1598adantr 479 . . . . . . . . 9 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → 𝐽 ∈ (TopOn‘𝑋))
16013adantr 479 . . . . . . . . 9 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → 𝐾 ∈ (TopOn‘(Base‘𝐻)))
161 simpr 483 . . . . . . . . 9 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → 𝑥𝑋)
162 iscnp 23232 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘(Base‘𝐻)) ∧ 𝑥𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ (𝐹:𝑋⟶(Base‘𝐻) ∧ ∀𝑦𝐾 ((𝐹𝑥) ∈ 𝑦 → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦)))))
163159, 160, 161, 162syl3anc 1368 . . . . . . . 8 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ (𝐹:𝑋⟶(Base‘𝐻) ∧ ∀𝑦𝐾 ((𝐹𝑥) ∈ 𝑦 → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦)))))
16417, 158, 163mpbir2and 711 . . . . . . 7 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
165164ralrimiva 3136 . . . . . 6 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
166 cncnp 23275 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘(Base‘𝐻))) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶(Base‘𝐻) ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))
1678, 13, 166syl2anc 582 . . . . . 6 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶(Base‘𝐻) ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))
16816, 165, 167mpbir2and 711 . . . . 5 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹 ∈ (𝐽 Cn 𝐾))
169168ex 411 . . . 4 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐹 ∈ (𝐽 Cn 𝐾)))
1703, 169jcad 511 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → (𝐴 𝐽𝐹 ∈ (𝐽 Cn 𝐾))))
1711cncnpi 23273 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 𝐽) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
172171ancoms 457 . . 3 ((𝐴 𝐽𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
173170, 172impbid1 224 . 2 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐴 𝐽𝐹 ∈ (𝐽 Cn 𝐾))))
1747, 28syl 17 . . . 4 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → 𝑋 = 𝐽)
175174eleq2d 2812 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐴𝑋𝐴 𝐽))
176175anbi1d 629 . 2 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → ((𝐴𝑋𝐹 ∈ (𝐽 Cn 𝐾)) ↔ (𝐴 𝐽𝐹 ∈ (𝐽 Cn 𝐾))))
177173, 176bitr4d 281 1 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐴𝑋𝐹 ∈ (𝐽 Cn 𝐾))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  wral 3051  wrex 3060  {crab 3419  wss 3947   cuni 4913  cmpt 5236  ccnv 5681  dom cdm 5682  cima 5685  Fun wfun 6548   Fn wfn 6549  wf 6550  cfv 6554  (class class class)co 7424  Basecbs 17213  +gcplusg 17266  TopOpenctopn 17436  0gc0g 17454  Grpcgrp 18928  invgcminusg 18929  -gcsg 18930   GrpHom cghm 19206  TopOnctopon 22903   Cn ccn 23219   CnP ccnp 23220  TopMndctmd 24065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-map 8857  df-0g 17456  df-topgen 17458  df-plusf 18632  df-mgm 18633  df-sgrp 18712  df-mnd 18728  df-grp 18931  df-minusg 18932  df-sbg 18933  df-ghm 19207  df-top 22887  df-topon 22904  df-topsp 22926  df-bases 22940  df-cn 23222  df-cnp 23223  df-tx 23557  df-tmd 24067
This theorem is referenced by:  qqhcn  33806
  Copyright terms: Public domain W3C validator