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Theorem ghmcnp 22127
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 2805 . . . . . 6 𝐽 = 𝐽
21cnprcl 21259 . . . . 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 22094 . . . . . . . . 9 (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝑋))
763ad2ant1 1156 . . . . . . . 8 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → 𝐽 ∈ (TopOn‘𝑋))
87adantr 468 . . . . . . 7 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐽 ∈ (TopOn‘𝑋))
9 simpl2 1237 . . . . . . . 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 2805 . . . . . . . . 9 (Base‘𝐻) = (Base‘𝐻)
1210, 11tmdtopon 22094 . . . . . . . 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 473 . . . . . . 7 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
15 cnpf2 21264 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘(Base‘𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶(Base‘𝐻))
168, 13, 14, 15syl3anc 1483 . . . . . 6 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶(Base‘𝐻))
1716adantr 468 . . . . . . . 8 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → 𝐹:𝑋⟶(Base‘𝐻))
1814adantr 468 . . . . . . . . . . . . 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 2805 . . . . . . . . . . . . . . 15 (𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤)) = (𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤))
2019mptpreima 5839 . . . . . . . . . . . . . 14 ((𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤)) “ 𝑦) = {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦}
219adantr 468 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝐻 ∈ TopMnd)
2216adantr 468 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝐹:𝑋⟶(Base‘𝐻))
23 simpll3 1266 . . . . . . . . . . . . . . . . . . 19 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
24 ghmgrp1 17860 . . . . . . . . . . . . . . . . . . 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 778 . . . . . . . . . . . . . . . . . 18 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝑥𝑋)
272adantl 469 . . . . . . . . . . . . . . . . . . . 20 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 𝐽)
28 toponuni 20928 . . . . . . . . . . . . . . . . . . . . 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 2887 . . . . . . . . . . . . . . . . . . 19 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴𝑋)
3130adantr 468 . . . . . . . . . . . . . . . . . 18 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝐴𝑋)
32 eqid 2805 . . . . . . . . . . . . . . . . . . 19 (-g𝐺) = (-g𝐺)
335, 32grpsubcl 17696 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ Grp ∧ 𝑥𝑋𝐴𝑋) → (𝑥(-g𝐺)𝐴) ∈ 𝑋)
3425, 26, 31, 33syl3anc 1483 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (𝑥(-g𝐺)𝐴) ∈ 𝑋)
3522, 34ffvelrnd 6579 . . . . . . . . . . . . . . . 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 2805 . . . . . . . . . . . . . . . . 17 (+g𝐻) = (+g𝐻)
3719, 11, 36, 10tmdlactcn 22115 . . . . . . . . . . . . . . . 16 ((𝐻 ∈ TopMnd ∧ (𝐹‘(𝑥(-g𝐺)𝐴)) ∈ (Base‘𝐻)) → (𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤)) ∈ (𝐾 Cn 𝐾))
3821, 35, 37syl2anc 575 . . . . . . . . . . . . . . 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 790 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝑦𝐾)
40 cnima 21279 . . . . . . . . . . . . . . 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 575 . . . . . . . . . . . . . 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, 41syl5eqelr 2889 . . . . . . . . . . . . 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)
4322, 31ffvelrnd 6579 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (𝐹𝐴) ∈ (Base‘𝐻))
44 eqid 2805 . . . . . . . . . . . . . . . . . . 19 (-g𝐻) = (-g𝐻)
455, 32, 44ghmsub 17866 . . . . . . . . . . . . . . . . . 18 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥𝑋𝐴𝑋) → (𝐹‘(𝑥(-g𝐺)𝐴)) = ((𝐹𝑥)(-g𝐻)(𝐹𝐴)))
4623, 26, 31, 45syl3anc 1483 . . . . . . . . . . . . . . . . 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)))
4746oveq1d 6886 . . . . . . . . . . . . . . . 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)))
48 ghmgrp2 17861 . . . . . . . . . . . . . . . . . 18 (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐻 ∈ Grp)
4923, 48syl 17 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝐻 ∈ Grp)
5022, 26ffvelrnd 6579 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (𝐹𝑥) ∈ (Base‘𝐻))
5111, 36, 44grpnpcan 17708 . . . . . . . . . . . . . . . . 17 ((𝐻 ∈ Grp ∧ (𝐹𝑥) ∈ (Base‘𝐻) ∧ (𝐹𝐴) ∈ (Base‘𝐻)) → (((𝐹𝑥)(-g𝐻)(𝐹𝐴))(+g𝐻)(𝐹𝐴)) = (𝐹𝑥))
5249, 50, 43, 51syl3anc 1483 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (((𝐹𝑥)(-g𝐻)(𝐹𝐴))(+g𝐻)(𝐹𝐴)) = (𝐹𝑥))
5347, 52eqtrd 2839 . . . . . . . . . . . . . . 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))
54 simprrr 791 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (𝐹𝑥) ∈ 𝑦)
5553, 54eqeltrd 2884 . . . . . . . . . . . . . 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)
56 oveq2 6879 . . . . . . . . . . . . . . . 16 (𝑤 = (𝐹𝐴) → ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) = ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝐴)))
5756eleq1d 2869 . . . . . . . . . . . . . . 15 (𝑤 = (𝐹𝐴) → (((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦 ↔ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝐴)) ∈ 𝑦))
5857elrab 3558 . . . . . . . . . . . . . 14 ((𝐹𝐴) ∈ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦} ↔ ((𝐹𝐴) ∈ (Base‘𝐻) ∧ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝐴)) ∈ 𝑦))
5943, 55, 58sylanbrc 574 . . . . . . . . . . . . 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})
60 cnpimaex 21270 . . . . . . . . . . . . 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}))
6118, 42, 59, 60syl3anc 1483 . . . . . . . . . . . 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}))
62 ssrab 3874 . . . . . . . . . . . . . . . 16 ((𝐹𝑧) ⊆ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦} ↔ ((𝐹𝑧) ⊆ (Base‘𝐻) ∧ ∀𝑤 ∈ (𝐹𝑧)((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦))
6362simprbi 486 . . . . . . . . . . . . . . 15 ((𝐹𝑧) ⊆ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦} → ∀𝑤 ∈ (𝐹𝑧)((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦)
6422adantr 468 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑧𝐽) → 𝐹:𝑋⟶(Base‘𝐻))
6564ffnd 6254 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑧𝐽) → 𝐹 Fn 𝑋)
668adantr 468 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝐽 ∈ (TopOn‘𝑋))
67 toponss 20941 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → 𝑧𝑋)
6866, 67sylan 571 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑧𝐽) → 𝑧𝑋)
69 oveq2 6879 . . . . . . . . . . . . . . . . . 18 (𝑤 = (𝐹𝑣) → ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) = ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)))
7069eleq1d 2869 . . . . . . . . . . . . . . . . 17 (𝑤 = (𝐹𝑣) → (((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦 ↔ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))
7170ralima 6720 . . . . . . . . . . . . . . . 16 ((𝐹 Fn 𝑋𝑧𝑋) → (∀𝑤 ∈ (𝐹𝑧)((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦 ↔ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))
7265, 68, 71syl2anc 575 . . . . . . . . . . . . . . 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))
7363, 72syl5ib 235 . . . . . . . . . . . . . 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))
74 eqid 2805 . . . . . . . . . . . . . . . . . 18 (𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)) = (𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))
7574mptpreima 5839 . . . . . . . . . . . . . . . . 17 ((𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)) “ 𝑧) = {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧}
76 simpl1 1235 . . . . . . . . . . . . . . . . . . . 20 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐺 ∈ TopMnd)
7776ad2antrr 708 . . . . . . . . . . . . . . . . . . 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)
7825adantr 468 . . . . . . . . . . . . . . . . . . . 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)
7931adantr 468 . . . . . . . . . . . . . . . . . . . 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)
8026adantr 468 . . . . . . . . . . . . . . . . . . . 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)
815, 32grpsubcl 17696 . . . . . . . . . . . . . . . . . . . 20 ((𝐺 ∈ Grp ∧ 𝐴𝑋𝑥𝑋) → (𝐴(-g𝐺)𝑥) ∈ 𝑋)
8278, 79, 80, 81syl3anc 1483 . . . . . . . . . . . . . . . . . . 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)
83 eqid 2805 . . . . . . . . . . . . . . . . . . . 20 (+g𝐺) = (+g𝐺)
8474, 5, 83, 4tmdlactcn 22115 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ TopMnd ∧ (𝐴(-g𝐺)𝑥) ∈ 𝑋) → (𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)) ∈ (𝐽 Cn 𝐽))
8577, 82, 84syl2anc 575 . . . . . . . . . . . . . . . . . 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))
86 simprl 778 . . . . . . . . . . . . . . . . . 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)
87 cnima 21279 . . . . . . . . . . . . . . . . . 18 (((𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)) ∈ (𝐽 Cn 𝐽) ∧ 𝑧𝐽) → ((𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)) “ 𝑧) ∈ 𝐽)
8885, 86, 87syl2anc 575 . . . . . . . . . . . . . . . . 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)
8975, 88syl5eqelr 2889 . . . . . . . . . . . . . . . 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)
905, 83, 32grpnpcan 17708 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ Grp ∧ 𝐴𝑋𝑥𝑋) → ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑥) = 𝐴)
9178, 79, 80, 90syl3anc 1483 . . . . . . . . . . . . . . . . . 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)
92 simprrl 790 . . . . . . . . . . . . . . . . . 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)
9391, 92eqeltrd 2884 . . . . . . . . . . . . . . . . 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)
94 oveq2 6879 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑥 → ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) = ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑥))
9594eleq1d 2869 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑥 → (((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧 ↔ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑥) ∈ 𝑧))
9695elrab 3558 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} ↔ (𝑥𝑋 ∧ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑥) ∈ 𝑧))
9780, 93, 96sylanbrc 574 . . . . . . . . . . . . . . . 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})
98 simprrr 791 . . . . . . . . . . . . . . . . . . . . . 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)
99 fveq2 6405 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑣 = ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) → (𝐹𝑣) = (𝐹‘((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)))
10099oveq2d 6887 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 = ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) → ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) = ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹‘((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))))
101100eleq1d 2869 . . . . . . . . . . . . . . . . . . . . . . 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))
102101rspccv 3498 . . . . . . . . . . . . . . . . . . . . . 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))
10398, 102syl 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))
104103adantr 468 . . . . . . . . . . . . . . . . . . . 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))
10523adantr 468 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
10634adantr 468 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → (𝑥(-g𝐺)𝐴) ∈ 𝑋)
107105, 24syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → 𝐺 ∈ Grp)
10831adantr 468 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → 𝐴𝑋)
10926adantr 468 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → 𝑥𝑋)
110107, 108, 109, 81syl3anc 1483 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → (𝐴(-g𝐺)𝑥) ∈ 𝑋)
111 simpr 473 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → 𝑤𝑋)
1125, 83grpcl 17631 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐺 ∈ Grp ∧ (𝐴(-g𝐺)𝑥) ∈ 𝑋𝑤𝑋) → ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑋)
113107, 110, 111, 112syl3anc 1483 . . . . . . . . . . . . . . . . . . . . . . . 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)
1145, 83, 36ghmlin 17863 . . . . . . . . . . . . . . . . . . . . . . . 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))))
115105, 106, 113, 114syl3anc 1483 . . . . . . . . . . . . . . . . . . . . . . 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))))
116 eqid 2805 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (invg𝐺) = (invg𝐺)
1175, 32, 116grpinvsub 17698 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐺 ∈ Grp ∧ 𝑥𝑋𝐴𝑋) → ((invg𝐺)‘(𝑥(-g𝐺)𝐴)) = (𝐴(-g𝐺)𝑥))
118107, 109, 108, 117syl3anc 1483 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
119118oveq2d 6887 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
120 eqid 2805 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (0g𝐺) = (0g𝐺)
1215, 83, 120, 116grprinv 17670 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐺 ∈ Grp ∧ (𝑥(-g𝐺)𝐴) ∈ 𝑋) → ((𝑥(-g𝐺)𝐴)(+g𝐺)((invg𝐺)‘(𝑥(-g𝐺)𝐴))) = (0g𝐺))
122107, 106, 121syl2anc 575 . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
123119, 122eqtr3d 2841 . . . . . . . . . . . . . . . . . . . . . . . . . 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))
124123oveq1d 6886 . . . . . . . . . . . . . . . . . . . . . . . . 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))
1255, 83grpass 17632 . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
126107, 106, 110, 111, 125syl13anc 1484 . . . . . . . . . . . . . . . . . . . . . . . . 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)))
1275, 83, 120grplid 17653 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐺 ∈ Grp ∧ 𝑤𝑋) → ((0g𝐺)(+g𝐺)𝑤) = 𝑤)
128107, 111, 127syl2anc 575 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → ((0g𝐺)(+g𝐺)𝑤) = 𝑤)
129124, 126, 1283eqtr3d 2847 . . . . . . . . . . . . . . . . . . . . . . . 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)
130129fveq2d 6409 . . . . . . . . . . . . . . . . . . . . . . 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))
131115, 130eqtr3d 2841 . . . . . . . . . . . . . . . . . . . . . 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))
132131adantlr 697 . . . . . . . . . . . . . . . . . . . . 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))
133132eleq1d 2869 . . . . . . . . . . . . . . . . . . . 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))
134104, 133sylibd 230 . . . . . . . . . . . . . . . . . . 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))
135134ralrimiva 3153 . . . . . . . . . . . . . . . . . 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))
136 fveq2 6405 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝑤 → (𝐹𝑣) = (𝐹𝑤))
137136eleq1d 2869 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝑤 → ((𝐹𝑣) ∈ 𝑦 ↔ (𝐹𝑤) ∈ 𝑦))
138137ralrab2 3567 . . . . . . . . . . . . . . . . . 18 (∀𝑣 ∈ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} (𝐹𝑣) ∈ 𝑦 ↔ ∀𝑤𝑋 (((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧 → (𝐹𝑤) ∈ 𝑦))
139135, 138sylibr 225 . . . . . . . . . . . . . . . . 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)
14022adantr 468 . . . . . . . . . . . . . . . . . . 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))
141140ffund 6257 . . . . . . . . . . . . . . . . . 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)
142 ssrab2 3881 . . . . . . . . . . . . . . . . . . 19 {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} ⊆ 𝑋
143140fdmd 6262 . . . . . . . . . . . . . . . . . . 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)
144142, 143syl5sseqr 3848 . . . . . . . . . . . . . . . . . 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)
145 funimass4 6465 . . . . . . . . . . . . . . . . . 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))
146141, 144, 145syl2anc 575 . . . . . . . . . . . . . . . . 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))
147139, 146mpbird 248 . . . . . . . . . . . . . . . 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)
148 eleq2 2873 . . . . . . . . . . . . . . . . . 18 (𝑢 = {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} → (𝑥𝑢𝑥 ∈ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧}))
149 imaeq2 5669 . . . . . . . . . . . . . . . . . . 19 (𝑢 = {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} → (𝐹𝑢) = (𝐹 “ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧}))
150149sseq1d 3826 . . . . . . . . . . . . . . . . . 18 (𝑢 = {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} → ((𝐹𝑢) ⊆ 𝑦 ↔ (𝐹 “ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧}) ⊆ 𝑦))
151148, 150anbi12d 618 . . . . . . . . . . . . . . . . 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)))
152151rspcev 3501 . . . . . . . . . . . . . . . 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))
15389, 97, 147, 152syl12anc 856 . . . . . . . . . . . . . . 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))
154153expr 446 . . . . . . . . . . . . . 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)))
15573, 154sylan2d 594 . . . . . . . . . . . . 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)))
156155rexlimdva 3218 . . . . . . . . . . . 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)))
15761, 156mpd 15 . . . . . . . . . . 11 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦))
158157anassrs 455 . . . . . . . . . 10 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦)) → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦))
159158expr 446 . . . . . . . . 9 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) ∧ 𝑦𝐾) → ((𝐹𝑥) ∈ 𝑦 → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦)))
160159ralrimiva 3153 . . . . . . . 8 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → ∀𝑦𝐾 ((𝐹𝑥) ∈ 𝑦 → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦)))
1618adantr 468 . . . . . . . . 9 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → 𝐽 ∈ (TopOn‘𝑋))
16213adantr 468 . . . . . . . . 9 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → 𝐾 ∈ (TopOn‘(Base‘𝐻)))
163 simpr 473 . . . . . . . . 9 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → 𝑥𝑋)
164 iscnp 21251 . . . . . . . . 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)))))
165161, 162, 163, 164syl3anc 1483 . . . . . . . 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)))))
16617, 160, 165mpbir2and 695 . . . . . . 7 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
167166ralrimiva 3153 . . . . . 6 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
168 cncnp 21294 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘(Base‘𝐻))) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶(Base‘𝐻) ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))
1698, 13, 168syl2anc 575 . . . . . 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))))
17016, 167, 169mpbir2and 695 . . . . 5 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹 ∈ (𝐽 Cn 𝐾))
171170ex 399 . . . 4 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐹 ∈ (𝐽 Cn 𝐾)))
1723, 171jcad 504 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → (𝐴 𝐽𝐹 ∈ (𝐽 Cn 𝐾))))
1731cncnpi 21292 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 𝐽) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
174173ancoms 448 . . 3 ((𝐴 𝐽𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
175172, 174impbid1 216 . 2 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐴 𝐽𝐹 ∈ (𝐽 Cn 𝐾))))
1767, 28syl 17 . . . 4 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → 𝑋 = 𝐽)
177176eleq2d 2870 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐴𝑋𝐴 𝐽))
178177anbi1d 617 . 2 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → ((𝐴𝑋𝐹 ∈ (𝐽 Cn 𝐾)) ↔ (𝐴 𝐽𝐹 ∈ (𝐽 Cn 𝐾))))
179175, 178bitr4d 273 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 197  wa 384  w3a 1100   = wceq 1637  wcel 2158  wral 3095  wrex 3096  {crab 3099  wss 3766   cuni 4626  cmpt 4919  ccnv 5307  dom cdm 5308  cima 5311  Fun wfun 6092   Fn wfn 6093  wf 6094  cfv 6098  (class class class)co 6871  Basecbs 16064  +gcplusg 16149  TopOpenctopn 16283  0gc0g 16301  Grpcgrp 17623  invgcminusg 17624  -gcsg 17625   GrpHom cghm 17855  TopOnctopon 20924   Cn ccn 21238   CnP ccnp 21239  TopMndctmd 22083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-8 2160  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-rep 4960  ax-sep 4971  ax-nul 4980  ax-pow 5032  ax-pr 5093  ax-un 7176
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ne 2978  df-ral 3100  df-rex 3101  df-reu 3102  df-rmo 3103  df-rab 3104  df-v 3392  df-sbc 3631  df-csb 3726  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-nul 4114  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-op 4374  df-uni 4627  df-iun 4710  df-br 4841  df-opab 4903  df-mpt 4920  df-id 5216  df-xp 5314  df-rel 5315  df-cnv 5316  df-co 5317  df-dm 5318  df-rn 5319  df-res 5320  df-ima 5321  df-iota 6061  df-fun 6100  df-fn 6101  df-f 6102  df-f1 6103  df-fo 6104  df-f1o 6105  df-fv 6106  df-riota 6832  df-ov 6874  df-oprab 6875  df-mpt2 6876  df-1st 7395  df-2nd 7396  df-map 8091  df-0g 16303  df-topgen 16305  df-plusf 17442  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-grp 17626  df-minusg 17627  df-sbg 17628  df-ghm 17856  df-top 20908  df-topon 20925  df-topsp 20947  df-bases 20960  df-cn 21241  df-cnp 21242  df-tx 21575  df-tmd 22085
This theorem is referenced by:  qqhcn  30357
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