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Theorem ghmcnp 22720
 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 2798 . . . . . 6 𝐽 = 𝐽
21cnprcl 21850 . . . . 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 22686 . . . . . . . . 9 (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝑋))
763ad2ant1 1130 . . . . . . . 8 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → 𝐽 ∈ (TopOn‘𝑋))
87adantr 484 . . . . . . 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 2798 . . . . . . . . 9 (Base‘𝐻) = (Base‘𝐻)
1210, 11tmdtopon 22686 . . . . . . . 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 488 . . . . . . 7 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
15 cnpf2 21855 . . . . . . 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 484 . . . . . . . 8 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → 𝐹:𝑋⟶(Base‘𝐻))
1814adantr 484 . . . . . . . . . . . . 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 2798 . . . . . . . . . . . . . . 15 (𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤)) = (𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤))
2019mptpreima 6059 . . . . . . . . . . . . . 14 ((𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤)) “ 𝑦) = {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦}
219adantr 484 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝐻 ∈ TopMnd)
2216adantr 484 . . . . . . . . . . . . . . . . 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 18352 . . . . . . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . . . . 18 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝑥𝑋)
272adantl 485 . . . . . . . . . . . . . . . . . . . 20 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 𝐽)
28 toponuni 21519 . . . . . . . . . . . . . . . . . . . . 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 2893 . . . . . . . . . . . . . . . . . . 19 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴𝑋)
3130adantr 484 . . . . . . . . . . . . . . . . . 18 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝐴𝑋)
32 eqid 2798 . . . . . . . . . . . . . . . . . . 19 (-g𝐺) = (-g𝐺)
335, 32grpsubcl 18171 . . . . . . . . . . . . . . . . . 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, 34ffvelrnd 6829 . . . . . . . . . . . . . . . 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 2798 . . . . . . . . . . . . . . . . 17 (+g𝐻) = (+g𝐻)
3719, 11, 36, 10tmdlactcn 22707 . . . . . . . . . . . . . . . 16 ((𝐻 ∈ TopMnd ∧ (𝐹‘(𝑥(-g𝐺)𝐴)) ∈ (Base‘𝐻)) → (𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤)) ∈ (𝐾 Cn 𝐾))
3821, 35, 37syl2anc 587 . . . . . . . . . . . . . . 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 780 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝑦𝐾)
40 cnima 21870 . . . . . . . . . . . . . . 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 587 . . . . . . . . . . . . . 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 2895 . . . . . . . . . . . . 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 7143 . . . . . . . . . . . . . . 15 (𝑤 = (𝐹𝐴) → ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) = ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝐴)))
4443eleq1d 2874 . . . . . . . . . . . . . 14 (𝑤 = (𝐹𝐴) → (((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦 ↔ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝐴)) ∈ 𝑦))
4522, 31ffvelrnd 6829 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (𝐹𝐴) ∈ (Base‘𝐻))
46 eqid 2798 . . . . . . . . . . . . . . . . . . 19 (-g𝐻) = (-g𝐻)
475, 32, 46ghmsub 18358 . . . . . . . . . . . . . . . . . 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 7150 . . . . . . . . . . . . . . . 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 18353 . . . . . . . . . . . . . . . . . 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, 26ffvelrnd 6829 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (𝐹𝑥) ∈ (Base‘𝐻))
5311, 36, 46grpnpcan 18183 . . . . . . . . . . . . . . . . 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 2833 . . . . . . . . . . . . . . 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 781 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (𝐹𝑥) ∈ 𝑦)
5755, 56eqeltrd 2890 . . . . . . . . . . . . . 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 3630 . . . . . . . . . . . . 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 21861 . . . . . . . . . . . . 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 4000 . . . . . . . . . . . . . . . 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 500 . . . . . . . . . . . . . . 15 ((𝐹𝑧) ⊆ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦} → ∀𝑤 ∈ (𝐹𝑧)((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦)
6322adantr 484 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑧𝐽) → 𝐹:𝑋⟶(Base‘𝐻))
6463ffnd 6488 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑧𝐽) → 𝐹 Fn 𝑋)
658adantr 484 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝐽 ∈ (TopOn‘𝑋))
66 toponss 21532 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → 𝑧𝑋)
6765, 66sylan 583 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑧𝐽) → 𝑧𝑋)
68 oveq2 7143 . . . . . . . . . . . . . . . . . 18 (𝑤 = (𝐹𝑣) → ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) = ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)))
6968eleq1d 2874 . . . . . . . . . . . . . . . . 17 (𝑤 = (𝐹𝑣) → (((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦 ↔ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))
7069ralima 6978 . . . . . . . . . . . . . . . 16 ((𝐹 Fn 𝑋𝑧𝑋) → (∀𝑤 ∈ (𝐹𝑧)((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦 ↔ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))
7164, 67, 70syl2anc 587 . . . . . . . . . . . . . . 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, 71syl5ib 247 . . . . . . . . . . . . . 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 2798 . . . . . . . . . . . . . . . . . 18 (𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)) = (𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))
7473mptpreima 6059 . . . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . . 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 18171 . . . . . . . . . . . . . . . . . . . 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 2798 . . . . . . . . . . . . . . . . . . . 20 (+g𝐺) = (+g𝐺)
8373, 5, 82, 4tmdlactcn 22707 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ TopMnd ∧ (𝐴(-g𝐺)𝑥) ∈ 𝑋) → (𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)) ∈ (𝐽 Cn 𝐽))
8476, 81, 83syl2anc 587 . . . . . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . . . . 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 21870 . . . . . . . . . . . . . . . . . 18 (((𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)) ∈ (𝐽 Cn 𝐽) ∧ 𝑧𝐽) → ((𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)) “ 𝑧) ∈ 𝐽)
8784, 85, 86syl2anc 587 . . . . . . . . . . . . . . . . 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 2895 . . . . . . . . . . . . . . . 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 7143 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑥 → ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) = ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑥))
9089eleq1d 2874 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑥 → (((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧 ↔ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑥) ∈ 𝑧))
915, 82, 32grpnpcan 18183 . . . . . . . . . . . . . . . . . . 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 780 . . . . . . . . . . . . . . . . . 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 2890 . . . . . . . . . . . . . . . . 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 3630 . . . . . . . . . . . . . . . 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 781 . . . . . . . . . . . . . . . . . . . . . 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 6645 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑣 = ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) → (𝐹𝑣) = (𝐹‘((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)))
9897oveq2d 7151 . . . . . . . . . . . . . . . . . . . . . . . 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 2874 . . . . . . . . . . . . . . . . . . . . . . 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 3568 . . . . . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
10434adantr 484 . . . . . . . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → 𝐴𝑋)
10726adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → 𝑤𝑋)
1105, 82grpcl 18103 . . . . . . . . . . . . . . . . . . . . . . . . 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 18355 . . . . . . . . . . . . . . . . . . . . . . . 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 2798 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (invg𝐺) = (invg𝐺)
1155, 32, 114grpinvsub 18173 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7151 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2798 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (0g𝐺) = (0g𝐺)
1195, 82, 118, 114grprinv 18145 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐺 ∈ Grp ∧ (𝑥(-g𝐺)𝐴) ∈ 𝑋) → ((𝑥(-g𝐺)𝐴)(+g𝐺)((invg𝐺)‘(𝑥(-g𝐺)𝐴))) = (0g𝐺))
120105, 104, 119syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2835 . . . . . . . . . . . . . . . . . . . . . . . . . 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 7150 . . . . . . . . . . . . . . . . . . . . . . . . 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 18104 . . . . . . . . . . . . . . . . . . . . . . . . . 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 18125 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐺 ∈ Grp ∧ 𝑤𝑋) → ((0g𝐺)(+g𝐺)𝑤) = 𝑤)
126105, 109, 125syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . 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 2841 . . . . . . . . . . . . . . . . . . . . . . . 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 6649 . . . . . . . . . . . . . . . . . . . . . . 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 2835 . . . . . . . . . . . . . . . . . . . . . 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 714 . . . . . . . . . . . . . . . . . . . . 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 2874 . . . . . . . . . . . . . . . . . . . 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 242 . . . . . . . . . . . . . . . . . . 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 3149 . . . . . . . . . . . . . . . . . 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 6645 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝑤 → (𝐹𝑣) = (𝐹𝑤))
135134eleq1d 2874 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝑤 → ((𝐹𝑣) ∈ 𝑦 ↔ (𝐹𝑤) ∈ 𝑦))
136135ralrab2 3638 . . . . . . . . . . . . . . . . . 18 (∀𝑣 ∈ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} (𝐹𝑣) ∈ 𝑦 ↔ ∀𝑤𝑋 (((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧 → (𝐹𝑤) ∈ 𝑦))
137133, 136sylibr 237 . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . 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 6491 . . . . . . . . . . . . . . . . . 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 4007 . . . . . . . . . . . . . . . . . . 19 {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} ⊆ 𝑋
141138fdmd 6497 . . . . . . . . . . . . . . . . . . 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 3968 . . . . . . . . . . . . . . . . . 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 6705 . . . . . . . . . . . . . . . . . 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 587 . . . . . . . . . . . . . . . . 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 260 . . . . . . . . . . . . . . . 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 2878 . . . . . . . . . . . . . . . . . 18 (𝑢 = {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} → (𝑥𝑢𝑥 ∈ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧}))
147 imaeq2 5892 . . . . . . . . . . . . . . . . . . 19 (𝑢 = {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} → (𝐹𝑢) = (𝐹 “ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧}))
148147sseq1d 3946 . . . . . . . . . . . . . . . . . 18 (𝑢 = {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} → ((𝐹𝑢) ⊆ 𝑦 ↔ (𝐹 “ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧}) ⊆ 𝑦))
149146, 148anbi12d 633 . . . . . . . . . . . . . . . . 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 3571 . . . . . . . . . . . . . . . 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 460 . . . . . . . . . . . . . 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 607 . . . . . . . . . . . . 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 3243 . . . . . . . . . . . 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 471 . . . . . . . . . 10 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦)) → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦))
157156expr 460 . . . . . . . . 9 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) ∧ 𝑦𝐾) → ((𝐹𝑥) ∈ 𝑦 → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦)))
158157ralrimiva 3149 . . . . . . . 8 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → ∀𝑦𝐾 ((𝐹𝑥) ∈ 𝑦 → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦)))
1598adantr 484 . . . . . . . . 9 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → 𝐽 ∈ (TopOn‘𝑋))
16013adantr 484 . . . . . . . . 9 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → 𝐾 ∈ (TopOn‘(Base‘𝐻)))
161 simpr 488 . . . . . . . . 9 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → 𝑥𝑋)
162 iscnp 21842 . . . . . . . . 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 712 . . . . . . 7 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
165164ralrimiva 3149 . . . . . 6 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
166 cncnp 21885 . . . . . . 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 587 . . . . . 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 712 . . . . 5 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹 ∈ (𝐽 Cn 𝐾))
169168ex 416 . . . 4 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐹 ∈ (𝐽 Cn 𝐾)))
1703, 169jcad 516 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → (𝐴 𝐽𝐹 ∈ (𝐽 Cn 𝐾))))
1711cncnpi 21883 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 𝐽) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
172171ancoms 462 . . 3 ((𝐴 𝐽𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
173170, 172impbid1 228 . 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 2875 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐴𝑋𝐴 𝐽))
176175anbi1d 632 . 2 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → ((𝐴𝑋𝐹 ∈ (𝐽 Cn 𝐾)) ↔ (𝐴 𝐽𝐹 ∈ (𝐽 Cn 𝐾))))
177173, 176bitr4d 285 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  {crab 3110   ⊆ wss 3881  ∪ cuni 4800   ↦ cmpt 5110  ◡ccnv 5518  dom cdm 5519   “ cima 5522  Fun wfun 6318   Fn wfn 6319  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  +gcplusg 16557  TopOpenctopn 16687  0gc0g 16705  Grpcgrp 18095  invgcminusg 18096  -gcsg 18097   GrpHom cghm 18347  TopOnctopon 21515   Cn ccn 21829   CnP ccnp 21830  TopMndctmd 22675 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441 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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-map 8391  df-0g 16707  df-topgen 16709  df-plusf 17843  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098  df-minusg 18099  df-sbg 18100  df-ghm 18348  df-top 21499  df-topon 21516  df-topsp 21538  df-bases 21551  df-cn 21832  df-cnp 21833  df-tx 22167  df-tmd 22677 This theorem is referenced by:  qqhcn  31342
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