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Theorem ghmqusker 19318
Description: A surjective group homomorphism 𝐹 from 𝐺 to 𝐻 induces an isomorphism 𝐽 from 𝑄 to 𝐻, where 𝑄 is the factor group of 𝐺 by 𝐹's kernel 𝐾. (Contributed by Thierry Arnoux, 15-Feb-2025.)
Hypotheses
Ref Expression
ghmqusker.1 0 = (0g𝐻)
ghmqusker.f (𝜑𝐹 ∈ (𝐺 GrpHom 𝐻))
ghmqusker.k 𝐾 = (𝐹 “ { 0 })
ghmqusker.q 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))
ghmqusker.j 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ (𝐹𝑞))
ghmqusker.s (𝜑 → ran 𝐹 = (Base‘𝐻))
Assertion
Ref Expression
ghmqusker (𝜑𝐽 ∈ (𝑄 GrpIso 𝐻))
Distinct variable groups:   𝐹,𝑞   𝐺,𝑞   𝐻,𝑞   𝐽,𝑞   𝐾,𝑞   𝑄,𝑞   𝜑,𝑞
Allowed substitution hint:   0 (𝑞)

Proof of Theorem ghmqusker
Dummy variables 𝑟 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ghmqusker.1 . . 3 0 = (0g𝐻)
2 ghmqusker.f . . 3 (𝜑𝐹 ∈ (𝐺 GrpHom 𝐻))
3 ghmqusker.k . . 3 𝐾 = (𝐹 “ { 0 })
4 ghmqusker.q . . 3 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))
5 ghmqusker.j . . 3 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ (𝐹𝑞))
61, 2, 3, 4, 5ghmquskerlem3 19317 . 2 (𝜑𝐽 ∈ (𝑄 GrpHom 𝐻))
7 ghmgrp1 19249 . . . . . . . . . . . . . . . 16 (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp)
82, 7syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐺 ∈ Grp)
98ad4antr 732 . . . . . . . . . . . . . 14 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ (𝐽𝑟) = 0 ) → 𝐺 ∈ Grp)
101ghmker 19273 . . . . . . . . . . . . . . . . . 18 (𝐹 ∈ (𝐺 GrpHom 𝐻) → (𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
112, 10syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
123, 11eqeltrid 2843 . . . . . . . . . . . . . . . 16 (𝜑𝐾 ∈ (NrmSGrp‘𝐺))
13 nsgsubg 19189 . . . . . . . . . . . . . . . 16 (𝐾 ∈ (NrmSGrp‘𝐺) → 𝐾 ∈ (SubGrp‘𝐺))
1412, 13syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐾 ∈ (SubGrp‘𝐺))
1514ad4antr 732 . . . . . . . . . . . . . 14 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ (𝐽𝑟) = 0 ) → 𝐾 ∈ (SubGrp‘𝐺))
16 eqid 2735 . . . . . . . . . . . . . . . . . . . . 21 (Base‘𝐺) = (Base‘𝐺)
17 eqid 2735 . . . . . . . . . . . . . . . . . . . . 21 (Base‘𝐻) = (Base‘𝐻)
1816, 17ghmf 19251 . . . . . . . . . . . . . . . . . . . 20 (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
192, 18syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐹:(Base‘𝐺)⟶(Base‘𝐻))
2019ffnd 6738 . . . . . . . . . . . . . . . . . 18 (𝜑𝐹 Fn (Base‘𝐺))
2120ad3antrrr 730 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → 𝐹 Fn (Base‘𝐺))
2221adantr 480 . . . . . . . . . . . . . . . 16 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ (𝐽𝑟) = 0 ) → 𝐹 Fn (Base‘𝐺))
234a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑄 = (𝐺 /s (𝐺 ~QG 𝐾)))
24 eqidd 2736 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (Base‘𝐺) = (Base‘𝐺))
25 ovexd 7466 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝐺 ~QG 𝐾) ∈ V)
2623, 24, 25, 8qusbas 17592 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) = (Base‘𝑄))
27 eqid 2735 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐺 ~QG 𝐾) = (𝐺 ~QG 𝐾)
2816, 27eqger 19209 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐾 ∈ (SubGrp‘𝐺) → (𝐺 ~QG 𝐾) Er (Base‘𝐺))
2912, 13, 283syl 18 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝐺 ~QG 𝐾) Er (Base‘𝐺))
3029qsss 8817 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) ⊆ 𝒫 (Base‘𝐺))
3126, 30eqsstrrd 4035 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (Base‘𝑄) ⊆ 𝒫 (Base‘𝐺))
3231sselda 3995 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ 𝒫 (Base‘𝐺))
3332elpwid 4614 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑟 ∈ (Base‘𝑄)) → 𝑟 ⊆ (Base‘𝐺))
3433sselda 3995 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) → 𝑥 ∈ (Base‘𝐺))
3534adantr 480 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → 𝑥 ∈ (Base‘𝐺))
3635adantr 480 . . . . . . . . . . . . . . . 16 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ (𝐽𝑟) = 0 ) → 𝑥 ∈ (Base‘𝐺))
37 simpr 484 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → (𝐽𝑟) = (𝐹𝑥))
3837eqeq1d 2737 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → ((𝐽𝑟) = 0 ↔ (𝐹𝑥) = 0 ))
3938biimpa 476 . . . . . . . . . . . . . . . 16 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ (𝐽𝑟) = 0 ) → (𝐹𝑥) = 0 )
40 fniniseg 7080 . . . . . . . . . . . . . . . . 17 (𝐹 Fn (Base‘𝐺) → (𝑥 ∈ (𝐹 “ { 0 }) ↔ (𝑥 ∈ (Base‘𝐺) ∧ (𝐹𝑥) = 0 )))
4140biimpar 477 . . . . . . . . . . . . . . . 16 ((𝐹 Fn (Base‘𝐺) ∧ (𝑥 ∈ (Base‘𝐺) ∧ (𝐹𝑥) = 0 )) → 𝑥 ∈ (𝐹 “ { 0 }))
4222, 36, 39, 41syl12anc 837 . . . . . . . . . . . . . . 15 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ (𝐽𝑟) = 0 ) → 𝑥 ∈ (𝐹 “ { 0 }))
4342, 3eleqtrrdi 2850 . . . . . . . . . . . . . 14 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ (𝐽𝑟) = 0 ) → 𝑥𝐾)
4427eqg0el 19214 . . . . . . . . . . . . . . 15 ((𝐺 ∈ Grp ∧ 𝐾 ∈ (SubGrp‘𝐺)) → ([𝑥](𝐺 ~QG 𝐾) = 𝐾𝑥𝐾))
4544biimpar 477 . . . . . . . . . . . . . 14 (((𝐺 ∈ Grp ∧ 𝐾 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝐾) → [𝑥](𝐺 ~QG 𝐾) = 𝐾)
469, 15, 43, 45syl21anc 838 . . . . . . . . . . . . 13 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ (𝐽𝑟) = 0 ) → [𝑥](𝐺 ~QG 𝐾) = 𝐾)
4729ad4antr 732 . . . . . . . . . . . . . 14 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ (𝐽𝑟) = 0 ) → (𝐺 ~QG 𝐾) Er (Base‘𝐺))
48 simpr 484 . . . . . . . . . . . . . . . 16 ((𝜑𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ (Base‘𝑄))
4926adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑟 ∈ (Base‘𝑄)) → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) = (Base‘𝑄))
5048, 49eleqtrrd 2842 . . . . . . . . . . . . . . 15 ((𝜑𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)))
5150ad3antrrr 730 . . . . . . . . . . . . . 14 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ (𝐽𝑟) = 0 ) → 𝑟 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)))
52 simpllr 776 . . . . . . . . . . . . . 14 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ (𝐽𝑟) = 0 ) → 𝑥𝑟)
53 qsel 8835 . . . . . . . . . . . . . 14 (((𝐺 ~QG 𝐾) Er (Base‘𝐺) ∧ 𝑟 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)) ∧ 𝑥𝑟) → 𝑟 = [𝑥](𝐺 ~QG 𝐾))
5447, 51, 52, 53syl3anc 1370 . . . . . . . . . . . . 13 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ (𝐽𝑟) = 0 ) → 𝑟 = [𝑥](𝐺 ~QG 𝐾))
55 eqid 2735 . . . . . . . . . . . . . . . 16 (0g𝐺) = (0g𝐺)
5616, 27, 55eqgid 19211 . . . . . . . . . . . . . . 15 (𝐾 ∈ (SubGrp‘𝐺) → [(0g𝐺)](𝐺 ~QG 𝐾) = 𝐾)
5714, 56syl 17 . . . . . . . . . . . . . 14 (𝜑 → [(0g𝐺)](𝐺 ~QG 𝐾) = 𝐾)
5857ad4antr 732 . . . . . . . . . . . . 13 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ (𝐽𝑟) = 0 ) → [(0g𝐺)](𝐺 ~QG 𝐾) = 𝐾)
5946, 54, 583eqtr4d 2785 . . . . . . . . . . . 12 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ (𝐽𝑟) = 0 ) → 𝑟 = [(0g𝐺)](𝐺 ~QG 𝐾))
604, 55qus0 19220 . . . . . . . . . . . . . . 15 (𝐾 ∈ (NrmSGrp‘𝐺) → [(0g𝐺)](𝐺 ~QG 𝐾) = (0g𝑄))
6112, 60syl 17 . . . . . . . . . . . . . 14 (𝜑 → [(0g𝐺)](𝐺 ~QG 𝐾) = (0g𝑄))
6261ad3antrrr 730 . . . . . . . . . . . . 13 ((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → [(0g𝐺)](𝐺 ~QG 𝐾) = (0g𝑄))
6362adantr 480 . . . . . . . . . . . 12 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ (𝐽𝑟) = 0 ) → [(0g𝐺)](𝐺 ~QG 𝐾) = (0g𝑄))
6459, 63eqtrd 2775 . . . . . . . . . . 11 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ (𝐽𝑟) = 0 ) → 𝑟 = (0g𝑄))
6562eqeq2d 2746 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → (𝑟 = [(0g𝐺)](𝐺 ~QG 𝐾) ↔ 𝑟 = (0g𝑄)))
6665biimpar 477 . . . . . . . . . . . . 13 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑟 = (0g𝑄)) → 𝑟 = [(0g𝐺)](𝐺 ~QG 𝐾))
6766fveq2d 6911 . . . . . . . . . . . 12 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑟 = (0g𝑄)) → (𝐽𝑟) = (𝐽‘[(0g𝐺)](𝐺 ~QG 𝐾)))
682adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑟 ∈ (Base‘𝑄)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
6968ad3antrrr 730 . . . . . . . . . . . . 13 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑟 = (0g𝑄)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
7016, 55grpidcl 18996 . . . . . . . . . . . . . . 15 (𝐺 ∈ Grp → (0g𝐺) ∈ (Base‘𝐺))
718, 70syl 17 . . . . . . . . . . . . . 14 (𝜑 → (0g𝐺) ∈ (Base‘𝐺))
7271ad4antr 732 . . . . . . . . . . . . 13 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑟 = (0g𝑄)) → (0g𝐺) ∈ (Base‘𝐺))
731, 69, 3, 4, 5, 72ghmquskerlem1 19314 . . . . . . . . . . . 12 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑟 = (0g𝑄)) → (𝐽‘[(0g𝐺)](𝐺 ~QG 𝐾)) = (𝐹‘(0g𝐺)))
7455, 1ghmid 19253 . . . . . . . . . . . . . 14 (𝐹 ∈ (𝐺 GrpHom 𝐻) → (𝐹‘(0g𝐺)) = 0 )
752, 74syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝐹‘(0g𝐺)) = 0 )
7675ad4antr 732 . . . . . . . . . . . 12 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑟 = (0g𝑄)) → (𝐹‘(0g𝐺)) = 0 )
7767, 73, 763eqtrd 2779 . . . . . . . . . . 11 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑟 = (0g𝑄)) → (𝐽𝑟) = 0 )
7864, 77impbida 801 . . . . . . . . . 10 ((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → ((𝐽𝑟) = 0𝑟 = (0g𝑄)))
791, 68, 3, 4, 5, 48ghmquskerlem2 19316 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝑄)) → ∃𝑥𝑟 (𝐽𝑟) = (𝐹𝑥))
8078, 79r19.29a 3160 . . . . . . . . 9 ((𝜑𝑟 ∈ (Base‘𝑄)) → ((𝐽𝑟) = 0𝑟 = (0g𝑄)))
8180pm5.32da 579 . . . . . . . 8 (𝜑 → ((𝑟 ∈ (Base‘𝑄) ∧ (𝐽𝑟) = 0 ) ↔ (𝑟 ∈ (Base‘𝑄) ∧ 𝑟 = (0g𝑄))))
82 simpr 484 . . . . . . . . . . 11 ((𝜑𝑟 = (0g𝑄)) → 𝑟 = (0g𝑄))
834qusgrp 19217 . . . . . . . . . . . . . 14 (𝐾 ∈ (NrmSGrp‘𝐺) → 𝑄 ∈ Grp)
8412, 83syl 17 . . . . . . . . . . . . 13 (𝜑𝑄 ∈ Grp)
85 eqid 2735 . . . . . . . . . . . . . 14 (Base‘𝑄) = (Base‘𝑄)
86 eqid 2735 . . . . . . . . . . . . . 14 (0g𝑄) = (0g𝑄)
8785, 86grpidcl 18996 . . . . . . . . . . . . 13 (𝑄 ∈ Grp → (0g𝑄) ∈ (Base‘𝑄))
8884, 87syl 17 . . . . . . . . . . . 12 (𝜑 → (0g𝑄) ∈ (Base‘𝑄))
8988adantr 480 . . . . . . . . . . 11 ((𝜑𝑟 = (0g𝑄)) → (0g𝑄) ∈ (Base‘𝑄))
9082, 89eqeltrd 2839 . . . . . . . . . 10 ((𝜑𝑟 = (0g𝑄)) → 𝑟 ∈ (Base‘𝑄))
9190ex 412 . . . . . . . . 9 (𝜑 → (𝑟 = (0g𝑄) → 𝑟 ∈ (Base‘𝑄)))
9291pm4.71rd 562 . . . . . . . 8 (𝜑 → (𝑟 = (0g𝑄) ↔ (𝑟 ∈ (Base‘𝑄) ∧ 𝑟 = (0g𝑄))))
9381, 92bitr4d 282 . . . . . . 7 (𝜑 → ((𝑟 ∈ (Base‘𝑄) ∧ (𝐽𝑟) = 0 ) ↔ 𝑟 = (0g𝑄)))
942adantr 480 . . . . . . . . . . . 12 ((𝜑𝑞 ∈ (Base‘𝑄)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
9594imaexd 7939 . . . . . . . . . . 11 ((𝜑𝑞 ∈ (Base‘𝑄)) → (𝐹𝑞) ∈ V)
9695uniexd 7761 . . . . . . . . . 10 ((𝜑𝑞 ∈ (Base‘𝑄)) → (𝐹𝑞) ∈ V)
975a1i 11 . . . . . . . . . 10 (𝜑𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ (𝐹𝑞)))
9821, 35fnfvelrnd 7102 . . . . . . . . . . . . 13 ((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → (𝐹𝑥) ∈ ran 𝐹)
99 ghmqusker.s . . . . . . . . . . . . . 14 (𝜑 → ran 𝐹 = (Base‘𝐻))
10099ad3antrrr 730 . . . . . . . . . . . . 13 ((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → ran 𝐹 = (Base‘𝐻))
10198, 100eleqtrd 2841 . . . . . . . . . . . 12 ((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → (𝐹𝑥) ∈ (Base‘𝐻))
10237, 101eqeltrd 2839 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → (𝐽𝑟) ∈ (Base‘𝐻))
103102, 79r19.29a 3160 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝑄)) → (𝐽𝑟) ∈ (Base‘𝐻))
10496, 97, 103fmpt2d 7144 . . . . . . . . 9 (𝜑𝐽:(Base‘𝑄)⟶(Base‘𝐻))
105104ffnd 6738 . . . . . . . 8 (𝜑𝐽 Fn (Base‘𝑄))
106 fniniseg 7080 . . . . . . . 8 (𝐽 Fn (Base‘𝑄) → (𝑟 ∈ (𝐽 “ { 0 }) ↔ (𝑟 ∈ (Base‘𝑄) ∧ (𝐽𝑟) = 0 )))
107105, 106syl 17 . . . . . . 7 (𝜑 → (𝑟 ∈ (𝐽 “ { 0 }) ↔ (𝑟 ∈ (Base‘𝑄) ∧ (𝐽𝑟) = 0 )))
108 velsn 4647 . . . . . . . 8 (𝑟 ∈ {(0g𝑄)} ↔ 𝑟 = (0g𝑄))
109108a1i 11 . . . . . . 7 (𝜑 → (𝑟 ∈ {(0g𝑄)} ↔ 𝑟 = (0g𝑄)))
11093, 107, 1093bitr4d 311 . . . . . 6 (𝜑 → (𝑟 ∈ (𝐽 “ { 0 }) ↔ 𝑟 ∈ {(0g𝑄)}))
111110eqrdv 2733 . . . . 5 (𝜑 → (𝐽 “ { 0 }) = {(0g𝑄)})
11285, 17, 86, 1kerf1ghm 19278 . . . . . 6 (𝐽 ∈ (𝑄 GrpHom 𝐻) → (𝐽:(Base‘𝑄)–1-1→(Base‘𝐻) ↔ (𝐽 “ { 0 }) = {(0g𝑄)}))
113112biimpar 477 . . . . 5 ((𝐽 ∈ (𝑄 GrpHom 𝐻) ∧ (𝐽 “ { 0 }) = {(0g𝑄)}) → 𝐽:(Base‘𝑄)–1-1→(Base‘𝐻))
1146, 111, 113syl2anc 584 . . . 4 (𝜑𝐽:(Base‘𝑄)–1-1→(Base‘𝐻))
115 f1f1orn 6860 . . . 4 (𝐽:(Base‘𝑄)–1-1→(Base‘𝐻) → 𝐽:(Base‘𝑄)–1-1-onto→ran 𝐽)
116114, 115syl 17 . . 3 (𝜑𝐽:(Base‘𝑄)–1-1-onto→ran 𝐽)
117 simpr 484 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝐺)) → 𝑥 ∈ (Base‘𝐺))
118 ovex 7464 . . . . . . . . . . 11 (𝐺 ~QG 𝐾) ∈ V
119118ecelqsi 8812 . . . . . . . . . 10 (𝑥 ∈ (Base‘𝐺) → [𝑥](𝐺 ~QG 𝐾) ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)))
120117, 119syl 17 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐺)) → [𝑥](𝐺 ~QG 𝐾) ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)))
12126adantr 480 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐺)) → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) = (Base‘𝑄))
122120, 121eleqtrd 2841 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐺)) → [𝑥](𝐺 ~QG 𝐾) ∈ (Base‘𝑄))
123 elqsi 8809 . . . . . . . . 9 (𝑟 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)) → ∃𝑥 ∈ (Base‘𝐺)𝑟 = [𝑥](𝐺 ~QG 𝐾))
12450, 123syl 17 . . . . . . . 8 ((𝜑𝑟 ∈ (Base‘𝑄)) → ∃𝑥 ∈ (Base‘𝐺)𝑟 = [𝑥](𝐺 ~QG 𝐾))
125 simpr 484 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝐺)) ∧ 𝑟 = [𝑥](𝐺 ~QG 𝐾)) → 𝑟 = [𝑥](𝐺 ~QG 𝐾))
126125fveq2d 6911 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝐺)) ∧ 𝑟 = [𝑥](𝐺 ~QG 𝐾)) → (𝐽𝑟) = (𝐽‘[𝑥](𝐺 ~QG 𝐾)))
1272adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝐺)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
1281, 127, 3, 4, 5, 117ghmquskerlem1 19314 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (Base‘𝐺)) → (𝐽‘[𝑥](𝐺 ~QG 𝐾)) = (𝐹𝑥))
129128adantr 480 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝐺)) ∧ 𝑟 = [𝑥](𝐺 ~QG 𝐾)) → (𝐽‘[𝑥](𝐺 ~QG 𝐾)) = (𝐹𝑥))
130126, 129eqtrd 2775 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝐺)) ∧ 𝑟 = [𝑥](𝐺 ~QG 𝐾)) → (𝐽𝑟) = (𝐹𝑥))
1311303impa 1109 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐺) ∧ 𝑟 = [𝑥](𝐺 ~QG 𝐾)) → (𝐽𝑟) = (𝐹𝑥))
132131eqeq1d 2737 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐺) ∧ 𝑟 = [𝑥](𝐺 ~QG 𝐾)) → ((𝐽𝑟) = 𝑦 ↔ (𝐹𝑥) = 𝑦))
133122, 124, 132rexxfrd2 5419 . . . . . . 7 (𝜑 → (∃𝑟 ∈ (Base‘𝑄)(𝐽𝑟) = 𝑦 ↔ ∃𝑥 ∈ (Base‘𝐺)(𝐹𝑥) = 𝑦))
134 fvelrnb 6969 . . . . . . . 8 (𝐽 Fn (Base‘𝑄) → (𝑦 ∈ ran 𝐽 ↔ ∃𝑟 ∈ (Base‘𝑄)(𝐽𝑟) = 𝑦))
135105, 134syl 17 . . . . . . 7 (𝜑 → (𝑦 ∈ ran 𝐽 ↔ ∃𝑟 ∈ (Base‘𝑄)(𝐽𝑟) = 𝑦))
136 fvelrnb 6969 . . . . . . . 8 (𝐹 Fn (Base‘𝐺) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ (Base‘𝐺)(𝐹𝑥) = 𝑦))
13720, 136syl 17 . . . . . . 7 (𝜑 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ (Base‘𝐺)(𝐹𝑥) = 𝑦))
138133, 135, 1373bitr4rd 312 . . . . . 6 (𝜑 → (𝑦 ∈ ran 𝐹𝑦 ∈ ran 𝐽))
139138eqrdv 2733 . . . . 5 (𝜑 → ran 𝐹 = ran 𝐽)
140139, 99eqtr3d 2777 . . . 4 (𝜑 → ran 𝐽 = (Base‘𝐻))
141140f1oeq3d 6846 . . 3 (𝜑 → (𝐽:(Base‘𝑄)–1-1-onto→ran 𝐽𝐽:(Base‘𝑄)–1-1-onto→(Base‘𝐻)))
142116, 141mpbid 232 . 2 (𝜑𝐽:(Base‘𝑄)–1-1-onto→(Base‘𝐻))
14385, 17isgim 19293 . 2 (𝐽 ∈ (𝑄 GrpIso 𝐻) ↔ (𝐽 ∈ (𝑄 GrpHom 𝐻) ∧ 𝐽:(Base‘𝑄)–1-1-onto→(Base‘𝐻)))
1446, 142, 143sylanbrc 583 1 (𝜑𝐽 ∈ (𝑄 GrpIso 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wrex 3068  Vcvv 3478  𝒫 cpw 4605  {csn 4631   cuni 4912  cmpt 5231  ccnv 5688  ran crn 5690  cima 5692   Fn wfn 6558  wf 6559  1-1wf1 6560  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431   Er wer 8741  [cec 8742   / cqs 8743  Basecbs 17245  0gc0g 17486   /s cqus 17552  Grpcgrp 18964  SubGrpcsubg 19151  NrmSGrpcnsg 19152   ~QG cqg 19153   GrpHom cghm 19243   GrpIso cgim 19288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-ec 8746  df-qs 8750  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-0g 17488  df-imas 17555  df-qus 17556  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-nsg 19155  df-eqg 19156  df-ghm 19244  df-gim 19290
This theorem is referenced by:  gicqusker  19319  lmhmqusker  33425  rhmqusker  33434  aks6d1c6lem5  42159
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