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Theorem ghmquskerlem3 19317
Description: The mapping 𝐻 induced by a surjective group homomorphism 𝐹 from the quotient group 𝑄 over 𝐹's kernel 𝐾 is a group isomorphism. In this case, one says that 𝐹 factors through 𝑄, which is also called the factor group. (Contributed by Thierry Arnoux, 22-Mar-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))
Assertion
Ref Expression
ghmquskerlem3 (𝜑𝐽 ∈ (𝑄 GrpHom 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 ghmquskerlem3
Dummy variables 𝑟 𝑥 𝑦 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2735 . 2 (Base‘𝑄) = (Base‘𝑄)
2 eqid 2735 . 2 (Base‘𝐻) = (Base‘𝐻)
3 eqid 2735 . 2 (+g𝑄) = (+g𝑄)
4 eqid 2735 . 2 (+g𝐻) = (+g𝐻)
5 ghmqusker.k . . . 4 𝐾 = (𝐹 “ { 0 })
6 ghmqusker.f . . . . 5 (𝜑𝐹 ∈ (𝐺 GrpHom 𝐻))
7 ghmqusker.1 . . . . . 6 0 = (0g𝐻)
87ghmker 19273 . . . . 5 (𝐹 ∈ (𝐺 GrpHom 𝐻) → (𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
96, 8syl 17 . . . 4 (𝜑 → (𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
105, 9eqeltrid 2843 . . 3 (𝜑𝐾 ∈ (NrmSGrp‘𝐺))
11 ghmqusker.q . . . 4 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))
1211qusgrp 19217 . . 3 (𝐾 ∈ (NrmSGrp‘𝐺) → 𝑄 ∈ Grp)
1310, 12syl 17 . 2 (𝜑𝑄 ∈ Grp)
14 ghmrn 19260 . . 3 (𝐹 ∈ (𝐺 GrpHom 𝐻) → ran 𝐹 ∈ (SubGrp‘𝐻))
15 subgrcl 19162 . . 3 (ran 𝐹 ∈ (SubGrp‘𝐻) → 𝐻 ∈ Grp)
166, 14, 153syl 18 . 2 (𝜑𝐻 ∈ Grp)
176adantr 480 . . . . 5 ((𝜑𝑞 ∈ (Base‘𝑄)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
1817imaexd 7939 . . . 4 ((𝜑𝑞 ∈ (Base‘𝑄)) → (𝐹𝑞) ∈ V)
1918uniexd 7761 . . 3 ((𝜑𝑞 ∈ (Base‘𝑄)) → (𝐹𝑞) ∈ V)
20 ghmqusker.j . . . 4 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ (𝐹𝑞))
2120a1i 11 . . 3 (𝜑𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ (𝐹𝑞)))
22 simpr 484 . . . . 5 ((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → (𝐽𝑟) = (𝐹𝑥))
23 eqid 2735 . . . . . . . . . 10 (Base‘𝐺) = (Base‘𝐺)
2423, 2ghmf 19251 . . . . . . . . 9 (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
256, 24syl 17 . . . . . . . 8 (𝜑𝐹:(Base‘𝐺)⟶(Base‘𝐻))
2625frnd 6745 . . . . . . 7 (𝜑 → ran 𝐹 ⊆ (Base‘𝐻))
2726ad3antrrr 730 . . . . . 6 ((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → ran 𝐹 ⊆ (Base‘𝐻))
2825ffnd 6738 . . . . . . . 8 (𝜑𝐹 Fn (Base‘𝐺))
2928ad3antrrr 730 . . . . . . 7 ((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → 𝐹 Fn (Base‘𝐺))
3011a1i 11 . . . . . . . . . . . . 13 (𝜑𝑄 = (𝐺 /s (𝐺 ~QG 𝐾)))
31 eqidd 2736 . . . . . . . . . . . . 13 (𝜑 → (Base‘𝐺) = (Base‘𝐺))
32 ovexd 7466 . . . . . . . . . . . . 13 (𝜑 → (𝐺 ~QG 𝐾) ∈ V)
33 ghmgrp1 19249 . . . . . . . . . . . . . 14 (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp)
346, 33syl 17 . . . . . . . . . . . . 13 (𝜑𝐺 ∈ Grp)
3530, 31, 32, 34qusbas 17592 . . . . . . . . . . . 12 (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) = (Base‘𝑄))
36 nsgsubg 19189 . . . . . . . . . . . . . 14 (𝐾 ∈ (NrmSGrp‘𝐺) → 𝐾 ∈ (SubGrp‘𝐺))
37 eqid 2735 . . . . . . . . . . . . . . 15 (𝐺 ~QG 𝐾) = (𝐺 ~QG 𝐾)
3823, 37eqger 19209 . . . . . . . . . . . . . 14 (𝐾 ∈ (SubGrp‘𝐺) → (𝐺 ~QG 𝐾) Er (Base‘𝐺))
3910, 36, 383syl 18 . . . . . . . . . . . . 13 (𝜑 → (𝐺 ~QG 𝐾) Er (Base‘𝐺))
4039qsss 8817 . . . . . . . . . . . 12 (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) ⊆ 𝒫 (Base‘𝐺))
4135, 40eqsstrrd 4035 . . . . . . . . . . 11 (𝜑 → (Base‘𝑄) ⊆ 𝒫 (Base‘𝐺))
4241sselda 3995 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ 𝒫 (Base‘𝐺))
4342elpwid 4614 . . . . . . . . 9 ((𝜑𝑟 ∈ (Base‘𝑄)) → 𝑟 ⊆ (Base‘𝐺))
4443sselda 3995 . . . . . . . 8 (((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) → 𝑥 ∈ (Base‘𝐺))
4544adantr 480 . . . . . . 7 ((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → 𝑥 ∈ (Base‘𝐺))
4629, 45fnfvelrnd 7102 . . . . . 6 ((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → (𝐹𝑥) ∈ ran 𝐹)
4727, 46sseldd 3996 . . . . 5 ((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → (𝐹𝑥) ∈ (Base‘𝐻))
4822, 47eqeltrd 2839 . . . 4 ((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → (𝐽𝑟) ∈ (Base‘𝐻))
496adantr 480 . . . . 5 ((𝜑𝑟 ∈ (Base‘𝑄)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
50 simpr 484 . . . . 5 ((𝜑𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ (Base‘𝑄))
517, 49, 5, 11, 20, 50ghmquskerlem2 19316 . . . 4 ((𝜑𝑟 ∈ (Base‘𝑄)) → ∃𝑥𝑟 (𝐽𝑟) = (𝐹𝑥))
5248, 51r19.29a 3160 . . 3 ((𝜑𝑟 ∈ (Base‘𝑄)) → (𝐽𝑟) ∈ (Base‘𝐻))
5319, 21, 52fmpt2d 7144 . 2 (𝜑𝐽:(Base‘𝑄)⟶(Base‘𝐻))
5439ad6antr 736 . . . . . . . . . . 11 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐺 ~QG 𝐾) Er (Base‘𝐺))
5550ad5antr 734 . . . . . . . . . . . 12 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑟 ∈ (Base‘𝑄))
5635ad6antr 736 . . . . . . . . . . . 12 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) = (Base‘𝑄))
5755, 56eleqtrrd 2842 . . . . . . . . . . 11 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑟 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)))
58 simp-4r 784 . . . . . . . . . . 11 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑥𝑟)
59 qsel 8835 . . . . . . . . . . 11 (((𝐺 ~QG 𝐾) Er (Base‘𝐺) ∧ 𝑟 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)) ∧ 𝑥𝑟) → 𝑟 = [𝑥](𝐺 ~QG 𝐾))
6054, 57, 58, 59syl3anc 1370 . . . . . . . . . 10 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑟 = [𝑥](𝐺 ~QG 𝐾))
61 simp-5r 786 . . . . . . . . . . . 12 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑠 ∈ (Base‘𝑄))
6261, 56eleqtrrd 2842 . . . . . . . . . . 11 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑠 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)))
63 simplr 769 . . . . . . . . . . 11 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑦𝑠)
64 qsel 8835 . . . . . . . . . . 11 (((𝐺 ~QG 𝐾) Er (Base‘𝐺) ∧ 𝑠 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)) ∧ 𝑦𝑠) → 𝑠 = [𝑦](𝐺 ~QG 𝐾))
6554, 62, 63, 64syl3anc 1370 . . . . . . . . . 10 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑠 = [𝑦](𝐺 ~QG 𝐾))
6660, 65oveq12d 7449 . . . . . . . . 9 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝑟(+g𝑄)𝑠) = ([𝑥](𝐺 ~QG 𝐾)(+g𝑄)[𝑦](𝐺 ~QG 𝐾)))
6710ad6antr 736 . . . . . . . . . 10 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝐾 ∈ (NrmSGrp‘𝐺))
6843ad5antr 734 . . . . . . . . . . 11 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑟 ⊆ (Base‘𝐺))
6968, 58sseldd 3996 . . . . . . . . . 10 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑥 ∈ (Base‘𝐺))
7041sselda 3995 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ (Base‘𝑄)) → 𝑠 ∈ 𝒫 (Base‘𝐺))
7170elpwid 4614 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ (Base‘𝑄)) → 𝑠 ⊆ (Base‘𝐺))
7271adantlr 715 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑠 ⊆ (Base‘𝐺))
7372ad4antr 732 . . . . . . . . . . 11 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑠 ⊆ (Base‘𝐺))
7473, 63sseldd 3996 . . . . . . . . . 10 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑦 ∈ (Base‘𝐺))
75 eqid 2735 . . . . . . . . . . 11 (+g𝐺) = (+g𝐺)
7611, 23, 75, 3qusadd 19219 . . . . . . . . . 10 ((𝐾 ∈ (NrmSGrp‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → ([𝑥](𝐺 ~QG 𝐾)(+g𝑄)[𝑦](𝐺 ~QG 𝐾)) = [(𝑥(+g𝐺)𝑦)](𝐺 ~QG 𝐾))
7767, 69, 74, 76syl3anc 1370 . . . . . . . . 9 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → ([𝑥](𝐺 ~QG 𝐾)(+g𝑄)[𝑦](𝐺 ~QG 𝐾)) = [(𝑥(+g𝐺)𝑦)](𝐺 ~QG 𝐾))
7866, 77eqtrd 2775 . . . . . . . 8 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝑟(+g𝑄)𝑠) = [(𝑥(+g𝐺)𝑦)](𝐺 ~QG 𝐾))
7978fveq2d 6911 . . . . . . 7 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐽‘(𝑟(+g𝑄)𝑠)) = (𝐽‘[(𝑥(+g𝐺)𝑦)](𝐺 ~QG 𝐾)))
806ad6antr 736 . . . . . . . 8 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
8180, 33syl 17 . . . . . . . . 9 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝐺 ∈ Grp)
8223, 75, 81, 69, 74grpcld 18978 . . . . . . . 8 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺))
837, 80, 5, 11, 20, 82ghmquskerlem1 19314 . . . . . . 7 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐽‘[(𝑥(+g𝐺)𝑦)](𝐺 ~QG 𝐾)) = (𝐹‘(𝑥(+g𝐺)𝑦)))
8423, 75, 4ghmlin 19252 . . . . . . . 8 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝐹‘(𝑥(+g𝐺)𝑦)) = ((𝐹𝑥)(+g𝐻)(𝐹𝑦)))
8580, 69, 74, 84syl3anc 1370 . . . . . . 7 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐹‘(𝑥(+g𝐺)𝑦)) = ((𝐹𝑥)(+g𝐻)(𝐹𝑦)))
8679, 83, 853eqtrd 2779 . . . . . 6 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐽‘(𝑟(+g𝑄)𝑠)) = ((𝐹𝑥)(+g𝐻)(𝐹𝑦)))
87 simpllr 776 . . . . . . 7 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐽𝑟) = (𝐹𝑥))
88 simpr 484 . . . . . . 7 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐽𝑠) = (𝐹𝑦))
8987, 88oveq12d 7449 . . . . . 6 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → ((𝐽𝑟)(+g𝐻)(𝐽𝑠)) = ((𝐹𝑥)(+g𝐻)(𝐹𝑦)))
9086, 89eqtr4d 2778 . . . . 5 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐽‘(𝑟(+g𝑄)𝑠)) = ((𝐽𝑟)(+g𝐻)(𝐽𝑠)))
916ad4antr 732 . . . . . 6 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
92 simpllr 776 . . . . . 6 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → 𝑠 ∈ (Base‘𝑄))
937, 91, 5, 11, 20, 92ghmquskerlem2 19316 . . . . 5 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → ∃𝑦𝑠 (𝐽𝑠) = (𝐹𝑦))
9490, 93r19.29a 3160 . . . 4 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → (𝐽‘(𝑟(+g𝑄)𝑠)) = ((𝐽𝑟)(+g𝐻)(𝐽𝑠)))
9551adantr 480 . . . 4 (((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → ∃𝑥𝑟 (𝐽𝑟) = (𝐹𝑥))
9694, 95r19.29a 3160 . . 3 (((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → (𝐽‘(𝑟(+g𝑄)𝑠)) = ((𝐽𝑟)(+g𝐻)(𝐽𝑠)))
9796anasss 466 . 2 ((𝜑 ∧ (𝑟 ∈ (Base‘𝑄) ∧ 𝑠 ∈ (Base‘𝑄))) → (𝐽‘(𝑟(+g𝑄)𝑠)) = ((𝐽𝑟)(+g𝐻)(𝐽𝑠)))
981, 2, 3, 4, 13, 16, 53, 97isghmd 19256 1 (𝜑𝐽 ∈ (𝑄 GrpHom 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wrex 3068  Vcvv 3478  wss 3963  𝒫 cpw 4605  {csn 4631   cuni 4912  cmpt 5231  ccnv 5688  ran crn 5690  cima 5692   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431   Er wer 8741  [cec 8742   / cqs 8743  Basecbs 17245  +gcplusg 17298  0gc0g 17486   /s cqus 17552  Grpcgrp 18964  SubGrpcsubg 19151  NrmSGrpcnsg 19152   ~QG cqg 19153   GrpHom cghm 19243
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
This theorem is referenced by:  ghmqusker  19318  rhmquskerlem  33433
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