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Theorem rhmqusnsg 21296
Description: The mapping 𝐽 induced by a ring homomorphism 𝐹 from a subring 𝑁 of the quotient group 𝑄 over 𝐹's kernel 𝐾 is a ring homomorphism. (Contributed by Thierry Arnoux, 13-May-2025.)
Hypotheses
Ref Expression
rhmqusnsg.0 0 = (0g𝐻)
rhmqusnsg.f (𝜑𝐹 ∈ (𝐺 RingHom 𝐻))
rhmqusnsg.k 𝐾 = (𝐹 “ { 0 })
rhmqusnsg.q 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))
rhmqusnsg.j 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ (𝐹𝑞))
rhmqusnsg.g (𝜑𝐺 ∈ CRing)
rhmqusnsg.n (𝜑𝑁𝐾)
rhmqusnsg.1 (𝜑𝑁 ∈ (LIdeal‘𝐺))
Assertion
Ref Expression
rhmqusnsg (𝜑𝐽 ∈ (𝑄 RingHom 𝐻))
Distinct variable groups:   𝐹,𝑞   𝐺,𝑞   𝐻,𝑞   𝐽,𝑞   𝐾,𝑞   𝑁,𝑞   𝑄,𝑞   𝜑,𝑞
Allowed substitution hint:   0 (𝑞)

Proof of Theorem rhmqusnsg
Dummy variables 𝑟 𝑥 𝑦 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2736 . 2 (Base‘𝑄) = (Base‘𝑄)
2 eqid 2736 . 2 (1r𝑄) = (1r𝑄)
3 eqid 2736 . 2 (1r𝐻) = (1r𝐻)
4 eqid 2736 . 2 (.r𝑄) = (.r𝑄)
5 eqid 2736 . 2 (.r𝐻) = (.r𝐻)
6 rhmqusnsg.g . . . . 5 (𝜑𝐺 ∈ CRing)
76crngringd 20244 . . . 4 (𝜑𝐺 ∈ Ring)
8 rhmqusnsg.1 . . . . 5 (𝜑𝑁 ∈ (LIdeal‘𝐺))
9 eqid 2736 . . . . . . 7 (LIdeal‘𝐺) = (LIdeal‘𝐺)
109crng2idl 21292 . . . . . 6 (𝐺 ∈ CRing → (LIdeal‘𝐺) = (2Ideal‘𝐺))
116, 10syl 17 . . . . 5 (𝜑 → (LIdeal‘𝐺) = (2Ideal‘𝐺))
128, 11eleqtrd 2842 . . . 4 (𝜑𝑁 ∈ (2Ideal‘𝐺))
13 rhmqusnsg.q . . . . 5 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))
14 eqid 2736 . . . . 5 (2Ideal‘𝐺) = (2Ideal‘𝐺)
15 eqid 2736 . . . . 5 (1r𝐺) = (1r𝐺)
1613, 14, 15qus1 21285 . . . 4 ((𝐺 ∈ Ring ∧ 𝑁 ∈ (2Ideal‘𝐺)) → (𝑄 ∈ Ring ∧ [(1r𝐺)](𝐺 ~QG 𝑁) = (1r𝑄)))
177, 12, 16syl2anc 584 . . 3 (𝜑 → (𝑄 ∈ Ring ∧ [(1r𝐺)](𝐺 ~QG 𝑁) = (1r𝑄)))
1817simpld 494 . 2 (𝜑𝑄 ∈ Ring)
19 rhmqusnsg.f . . 3 (𝜑𝐹 ∈ (𝐺 RingHom 𝐻))
20 rhmrcl2 20478 . . 3 (𝐹 ∈ (𝐺 RingHom 𝐻) → 𝐻 ∈ Ring)
2119, 20syl 17 . 2 (𝜑𝐻 ∈ Ring)
22 rhmqusnsg.0 . . . 4 0 = (0g𝐻)
23 rhmghm 20485 . . . . 5 (𝐹 ∈ (𝐺 RingHom 𝐻) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
2419, 23syl 17 . . . 4 (𝜑𝐹 ∈ (𝐺 GrpHom 𝐻))
25 rhmqusnsg.k . . . 4 𝐾 = (𝐹 “ { 0 })
26 rhmqusnsg.j . . . 4 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ (𝐹𝑞))
27 rhmqusnsg.n . . . 4 (𝜑𝑁𝐾)
28 lidlnsg 21259 . . . . 5 ((𝐺 ∈ Ring ∧ 𝑁 ∈ (LIdeal‘𝐺)) → 𝑁 ∈ (NrmSGrp‘𝐺))
297, 8, 28syl2anc 584 . . . 4 (𝜑𝑁 ∈ (NrmSGrp‘𝐺))
30 eqid 2736 . . . . . 6 (Base‘𝐺) = (Base‘𝐺)
3130, 15ringidcl 20263 . . . . 5 (𝐺 ∈ Ring → (1r𝐺) ∈ (Base‘𝐺))
327, 31syl 17 . . . 4 (𝜑 → (1r𝐺) ∈ (Base‘𝐺))
3322, 24, 25, 13, 26, 27, 29, 32ghmqusnsglem1 19299 . . 3 (𝜑 → (𝐽‘[(1r𝐺)](𝐺 ~QG 𝑁)) = (𝐹‘(1r𝐺)))
3417simprd 495 . . . 4 (𝜑 → [(1r𝐺)](𝐺 ~QG 𝑁) = (1r𝑄))
3534fveq2d 6909 . . 3 (𝜑 → (𝐽‘[(1r𝐺)](𝐺 ~QG 𝑁)) = (𝐽‘(1r𝑄)))
3615, 3rhm1 20490 . . . 4 (𝐹 ∈ (𝐺 RingHom 𝐻) → (𝐹‘(1r𝐺)) = (1r𝐻))
3719, 36syl 17 . . 3 (𝜑 → (𝐹‘(1r𝐺)) = (1r𝐻))
3833, 35, 373eqtr3d 2784 . 2 (𝜑 → (𝐽‘(1r𝑄)) = (1r𝐻))
3919ad6antr 736 . . . . . . 7 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝐹 ∈ (𝐺 RingHom 𝐻))
4013a1i 11 . . . . . . . . . . . . 13 (𝜑𝑄 = (𝐺 /s (𝐺 ~QG 𝑁)))
41 eqidd 2737 . . . . . . . . . . . . 13 (𝜑 → (Base‘𝐺) = (Base‘𝐺))
42 ovexd 7467 . . . . . . . . . . . . 13 (𝜑 → (𝐺 ~QG 𝑁) ∈ V)
4340, 41, 42, 6qusbas 17591 . . . . . . . . . . . 12 (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝑁)) = (Base‘𝑄))
44 nsgsubg 19177 . . . . . . . . . . . . . 14 (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺))
45 eqid 2736 . . . . . . . . . . . . . . 15 (𝐺 ~QG 𝑁) = (𝐺 ~QG 𝑁)
4630, 45eqger 19197 . . . . . . . . . . . . . 14 (𝑁 ∈ (SubGrp‘𝐺) → (𝐺 ~QG 𝑁) Er (Base‘𝐺))
4729, 44, 463syl 18 . . . . . . . . . . . . 13 (𝜑 → (𝐺 ~QG 𝑁) Er (Base‘𝐺))
4847qsss 8819 . . . . . . . . . . . 12 (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝑁)) ⊆ 𝒫 (Base‘𝐺))
4943, 48eqsstrrd 4018 . . . . . . . . . . 11 (𝜑 → (Base‘𝑄) ⊆ 𝒫 (Base‘𝐺))
5049sselda 3982 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ 𝒫 (Base‘𝐺))
5150elpwid 4608 . . . . . . . . 9 ((𝜑𝑟 ∈ (Base‘𝑄)) → 𝑟 ⊆ (Base‘𝐺))
5251ad5antr 734 . . . . . . . 8 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑟 ⊆ (Base‘𝐺))
53 simp-4r 783 . . . . . . . 8 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑥𝑟)
5452, 53sseldd 3983 . . . . . . 7 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑥 ∈ (Base‘𝐺))
5549sselda 3982 . . . . . . . . . . 11 ((𝜑𝑠 ∈ (Base‘𝑄)) → 𝑠 ∈ 𝒫 (Base‘𝐺))
5655elpwid 4608 . . . . . . . . . 10 ((𝜑𝑠 ∈ (Base‘𝑄)) → 𝑠 ⊆ (Base‘𝐺))
5756adantlr 715 . . . . . . . . 9 (((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑠 ⊆ (Base‘𝐺))
5857ad4antr 732 . . . . . . . 8 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑠 ⊆ (Base‘𝐺))
59 simplr 768 . . . . . . . 8 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑦𝑠)
6058, 59sseldd 3983 . . . . . . 7 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑦 ∈ (Base‘𝐺))
61 eqid 2736 . . . . . . . 8 (.r𝐺) = (.r𝐺)
6230, 61, 5rhmmul 20487 . . . . . . 7 ((𝐹 ∈ (𝐺 RingHom 𝐻) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝐹‘(𝑥(.r𝐺)𝑦)) = ((𝐹𝑥)(.r𝐻)(𝐹𝑦)))
6339, 54, 60, 62syl3anc 1372 . . . . . 6 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐹‘(𝑥(.r𝐺)𝑦)) = ((𝐹𝑥)(.r𝐻)(𝐹𝑦)))
6447ad6antr 736 . . . . . . . . . . 11 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐺 ~QG 𝑁) Er (Base‘𝐺))
65 simp-6r 787 . . . . . . . . . . . 12 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑟 ∈ (Base‘𝑄))
6643ad6antr 736 . . . . . . . . . . . 12 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → ((Base‘𝐺) / (𝐺 ~QG 𝑁)) = (Base‘𝑄))
6765, 66eleqtrrd 2843 . . . . . . . . . . 11 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑟 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑁)))
68 qsel 8837 . . . . . . . . . . 11 (((𝐺 ~QG 𝑁) Er (Base‘𝐺) ∧ 𝑟 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑁)) ∧ 𝑥𝑟) → 𝑟 = [𝑥](𝐺 ~QG 𝑁))
6964, 67, 53, 68syl3anc 1372 . . . . . . . . . 10 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑟 = [𝑥](𝐺 ~QG 𝑁))
70 simp-5r 785 . . . . . . . . . . . 12 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑠 ∈ (Base‘𝑄))
7170, 66eleqtrrd 2843 . . . . . . . . . . 11 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑠 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑁)))
72 qsel 8837 . . . . . . . . . . 11 (((𝐺 ~QG 𝑁) Er (Base‘𝐺) ∧ 𝑠 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑁)) ∧ 𝑦𝑠) → 𝑠 = [𝑦](𝐺 ~QG 𝑁))
7364, 71, 59, 72syl3anc 1372 . . . . . . . . . 10 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑠 = [𝑦](𝐺 ~QG 𝑁))
7469, 73oveq12d 7450 . . . . . . . . 9 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝑟(.r𝑄)𝑠) = ([𝑥](𝐺 ~QG 𝑁)(.r𝑄)[𝑦](𝐺 ~QG 𝑁)))
756ad6antr 736 . . . . . . . . . 10 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝐺 ∈ CRing)
768ad6antr 736 . . . . . . . . . 10 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑁 ∈ (LIdeal‘𝐺))
7713, 30, 61, 4, 75, 76, 54, 60qusmulcrng 21295 . . . . . . . . 9 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → ([𝑥](𝐺 ~QG 𝑁)(.r𝑄)[𝑦](𝐺 ~QG 𝑁)) = [(𝑥(.r𝐺)𝑦)](𝐺 ~QG 𝑁))
7874, 77eqtr2d 2777 . . . . . . . 8 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → [(𝑥(.r𝐺)𝑦)](𝐺 ~QG 𝑁) = (𝑟(.r𝑄)𝑠))
7978fveq2d 6909 . . . . . . 7 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐽‘[(𝑥(.r𝐺)𝑦)](𝐺 ~QG 𝑁)) = (𝐽‘(𝑟(.r𝑄)𝑠)))
8039, 23syl 17 . . . . . . . 8 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
8127ad6antr 736 . . . . . . . 8 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑁𝐾)
8229ad6antr 736 . . . . . . . 8 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑁 ∈ (NrmSGrp‘𝐺))
83 rhmrcl1 20477 . . . . . . . . . 10 (𝐹 ∈ (𝐺 RingHom 𝐻) → 𝐺 ∈ Ring)
8439, 83syl 17 . . . . . . . . 9 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝐺 ∈ Ring)
8530, 61, 84, 54, 60ringcld 20258 . . . . . . . 8 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝑥(.r𝐺)𝑦) ∈ (Base‘𝐺))
8622, 80, 25, 13, 26, 81, 82, 85ghmqusnsglem1 19299 . . . . . . 7 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐽‘[(𝑥(.r𝐺)𝑦)](𝐺 ~QG 𝑁)) = (𝐹‘(𝑥(.r𝐺)𝑦)))
8779, 86eqtr3d 2778 . . . . . 6 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐽‘(𝑟(.r𝑄)𝑠)) = (𝐹‘(𝑥(.r𝐺)𝑦)))
88 simpllr 775 . . . . . . 7 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐽𝑟) = (𝐹𝑥))
89 simpr 484 . . . . . . 7 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐽𝑠) = (𝐹𝑦))
9088, 89oveq12d 7450 . . . . . 6 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → ((𝐽𝑟)(.r𝐻)(𝐽𝑠)) = ((𝐹𝑥)(.r𝐻)(𝐹𝑦)))
9163, 87, 903eqtr4d 2786 . . . . 5 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐽‘(𝑟(.r𝑄)𝑠)) = ((𝐽𝑟)(.r𝐻)(𝐽𝑠)))
9224ad4antr 732 . . . . . 6 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
9327ad4antr 732 . . . . . 6 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → 𝑁𝐾)
9429ad4antr 732 . . . . . 6 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → 𝑁 ∈ (NrmSGrp‘𝐺))
95 simpllr 775 . . . . . 6 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → 𝑠 ∈ (Base‘𝑄))
9622, 92, 25, 13, 26, 93, 94, 95ghmqusnsglem2 19300 . . . . 5 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → ∃𝑦𝑠 (𝐽𝑠) = (𝐹𝑦))
9791, 96r19.29a 3161 . . . 4 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → (𝐽‘(𝑟(.r𝑄)𝑠)) = ((𝐽𝑟)(.r𝐻)(𝐽𝑠)))
9824ad2antrr 726 . . . . 5 (((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
9927ad2antrr 726 . . . . 5 (((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑁𝐾)
10029ad2antrr 726 . . . . 5 (((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑁 ∈ (NrmSGrp‘𝐺))
101 simplr 768 . . . . 5 (((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑟 ∈ (Base‘𝑄))
10222, 98, 25, 13, 26, 99, 100, 101ghmqusnsglem2 19300 . . . 4 (((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → ∃𝑥𝑟 (𝐽𝑟) = (𝐹𝑥))
10397, 102r19.29a 3161 . . 3 (((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → (𝐽‘(𝑟(.r𝑄)𝑠)) = ((𝐽𝑟)(.r𝐻)(𝐽𝑠)))
104103anasss 466 . 2 ((𝜑 ∧ (𝑟 ∈ (Base‘𝑄) ∧ 𝑠 ∈ (Base‘𝑄))) → (𝐽‘(𝑟(.r𝑄)𝑠)) = ((𝐽𝑟)(.r𝐻)(𝐽𝑠)))
10522, 24, 25, 13, 26, 27, 29ghmqusnsg 19301 . 2 (𝜑𝐽 ∈ (𝑄 GrpHom 𝐻))
1061, 2, 3, 4, 5, 18, 21, 38, 104, 105isrhm2d 20488 1 (𝜑𝐽 ∈ (𝑄 RingHom 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  Vcvv 3479  wss 3950  𝒫 cpw 4599  {csn 4625   cuni 4906  cmpt 5224  ccnv 5683  cima 5687  cfv 6560  (class class class)co 7432   Er wer 8743  [cec 8744   / cqs 8745  Basecbs 17248  .rcmulr 17299  0gc0g 17485   /s cqus 17551  SubGrpcsubg 19139  NrmSGrpcnsg 19140   ~QG cqg 19141   GrpHom cghm 19231  1rcur 20179  Ringcrg 20231  CRingccrg 20232   RingHom crh 20470  LIdealclidl 21217  2Idealc2idl 21260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-tpos 8252  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-ec 8748  df-qs 8752  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-sup 9483  df-inf 9484  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-fz 13549  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  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 17487  df-imas 17554  df-qus 17555  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-mhm 18797  df-grp 18955  df-minusg 18956  df-sbg 18957  df-subg 19142  df-nsg 19143  df-eqg 19144  df-ghm 19232  df-cmn 19801  df-abl 19802  df-mgp 20139  df-rng 20151  df-ur 20180  df-ring 20233  df-cring 20234  df-oppr 20335  df-rhm 20473  df-subrg 20571  df-lmod 20861  df-lss 20931  df-lsp 20971  df-sra 21173  df-rgmod 21174  df-lidl 21219  df-rsp 21220  df-2idl 21261
This theorem is referenced by:  zndvdchrrhm  41973  rhmqusspan  42187
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