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Theorem imasgrp2 18432

Description: The image structure of a group is a group. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)

**Hypotheses**
Ref	Expression
imasgrp.u	⊢ (𝜑 → 𝑈 = (𝐹 “_s 𝑅))
imasgrp.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
imasgrp.p	⊢ (𝜑 → + = (+_g‘𝑅))
imasgrp.f	⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
imasgrp.e	⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
imasgrp2.r	⊢ (𝜑 → 𝑅 ∈ 𝑊)
imasgrp2.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 + 𝑦) ∈ 𝑉)
imasgrp2.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘((𝑥 + 𝑦) + 𝑧)) = (𝐹‘(𝑥 + (𝑦 + 𝑧))))
imasgrp2.3	⊢ (𝜑 → 0 ∈ 𝑉)
imasgrp2.4	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹‘𝑥))
imasgrp2.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑁 ∈ 𝑉)
imasgrp2.6	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘(𝑁 + 𝑥)) = (𝐹‘ 0 ))

**Assertion**
Ref	Expression
imasgrp2	⊢ (𝜑 → (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0_g‘𝑈)))

Distinct variable groups: 𝑞,𝑝,𝑥,𝐵 𝑁,𝑝 𝑎,𝑏,𝑝,𝑞,𝑥,𝑦,𝑧,𝜑 𝑅,𝑝,𝑞 𝐹,𝑎,𝑏,𝑝,𝑞,𝑥,𝑦,𝑧 + ,𝑝,𝑞,𝑥,𝑦 𝑈,𝑎,𝑏,𝑝,𝑞,𝑥,𝑦,𝑧 𝑉,𝑎,𝑏,𝑝,𝑞,𝑥,𝑦,𝑧 0 ,𝑝,𝑞,𝑥 Allowed substitution hints: 𝐵(𝑦,𝑧,𝑎,𝑏) + (𝑧,𝑎,𝑏) 𝑅(𝑥,𝑦,𝑧,𝑎,𝑏) 𝑁(𝑥,𝑦,𝑧,𝑞,𝑎,𝑏) 𝑊(𝑥,𝑦,𝑧,𝑞,𝑝,𝑎,𝑏) 0 (𝑦,𝑧,𝑎,𝑏)

Proof of Theorem imasgrp2 Dummy variables 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imasgrp.u	. . . 4 ⊢ (𝜑 → 𝑈 = (𝐹 “_s 𝑅))
2		imasgrp.v	. . . 4 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
3		imasgrp.f	. . . 4 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
4		imasgrp2.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
5	1, 2, 3, 4	imasbas 16971	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝑈))
6		eqidd 2737	. . 3 ⊢ (𝜑 → (+_g‘𝑈) = (+_g‘𝑈))
7		imasgrp.e	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
8		imasgrp.p	. . . . . . . . . 10 ⊢ (𝜑 → + = (+_g‘𝑅))
9	8	oveqd 7208	. . . . . . . . 9 ⊢ (𝜑 → (𝑎 + 𝑏) = (𝑎(+_g‘𝑅)𝑏))
10	9	fveq2d 6699	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑎(+_g‘𝑅)𝑏)))
11	8	oveqd 7208	. . . . . . . . 9 ⊢ (𝜑 → (𝑝 + 𝑞) = (𝑝(+_g‘𝑅)𝑞))
12	11	fveq2d 6699	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘(𝑝 + 𝑞)) = (𝐹‘(𝑝(+_g‘𝑅)𝑞)))
13	10, 12	eqeq12d 2752	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)) ↔ (𝐹‘(𝑎(+_g‘𝑅)𝑏)) = (𝐹‘(𝑝(+_g‘𝑅)𝑞))))
14	13	3ad2ant1 1135	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → ((𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)) ↔ (𝐹‘(𝑎(+_g‘𝑅)𝑏)) = (𝐹‘(𝑝(+_g‘𝑅)𝑞))))
15	7, 14	sylibd 242	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎(+_g‘𝑅)𝑏)) = (𝐹‘(𝑝(+_g‘𝑅)𝑞))))
16		eqid 2736	. . . . 5 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
17		eqid 2736	. . . . 5 ⊢ (+_g‘𝑈) = (+_g‘𝑈)
18	11	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 + 𝑞) = (𝑝(+_g‘𝑅)𝑞))
19		imasgrp2.1	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 + 𝑦) ∈ 𝑉)
20	19	3expb 1122	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
21	20	caovclg 7378	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 + 𝑞) ∈ 𝑉)
22	18, 21	eqeltrrd 2832	. . . . 5 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝(+_g‘𝑅)𝑞) ∈ 𝑉)
23	3, 15, 1, 2, 4, 16, 17, 22	imasaddf 16992	. . . 4 ⊢ (𝜑 → (+_g‘𝑈):(𝐵 × 𝐵)⟶𝐵)
24		fovrn 7356	. . . 4 ⊢ (((+_g‘𝑈):(𝐵 × 𝐵)⟶𝐵 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → (𝑢(+_g‘𝑈)𝑣) ∈ 𝐵)
25	23, 24	syl3an1 1165	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → (𝑢(+_g‘𝑈)𝑣) ∈ 𝐵)
26		forn 6614	. . . . . . . . . 10 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵)
27	3, 26	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐹 = 𝐵)
28	27	eleq2d 2816	. . . . . . . 8 ⊢ (𝜑 → (𝑢 ∈ ran 𝐹 ↔ 𝑢 ∈ 𝐵))
29	27	eleq2d 2816	. . . . . . . 8 ⊢ (𝜑 → (𝑣 ∈ ran 𝐹 ↔ 𝑣 ∈ 𝐵))
30	27	eleq2d 2816	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ ran 𝐹 ↔ 𝑤 ∈ 𝐵))
31	28, 29, 30	3anbi123d 1438	. . . . . . 7 ⊢ (𝜑 → ((𝑢 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ∧ 𝑤 ∈ ran 𝐹) ↔ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)))
32		fofn 6613	. . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹 Fn 𝑉)
33	3, 32	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝑉)
34		fvelrnb 6751	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑉 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢))
35		fvelrnb 6751	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑉 → (𝑣 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣))
36		fvelrnb 6751	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑉 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤))
37	34, 35, 36	3anbi123d 1438	. . . . . . . 8 ⊢ (𝐹 Fn 𝑉 → ((𝑢 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ∧ 𝑤 ∈ ran 𝐹) ↔ (∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢 ∧ ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣 ∧ ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤)))
38	33, 37	syl 17	. . . . . . 7 ⊢ (𝜑 → ((𝑢 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ∧ 𝑤 ∈ ran 𝐹) ↔ (∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢 ∧ ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣 ∧ ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤)))
39	31, 38	bitr3d 284	. . . . . 6 ⊢ (𝜑 → ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ↔ (∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢 ∧ ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣 ∧ ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤)))
40		3reeanv 3270	. . . . . 6 ⊢ (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 ((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) ↔ (∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢 ∧ ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣 ∧ ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤))
41	39, 40	bitr4di 292	. . . . 5 ⊢ (𝜑 → ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ↔ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 ((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤)))
42		imasgrp2.2	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘((𝑥 + 𝑦) + 𝑧)) = (𝐹‘(𝑥 + (𝑦 + 𝑧))))
43	8	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → + = (+_g‘𝑅))
44	43	oveqd 7208	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 + 𝑦) + 𝑧) = ((𝑥 + 𝑦)(+_g‘𝑅)𝑧))
45	44	fveq2d 6699	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘((𝑥 + 𝑦) + 𝑧)) = (𝐹‘((𝑥 + 𝑦)(+_g‘𝑅)𝑧)))
46	43	oveqd 7208	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 + (𝑦 + 𝑧)) = (𝑥(+_g‘𝑅)(𝑦 + 𝑧)))
47	46	fveq2d 6699	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘(𝑥 + (𝑦 + 𝑧))) = (𝐹‘(𝑥(+_g‘𝑅)(𝑦 + 𝑧))))
48	42, 45, 47	3eqtr3d 2779	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘((𝑥 + 𝑦)(+_g‘𝑅)𝑧)) = (𝐹‘(𝑥(+_g‘𝑅)(𝑦 + 𝑧))))
49		simpl 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝜑)
50	19	3adant3r3 1186	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
51		simpr3 1198	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ 𝑉)
52	3, 15, 1, 2, 4, 16, 17	imasaddval 16991	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 + 𝑦) ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘(𝑥 + 𝑦))(+_g‘𝑈)(𝐹‘𝑧)) = (𝐹‘((𝑥 + 𝑦)(+_g‘𝑅)𝑧)))
53	49, 50, 51, 52	syl3anc 1373	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘(𝑥 + 𝑦))(+_g‘𝑈)(𝐹‘𝑧)) = (𝐹‘((𝑥 + 𝑦)(+_g‘𝑅)𝑧)))
54		simpr1 1196	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑥 ∈ 𝑉)
55	21	caovclg 7378	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
56	55	3adantr1 1171	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
57	3, 15, 1, 2, 4, 16, 17	imasaddval 16991	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ (𝑦 + 𝑧) ∈ 𝑉) → ((𝐹‘𝑥)(+_g‘𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥(+_g‘𝑅)(𝑦 + 𝑧))))
58	49, 54, 56, 57	syl3anc 1373	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(+_g‘𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥(+_g‘𝑅)(𝑦 + 𝑧))))
59	48, 53, 58	3eqtr4d 2781	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘(𝑥 + 𝑦))(+_g‘𝑈)(𝐹‘𝑧)) = ((𝐹‘𝑥)(+_g‘𝑈)(𝐹‘(𝑦 + 𝑧))))
60	3, 15, 1, 2, 4, 16, 17	imasaddval 16991	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ((𝐹‘𝑥)(+_g‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥(+_g‘𝑅)𝑦)))
61	60	3adant3r3 1186	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(+_g‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥(+_g‘𝑅)𝑦)))
62	43	oveqd 7208	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 + 𝑦) = (𝑥(+_g‘𝑅)𝑦))
63	62	fveq2d 6699	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝑥(+_g‘𝑅)𝑦)))
64	61, 63	eqtr4d 2774	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(+_g‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥 + 𝑦)))
65	64	oveq1d 7206	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (((𝐹‘𝑥)(+_g‘𝑈)(𝐹‘𝑦))(+_g‘𝑈)(𝐹‘𝑧)) = ((𝐹‘(𝑥 + 𝑦))(+_g‘𝑈)(𝐹‘𝑧)))
66	3, 15, 1, 2, 4, 16, 17	imasaddval 16991	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘𝑦)(+_g‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦(+_g‘𝑅)𝑧)))
67	66	3adant3r1 1184	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑦)(+_g‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦(+_g‘𝑅)𝑧)))
68	43	oveqd 7208	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 + 𝑧) = (𝑦(+_g‘𝑅)𝑧))
69	68	fveq2d 6699	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘(𝑦 + 𝑧)) = (𝐹‘(𝑦(+_g‘𝑅)𝑧)))
70	67, 69	eqtr4d 2774	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑦)(+_g‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦 + 𝑧)))
71	70	oveq2d 7207	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(+_g‘𝑈)((𝐹‘𝑦)(+_g‘𝑈)(𝐹‘𝑧))) = ((𝐹‘𝑥)(+_g‘𝑈)(𝐹‘(𝑦 + 𝑧))))
72	59, 65, 71	3eqtr4d 2781	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (((𝐹‘𝑥)(+_g‘𝑈)(𝐹‘𝑦))(+_g‘𝑈)(𝐹‘𝑧)) = ((𝐹‘𝑥)(+_g‘𝑈)((𝐹‘𝑦)(+_g‘𝑈)(𝐹‘𝑧))))
73		simp1 1138	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑥) = 𝑢)
74		simp2 1139	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑦) = 𝑣)
75	73, 74	oveq12d 7209	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑥)(+_g‘𝑈)(𝐹‘𝑦)) = (𝑢(+_g‘𝑈)𝑣))
76		simp3 1140	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑧) = 𝑤)
77	75, 76	oveq12d 7209	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (((𝐹‘𝑥)(+_g‘𝑈)(𝐹‘𝑦))(+_g‘𝑈)(𝐹‘𝑧)) = ((𝑢(+_g‘𝑈)𝑣)(+_g‘𝑈)𝑤))
78	74, 76	oveq12d 7209	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑦)(+_g‘𝑈)(𝐹‘𝑧)) = (𝑣(+_g‘𝑈)𝑤))
79	73, 78	oveq12d 7209	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑥)(+_g‘𝑈)((𝐹‘𝑦)(+_g‘𝑈)(𝐹‘𝑧))) = (𝑢(+_g‘𝑈)(𝑣(+_g‘𝑈)𝑤)))
80	77, 79	eqeq12d 2752	. . . . . . . . . 10 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((((𝐹‘𝑥)(+_g‘𝑈)(𝐹‘𝑦))(+_g‘𝑈)(𝐹‘𝑧)) = ((𝐹‘𝑥)(+_g‘𝑈)((𝐹‘𝑦)(+_g‘𝑈)(𝐹‘𝑧))) ↔ ((𝑢(+_g‘𝑈)𝑣)(+_g‘𝑈)𝑤) = (𝑢(+_g‘𝑈)(𝑣(+_g‘𝑈)𝑤))))
81	72, 80	syl5ibcom 248	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(+_g‘𝑈)𝑣)(+_g‘𝑈)𝑤) = (𝑢(+_g‘𝑈)(𝑣(+_g‘𝑈)𝑤))))
82	81	3exp2 1356	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑉 → (𝑧 ∈ 𝑉 → (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(+_g‘𝑈)𝑣)(+_g‘𝑈)𝑤) = (𝑢(+_g‘𝑈)(𝑣(+_g‘𝑈)𝑤)))))))
83	82	imp32 422	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑧 ∈ 𝑉 → (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(+_g‘𝑈)𝑣)(+_g‘𝑈)𝑤) = (𝑢(+_g‘𝑈)(𝑣(+_g‘𝑈)𝑤)))))
84	83	rexlimdv 3192	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (∃𝑧 ∈ 𝑉 ((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(+_g‘𝑈)𝑣)(+_g‘𝑈)𝑤) = (𝑢(+_g‘𝑈)(𝑣(+_g‘𝑈)𝑤))))
85	84	rexlimdvva 3203	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 ((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(+_g‘𝑈)𝑣)(+_g‘𝑈)𝑤) = (𝑢(+_g‘𝑈)(𝑣(+_g‘𝑈)𝑤))))
86	41, 85	sylbid 243	. . . 4 ⊢ (𝜑 → ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝑢(+_g‘𝑈)𝑣)(+_g‘𝑈)𝑤) = (𝑢(+_g‘𝑈)(𝑣(+_g‘𝑈)𝑤))))
87	86	imp 410	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢(+_g‘𝑈)𝑣)(+_g‘𝑈)𝑤) = (𝑢(+_g‘𝑈)(𝑣(+_g‘𝑈)𝑤)))
88		fof 6611	. . . . 5 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹:𝑉⟶𝐵)
89	3, 88	syl 17	. . . 4 ⊢ (𝜑 → 𝐹:𝑉⟶𝐵)
90		imasgrp2.3	. . . 4 ⊢ (𝜑 → 0 ∈ 𝑉)
91	89, 90	ffvelrnd 6883	. . 3 ⊢ (𝜑 → (𝐹‘ 0 ) ∈ 𝐵)
92	33, 34	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢))
93	28, 92	bitr3d 284	. . . . 5 ⊢ (𝜑 → (𝑢 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢))
94		simpl 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝜑)
95	90	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ∈ 𝑉)
96		simpr 488	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝑉)
97	3, 15, 1, 2, 4, 16, 17	imasaddval 16991	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘ 0 )(+_g‘𝑈)(𝐹‘𝑥)) = (𝐹‘( 0 (+_g‘𝑅)𝑥)))
98	94, 95, 96, 97	syl3anc 1373	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘ 0 )(+_g‘𝑈)(𝐹‘𝑥)) = (𝐹‘( 0 (+_g‘𝑅)𝑥)))
99	8	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → + = (+_g‘𝑅))
100	99	oveqd 7208	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ( 0 + 𝑥) = ( 0 (+_g‘𝑅)𝑥))
101	100	fveq2d 6699	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹‘( 0 (+_g‘𝑅)𝑥)))
102		imasgrp2.4	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹‘𝑥))
103	98, 101, 102	3eqtr2d 2777	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘ 0 )(+_g‘𝑈)(𝐹‘𝑥)) = (𝐹‘𝑥))
104		oveq2 7199	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑢 → ((𝐹‘ 0 )(+_g‘𝑈)(𝐹‘𝑥)) = ((𝐹‘ 0 )(+_g‘𝑈)𝑢))
105		id 22	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑢 → (𝐹‘𝑥) = 𝑢)
106	104, 105	eqeq12d 2752	. . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑢 → (((𝐹‘ 0 )(+_g‘𝑈)(𝐹‘𝑥)) = (𝐹‘𝑥) ↔ ((𝐹‘ 0 )(+_g‘𝑈)𝑢) = 𝑢))
107	103, 106	syl5ibcom 248	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘𝑥) = 𝑢 → ((𝐹‘ 0 )(+_g‘𝑈)𝑢) = 𝑢))
108	107	rexlimdva 3193	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢 → ((𝐹‘ 0 )(+_g‘𝑈)𝑢) = 𝑢))
109	93, 108	sylbid 243	. . . 4 ⊢ (𝜑 → (𝑢 ∈ 𝐵 → ((𝐹‘ 0 )(+_g‘𝑈)𝑢) = 𝑢))
110	109	imp 410	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵) → ((𝐹‘ 0 )(+_g‘𝑈)𝑢) = 𝑢)
111	89	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝐹:𝑉⟶𝐵)
112		imasgrp2.5	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑁 ∈ 𝑉)
113	111, 112	ffvelrnd 6883	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘𝑁) ∈ 𝐵)
114	3, 15, 1, 2, 4, 16, 17	imasaddval 16991	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘𝑁)(+_g‘𝑈)(𝐹‘𝑥)) = (𝐹‘(𝑁(+_g‘𝑅)𝑥)))
115	94, 112, 96, 114	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘𝑁)(+_g‘𝑈)(𝐹‘𝑥)) = (𝐹‘(𝑁(+_g‘𝑅)𝑥)))
116	99	oveqd 7208	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝑁 + 𝑥) = (𝑁(+_g‘𝑅)𝑥))
117	116	fveq2d 6699	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘(𝑁 + 𝑥)) = (𝐹‘(𝑁(+_g‘𝑅)𝑥)))
118		imasgrp2.6	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘(𝑁 + 𝑥)) = (𝐹‘ 0 ))
119	115, 117, 118	3eqtr2d 2777	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘𝑁)(+_g‘𝑈)(𝐹‘𝑥)) = (𝐹‘ 0 ))
120		oveq1 7198	. . . . . . . . . 10 ⊢ (𝑣 = (𝐹‘𝑁) → (𝑣(+_g‘𝑈)(𝐹‘𝑥)) = ((𝐹‘𝑁)(+_g‘𝑈)(𝐹‘𝑥)))
121	120	eqeq1d 2738	. . . . . . . . 9 ⊢ (𝑣 = (𝐹‘𝑁) → ((𝑣(+_g‘𝑈)(𝐹‘𝑥)) = (𝐹‘ 0 ) ↔ ((𝐹‘𝑁)(+_g‘𝑈)(𝐹‘𝑥)) = (𝐹‘ 0 )))
122	121	rspcev 3527	. . . . . . . 8 ⊢ (((𝐹‘𝑁) ∈ 𝐵 ∧ ((𝐹‘𝑁)(+_g‘𝑈)(𝐹‘𝑥)) = (𝐹‘ 0 )) → ∃𝑣 ∈ 𝐵 (𝑣(+_g‘𝑈)(𝐹‘𝑥)) = (𝐹‘ 0 ))
123	113, 119, 122	syl2anc 587	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ∃𝑣 ∈ 𝐵 (𝑣(+_g‘𝑈)(𝐹‘𝑥)) = (𝐹‘ 0 ))
124		oveq2 7199	. . . . . . . . 9 ⊢ ((𝐹‘𝑥) = 𝑢 → (𝑣(+_g‘𝑈)(𝐹‘𝑥)) = (𝑣(+_g‘𝑈)𝑢))
125	124	eqeq1d 2738	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑢 → ((𝑣(+_g‘𝑈)(𝐹‘𝑥)) = (𝐹‘ 0 ) ↔ (𝑣(+_g‘𝑈)𝑢) = (𝐹‘ 0 )))
126	125	rexbidv 3206	. . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑢 → (∃𝑣 ∈ 𝐵 (𝑣(+_g‘𝑈)(𝐹‘𝑥)) = (𝐹‘ 0 ) ↔ ∃𝑣 ∈ 𝐵 (𝑣(+_g‘𝑈)𝑢) = (𝐹‘ 0 )))
127	123, 126	syl5ibcom 248	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘𝑥) = 𝑢 → ∃𝑣 ∈ 𝐵 (𝑣(+_g‘𝑈)𝑢) = (𝐹‘ 0 )))
128	127	rexlimdva 3193	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢 → ∃𝑣 ∈ 𝐵 (𝑣(+_g‘𝑈)𝑢) = (𝐹‘ 0 )))
129	93, 128	sylbid 243	. . . 4 ⊢ (𝜑 → (𝑢 ∈ 𝐵 → ∃𝑣 ∈ 𝐵 (𝑣(+_g‘𝑈)𝑢) = (𝐹‘ 0 )))
130	129	imp 410	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵) → ∃𝑣 ∈ 𝐵 (𝑣(+_g‘𝑈)𝑢) = (𝐹‘ 0 ))
131	5, 6, 25, 87, 91, 110, 130	isgrpde 18342	. 2 ⊢ (𝜑 → 𝑈 ∈ Grp)
132	5, 6, 91, 110, 131	grpidd2 18359	. 2 ⊢ (𝜑 → (𝐹‘ 0 ) = (0_g‘𝑈))
133	131, 132	jca 515	1 ⊢ (𝜑 → (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0_g‘𝑈)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ∃wrex 3052 × cxp 5534 ran crn 5537 Fn wfn 6353 ⟶wf 6354 –onto→wfo 6356 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 +_gcplusg 16749 0_gc0g 16898 “_s cimas 16963 Grpcgrp 18319

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-sup 9036 df-inf 9037 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-fz 13061 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-plusg 16762 df-mulr 16763 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-ds 16771 df-0g 16900 df-imas 16967 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-grp 18322

This theorem is referenced by: imasgrp 18433 qusgrp2 18435

W3C validator