Theorem imasmnd2 18752

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

Hypotheses

Ref	Expression
imasmnd.u	⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))
imasmnd.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
imasmnd.p	⊢ + = (+g‘𝑅)
imasmnd.f	⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
imasmnd.e	⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
imasmnd2.r	⊢ (𝜑 → 𝑅 ∈ 𝑊)
imasmnd2.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 + 𝑦) ∈ 𝑉)
imasmnd2.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘((𝑥 + 𝑦) + 𝑧)) = (𝐹‘(𝑥 + (𝑦 + 𝑧))))
imasmnd2.3	⊢ (𝜑 → 0 ∈ 𝑉)
imasmnd2.4	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹‘𝑥))
imasmnd2.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘(𝑥 + 0 )) = (𝐹‘𝑥))

Assertion

Ref	Expression
imasmnd2	⊢ (𝜑 → (𝑈 ∈ Mnd ∧ (𝐹‘ 0 ) = (0g‘𝑈)))

Distinct variable groups:   𝑞,𝑝,𝑥,𝑦, +   𝑎,𝑏,𝑝,𝑞,𝑥,𝑦,𝑧,𝜑   𝑈,𝑎,𝑏,𝑝,𝑞,𝑥,𝑦,𝑧   0 ,𝑝,𝑞,𝑥   𝐵,𝑝,𝑞   𝐹,𝑎,𝑏,𝑝,𝑞,𝑥,𝑦,𝑧   𝑅,𝑝,𝑞   𝑉,𝑎,𝑏,𝑝,𝑞,𝑥,𝑦,𝑧

Allowed substitution hints:   𝐵(𝑥,𝑦,𝑧,𝑎,𝑏)   + (𝑧,𝑎,𝑏)   𝑅(𝑥,𝑦,𝑧,𝑎,𝑏)   𝑊(𝑥,𝑦,𝑧,𝑞,𝑝,𝑎,𝑏)   0 (𝑦,𝑧,𝑎,𝑏)

Proof of Theorem imasmnd2

Dummy variables 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imasmnd.u	. . . 4 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))
2		imasmnd.v	. . . 4 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
3		imasmnd.f	. . . 4 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
4		imasmnd2.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
5	1, 2, 3, 4	imasbas 17526	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝑈))
6		eqidd 2736	. . 3 ⊢ (𝜑 → (+g‘𝑈) = (+g‘𝑈))
7		imasmnd.e	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
8		imasmnd.p	. . . . 5 ⊢ + = (+g‘𝑅)
9		eqid 2735	. . . . 5 ⊢ (+g‘𝑈) = (+g‘𝑈)
10		imasmnd2.1	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 + 𝑦) ∈ 𝑉)
11	10	3expb 1120	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
12	11	caovclg 7599	. . . . 5 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 + 𝑞) ∈ 𝑉)
13	3, 7, 1, 2, 4, 8, 9, 12	imasaddf 17547	. . . 4 ⊢ (𝜑 → (+g‘𝑈):(𝐵 × 𝐵)⟶𝐵)
14		fovcdm 7577	. . . 4 ⊢ (((+g‘𝑈):(𝐵 × 𝐵)⟶𝐵 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → (𝑢(+g‘𝑈)𝑣) ∈ 𝐵)
15	13, 14	syl3an1 1163	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → (𝑢(+g‘𝑈)𝑣) ∈ 𝐵)
16		forn 6793	. . . . . . . . . 10 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵)
17	3, 16	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐹 = 𝐵)
18	17	eleq2d 2820	. . . . . . . 8 ⊢ (𝜑 → (𝑢 ∈ ran 𝐹 ↔ 𝑢 ∈ 𝐵))
19	17	eleq2d 2820	. . . . . . . 8 ⊢ (𝜑 → (𝑣 ∈ ran 𝐹 ↔ 𝑣 ∈ 𝐵))
20	17	eleq2d 2820	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ ran 𝐹 ↔ 𝑤 ∈ 𝐵))
21	18, 19, 20	3anbi123d 1438	. . . . . . 7 ⊢ (𝜑 → ((𝑢 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ∧ 𝑤 ∈ ran 𝐹) ↔ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)))
22		fofn 6792	. . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹 Fn 𝑉)
23	3, 22	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝑉)
24		fvelrnb 6939	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑉 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢))
25		fvelrnb 6939	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑉 → (𝑣 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣))
26		fvelrnb 6939	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑉 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤))
27	24, 25, 26	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)))
28	23, 27	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)))
29	21, 28	bitr3d 281	. . . . . 6 ⊢ (𝜑 → ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ↔ (∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢 ∧ ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣 ∧ ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤)))
30		3reeanv 3214	. . . . . 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))
31	29, 30	bitr4di 289	. . . . 5 ⊢ (𝜑 → ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ↔ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 ((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤)))
32		imasmnd2.2	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘((𝑥 + 𝑦) + 𝑧)) = (𝐹‘(𝑥 + (𝑦 + 𝑧))))
33		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝜑)
34	10	3adant3r3 1185	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
35		simpr3 1197	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ 𝑉)
36	3, 7, 1, 2, 4, 8, 9	imasaddval 17546	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 + 𝑦) ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘(𝑥 + 𝑦))(+g‘𝑈)(𝐹‘𝑧)) = (𝐹‘((𝑥 + 𝑦) + 𝑧)))
37	33, 34, 35, 36	syl3anc 1373	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘(𝑥 + 𝑦))(+g‘𝑈)(𝐹‘𝑧)) = (𝐹‘((𝑥 + 𝑦) + 𝑧)))
38		simpr1 1195	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑥 ∈ 𝑉)
39	12	caovclg 7599	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
40	39	3adantr1 1170	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
41	3, 7, 1, 2, 4, 8, 9	imasaddval 17546	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ (𝑦 + 𝑧) ∈ 𝑉) → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥 + (𝑦 + 𝑧))))
42	33, 38, 40, 41	syl3anc 1373	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥 + (𝑦 + 𝑧))))
43	32, 37, 42	3eqtr4d 2780	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘(𝑥 + 𝑦))(+g‘𝑈)(𝐹‘𝑧)) = ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘(𝑦 + 𝑧))))
44	3, 7, 1, 2, 4, 8, 9	imasaddval 17546	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥 + 𝑦)))
45	44	3adant3r3 1185	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥 + 𝑦)))
46	45	oveq1d 7420	. . . . . . . . . . 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)))
47	3, 7, 1, 2, 4, 8, 9	imasaddval 17546	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘𝑦)(+g‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦 + 𝑧)))
48	47	3adant3r1 1183	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑦)(+g‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦 + 𝑧)))
49	48	oveq2d 7421	. . . . . . . . . . 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))))
50	43, 46, 49	3eqtr4d 2780	. . . . . . . . . 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))))
51		simp1 1136	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑥) = 𝑢)
52		simp2 1137	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑦) = 𝑣)
53	51, 52	oveq12d 7423	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)) = (𝑢(+g‘𝑈)𝑣))
54		simp3 1138	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑧) = 𝑤)
55	53, 54	oveq12d 7423	. . . . . . . . . . 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))
56	52, 54	oveq12d 7423	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑦)(+g‘𝑈)(𝐹‘𝑧)) = (𝑣(+g‘𝑈)𝑤))
57	51, 56	oveq12d 7423	. . . . . . . . . . 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)))
58	55, 57	eqeq12d 2751	. . . . . . . . . 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))))
59	50, 58	syl5ibcom 245	. . . . . . . . 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))))
60	59	3exp2 1355	. . . . . . . 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)))))))
61	60	imp32 418	. . . . . . 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)))))
62	61	rexlimdv 3139	. . . . . 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))))
63	62	rexlimdvva 3198	. . . . 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))))
64	31, 63	sylbid 240	. . . 4 ⊢ (𝜑 → ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝑢(+g‘𝑈)𝑣)(+g‘𝑈)𝑤) = (𝑢(+g‘𝑈)(𝑣(+g‘𝑈)𝑤))))
65	64	imp 406	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢(+g‘𝑈)𝑣)(+g‘𝑈)𝑤) = (𝑢(+g‘𝑈)(𝑣(+g‘𝑈)𝑤)))
66		fof 6790	. . . . 5 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹:𝑉⟶𝐵)
67	3, 66	syl 17	. . . 4 ⊢ (𝜑 → 𝐹:𝑉⟶𝐵)
68		imasmnd2.3	. . . 4 ⊢ (𝜑 → 0 ∈ 𝑉)
69	67, 68	ffvelcdmd 7075	. . 3 ⊢ (𝜑 → (𝐹‘ 0 ) ∈ 𝐵)
70	23, 24	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢))
71	18, 70	bitr3d 281	. . . . 5 ⊢ (𝜑 → (𝑢 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢))
72		simpl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝜑)
73	68	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ∈ 𝑉)
74		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝑉)
75	3, 7, 1, 2, 4, 8, 9	imasaddval 17546	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘ 0 )(+g‘𝑈)(𝐹‘𝑥)) = (𝐹‘( 0 + 𝑥)))
76	72, 73, 74, 75	syl3anc 1373	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘ 0 )(+g‘𝑈)(𝐹‘𝑥)) = (𝐹‘( 0 + 𝑥)))
77		imasmnd2.4	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹‘𝑥))
78	76, 77	eqtrd 2770	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘ 0 )(+g‘𝑈)(𝐹‘𝑥)) = (𝐹‘𝑥))
79		oveq2 7413	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑢 → ((𝐹‘ 0 )(+g‘𝑈)(𝐹‘𝑥)) = ((𝐹‘ 0 )(+g‘𝑈)𝑢))
80		id 22	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑢 → (𝐹‘𝑥) = 𝑢)
81	79, 80	eqeq12d 2751	. . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑢 → (((𝐹‘ 0 )(+g‘𝑈)(𝐹‘𝑥)) = (𝐹‘𝑥) ↔ ((𝐹‘ 0 )(+g‘𝑈)𝑢) = 𝑢))
82	78, 81	syl5ibcom 245	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘𝑥) = 𝑢 → ((𝐹‘ 0 )(+g‘𝑈)𝑢) = 𝑢))
83	82	rexlimdva 3141	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢 → ((𝐹‘ 0 )(+g‘𝑈)𝑢) = 𝑢))
84	71, 83	sylbid 240	. . . 4 ⊢ (𝜑 → (𝑢 ∈ 𝐵 → ((𝐹‘ 0 )(+g‘𝑈)𝑢) = 𝑢))
85	84	imp 406	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵) → ((𝐹‘ 0 )(+g‘𝑈)𝑢) = 𝑢)
86	3, 7, 1, 2, 4, 8, 9	imasaddval 17546	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 0 ∈ 𝑉) → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘ 0 )) = (𝐹‘(𝑥 + 0 )))
87	73, 86	mpd3an3 1464	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘ 0 )) = (𝐹‘(𝑥 + 0 )))
88		imasmnd2.5	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘(𝑥 + 0 )) = (𝐹‘𝑥))
89	87, 88	eqtrd 2770	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘ 0 )) = (𝐹‘𝑥))
90		oveq1 7412	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑢 → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘ 0 )) = (𝑢(+g‘𝑈)(𝐹‘ 0 )))
91	90, 80	eqeq12d 2751	. . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑢 → (((𝐹‘𝑥)(+g‘𝑈)(𝐹‘ 0 )) = (𝐹‘𝑥) ↔ (𝑢(+g‘𝑈)(𝐹‘ 0 )) = 𝑢))
92	89, 91	syl5ibcom 245	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘𝑥) = 𝑢 → (𝑢(+g‘𝑈)(𝐹‘ 0 )) = 𝑢))
93	92	rexlimdva 3141	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢 → (𝑢(+g‘𝑈)(𝐹‘ 0 )) = 𝑢))
94	71, 93	sylbid 240	. . . 4 ⊢ (𝜑 → (𝑢 ∈ 𝐵 → (𝑢(+g‘𝑈)(𝐹‘ 0 )) = 𝑢))
95	94	imp 406	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵) → (𝑢(+g‘𝑈)(𝐹‘ 0 )) = 𝑢)
96	5, 6, 15, 65, 69, 85, 95	ismndd 18734	. 2 ⊢ (𝜑 → 𝑈 ∈ Mnd)
97	5, 6, 69, 85, 95	grpidd 18649	. 2 ⊢ (𝜑 → (𝐹‘ 0 ) = (0g‘𝑈))
98	96, 97	jca 511	1 ⊢ (𝜑 → (𝑈 ∈ Mnd ∧ (𝐹‘ 0 ) = (0g‘𝑈)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ∃wrex 3060   × cxp 5652  ran crn 5655   Fn wfn 6526  ⟶wf 6527  –onto→wfo 6529  ‘cfv 6531  (class class class)co 7405  Basecbs 17228  +_gcplusg 17271  0_gc0g 17453   “_s cimas 17518  Mndcmnd 18712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-sup 9454  df-inf 9455  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12502  df-z 12589  df-dec 12709  df-uz 12853  df-fz 13525  df-struct 17166  df-slot 17201  df-ndx 17213  df-base 17229  df-plusg 17284  df-mulr 17285  df-sca 17287  df-vsca 17288  df-ip 17289  df-tset 17290  df-ple 17291  df-ds 17293  df-0g 17455  df-imas 17522  df-mgm 18618  df-sgrp 18697  df-mnd 18713

This theorem is referenced by:  imasmnd  18753