Theorem imasmnd2 17679

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 16524	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝑈))
6		eqidd 2825	. . 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 2824	. . . . 5 ⊢ (+g‘𝑈) = (+g‘𝑈)
10		imasmnd2.1	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 + 𝑦) ∈ 𝑉)
11	10	3expb 1155	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
12	11	caovclg 7085	. . . . 5 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 + 𝑞) ∈ 𝑉)
13	3, 7, 1, 2, 4, 8, 9, 12	imasaddf 16545	. . . 4 ⊢ (𝜑 → (+g‘𝑈):(𝐵 × 𝐵)⟶𝐵)
14		fovrn 7063	. . . 4 ⊢ (((+g‘𝑈):(𝐵 × 𝐵)⟶𝐵 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → (𝑢(+g‘𝑈)𝑣) ∈ 𝐵)
15	13, 14	syl3an1 1208	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → (𝑢(+g‘𝑈)𝑣) ∈ 𝐵)
16		forn 6355	. . . . . . . . . 10 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵)
17	3, 16	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐹 = 𝐵)
18	17	eleq2d 2891	. . . . . . . 8 ⊢ (𝜑 → (𝑢 ∈ ran 𝐹 ↔ 𝑢 ∈ 𝐵))
19	17	eleq2d 2891	. . . . . . . 8 ⊢ (𝜑 → (𝑣 ∈ ran 𝐹 ↔ 𝑣 ∈ 𝐵))
20	17	eleq2d 2891	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ ran 𝐹 ↔ 𝑤 ∈ 𝐵))
21	18, 19, 20	3anbi123d 1566	. . . . . . 7 ⊢ (𝜑 → ((𝑢 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ∧ 𝑤 ∈ ran 𝐹) ↔ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)))
22		fofn 6354	. . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹 Fn 𝑉)
23	3, 22	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝑉)
24		fvelrnb 6489	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑉 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢))
25		fvelrnb 6489	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑉 → (𝑣 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣))
26		fvelrnb 6489	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑉 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤))
27	24, 25, 26	3anbi123d 1566	. . . . . . . 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 273	. . . . . 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 3317	. . . . . 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	syl6bbr 281	. . . . 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 476	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝜑)
34	10	3adant3r3 1241	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
35		simpr3 1258	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ 𝑉)
36	3, 7, 1, 2, 4, 8, 9	imasaddval 16544	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 + 𝑦) ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘(𝑥 + 𝑦))(+g‘𝑈)(𝐹‘𝑧)) = (𝐹‘((𝑥 + 𝑦) + 𝑧)))
37	33, 34, 35, 36	syl3anc 1496	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘(𝑥 + 𝑦))(+g‘𝑈)(𝐹‘𝑧)) = (𝐹‘((𝑥 + 𝑦) + 𝑧)))
38		simpr1 1254	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑥 ∈ 𝑉)
39	12	caovclg 7085	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
40	39	3adantr1 1216	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
41	3, 7, 1, 2, 4, 8, 9	imasaddval 16544	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ (𝑦 + 𝑧) ∈ 𝑉) → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥 + (𝑦 + 𝑧))))
42	33, 38, 40, 41	syl3anc 1496	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥 + (𝑦 + 𝑧))))
43	32, 37, 42	3eqtr4d 2870	. . . . . . . . . . 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 16544	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥 + 𝑦)))
45	44	3adant3r3 1241	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥 + 𝑦)))
46	45	oveq1d 6919	. . . . . . . . . . 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 16544	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘𝑦)(+g‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦 + 𝑧)))
48	47	3adant3r1 1239	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑦)(+g‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦 + 𝑧)))
49	48	oveq2d 6920	. . . . . . . . . . 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 2870	. . . . . . . . . 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 1172	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑥) = 𝑢)
52		simp2 1173	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑦) = 𝑣)
53	51, 52	oveq12d 6922	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)) = (𝑢(+g‘𝑈)𝑣))
54		simp3 1174	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑧) = 𝑤)
55	53, 54	oveq12d 6922	. . . . . . . . . . 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 6922	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑦)(+g‘𝑈)(𝐹‘𝑧)) = (𝑣(+g‘𝑈)𝑤))
57	51, 56	oveq12d 6922	. . . . . . . . . . 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 2839	. . . . . . . . . 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 237	. . . . . . . . 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 1469	. . . . . . . 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 411	. . . . . . 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 3238	. . . . . 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 3247	. . . . 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 232	. . . 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 397	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢(+g‘𝑈)𝑣)(+g‘𝑈)𝑤) = (𝑢(+g‘𝑈)(𝑣(+g‘𝑈)𝑤)))
66		fof 6352	. . . . 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	ffvelrnd 6608	. . 3 ⊢ (𝜑 → (𝐹‘ 0 ) ∈ 𝐵)
70	23, 24	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢))
71	18, 70	bitr3d 273	. . . . 5 ⊢ (𝜑 → (𝑢 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢))
72		simpl 476	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝜑)
73	68	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ∈ 𝑉)
74		simpr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝑉)
75	3, 7, 1, 2, 4, 8, 9	imasaddval 16544	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘ 0 )(+g‘𝑈)(𝐹‘𝑥)) = (𝐹‘( 0 + 𝑥)))
76	72, 73, 74, 75	syl3anc 1496	. . . . . . . 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 2860	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘ 0 )(+g‘𝑈)(𝐹‘𝑥)) = (𝐹‘𝑥))
79		oveq2 6912	. . . . . . . 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 2839	. . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑢 → (((𝐹‘ 0 )(+g‘𝑈)(𝐹‘𝑥)) = (𝐹‘𝑥) ↔ ((𝐹‘ 0 )(+g‘𝑈)𝑢) = 𝑢))
82	78, 81	syl5ibcom 237	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘𝑥) = 𝑢 → ((𝐹‘ 0 )(+g‘𝑈)𝑢) = 𝑢))
83	82	rexlimdva 3239	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢 → ((𝐹‘ 0 )(+g‘𝑈)𝑢) = 𝑢))
84	71, 83	sylbid 232	. . . 4 ⊢ (𝜑 → (𝑢 ∈ 𝐵 → ((𝐹‘ 0 )(+g‘𝑈)𝑢) = 𝑢))
85	84	imp 397	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵) → ((𝐹‘ 0 )(+g‘𝑈)𝑢) = 𝑢)
86	3, 7, 1, 2, 4, 8, 9	imasaddval 16544	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 0 ∈ 𝑉) → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘ 0 )) = (𝐹‘(𝑥 + 0 )))
87	73, 86	mpd3an3 1592	. . . . . . . 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 2860	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘ 0 )) = (𝐹‘𝑥))
90		oveq1 6911	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑢 → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘ 0 )) = (𝑢(+g‘𝑈)(𝐹‘ 0 )))
91	90, 80	eqeq12d 2839	. . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑢 → (((𝐹‘𝑥)(+g‘𝑈)(𝐹‘ 0 )) = (𝐹‘𝑥) ↔ (𝑢(+g‘𝑈)(𝐹‘ 0 )) = 𝑢))
92	89, 91	syl5ibcom 237	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘𝑥) = 𝑢 → (𝑢(+g‘𝑈)(𝐹‘ 0 )) = 𝑢))
93	92	rexlimdva 3239	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢 → (𝑢(+g‘𝑈)(𝐹‘ 0 )) = 𝑢))
94	71, 93	sylbid 232	. . . 4 ⊢ (𝜑 → (𝑢 ∈ 𝐵 → (𝑢(+g‘𝑈)(𝐹‘ 0 )) = 𝑢))
95	94	imp 397	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵) → (𝑢(+g‘𝑈)(𝐹‘ 0 )) = 𝑢)
96	5, 6, 15, 65, 69, 85, 95	ismndd 17665	. 2 ⊢ (𝜑 → 𝑈 ∈ Mnd)
97	5, 6, 69, 85, 95	grpidd 17620	. 2 ⊢ (𝜑 → (𝐹‘ 0 ) = (0g‘𝑈))
98	96, 97	jca 509	1 ⊢ (𝜑 → (𝑈 ∈ Mnd ∧ (𝐹‘ 0 ) = (0g‘𝑈)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1113   = wceq 1658   ∈ wcel 2166  ∃wrex 3117   × cxp 5339  ran crn 5342   Fn wfn 6117  ⟶wf 6118  –onto→wfo 6120  ‘cfv 6122  (class class class)co 6904  Basecbs 16221  +_gcplusg 16304  0_gc0g 16452   “_s cimas 16516  Mndcmnd 17646

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-rep 4993  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208  ax-cnex 10307  ax-resscn 10308  ax-1cn 10309  ax-icn 10310  ax-addcl 10311  ax-addrcl 10312  ax-mulcl 10313  ax-mulrcl 10314  ax-mulcom 10315  ax-addass 10316  ax-mulass 10317  ax-distr 10318  ax-i2m1 10319  ax-1ne0 10320  ax-1rid 10321  ax-rnegex 10322  ax-rrecex 10323  ax-cnre 10324  ax-pre-lttri 10325  ax-pre-lttrn 10326  ax-pre-ltadd 10327  ax-pre-mulgt0 10328

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-nel 3102  df-ral 3121  df-rex 3122  df-reu 3123  df-rmo 3124  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-pss 3813  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-tp 4401  df-op 4403  df-uni 4658  df-int 4697  df-iun 4741  df-br 4873  df-opab 4935  df-mpt 4952  df-tr 4975  df-id 5249  df-eprel 5254  df-po 5262  df-so 5263  df-fr 5300  df-we 5302  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-pred 5919  df-ord 5965  df-on 5966  df-lim 5967  df-suc 5968  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-fv 6130  df-riota 6865  df-ov 6907  df-oprab 6908  df-mpt2 6909  df-om 7326  df-1st 7427  df-2nd 7428  df-wrecs 7671  df-recs 7733  df-rdg 7771  df-1o 7825  df-oadd 7829  df-er 8008  df-en 8222  df-dom 8223  df-sdom 8224  df-fin 8225  df-sup 8616  df-inf 8617  df-pnf 10392  df-mnf 10393  df-xr 10394  df-ltxr 10395  df-le 10396  df-sub 10586  df-neg 10587  df-nn 11350  df-2 11413  df-3 11414  df-4 11415  df-5 11416  df-6 11417  df-7 11418  df-8 11419  df-9 11420  df-n0 11618  df-z 11704  df-dec 11821  df-uz 11968  df-fz 12619  df-struct 16223  df-ndx 16224  df-slot 16225  df-base 16227  df-plusg 16317  df-mulr 16318  df-sca 16320  df-vsca 16321  df-ip 16322  df-tset 16323  df-ple 16324  df-ds 16326  df-0g 16454  df-imas 16520  df-mgm 17594  df-sgrp 17636  df-mnd 17647

This theorem is referenced by:  imasmnd  17680