Theorem imasmnd2 17150

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 15995	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝑈))
6		eqidd 2611	. . 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 2610	. . . . 5 ⊢ (+g‘𝑈) = (+g‘𝑈)
10		imasmnd2.1	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 + 𝑦) ∈ 𝑉)
11	10	3expb 1258	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
12	11	caovclg 6724	. . . . 5 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 + 𝑞) ∈ 𝑉)
13	3, 7, 1, 2, 4, 8, 9, 12	imasaddf 16016	. . . 4 ⊢ (𝜑 → (+g‘𝑈):(𝐵 × 𝐵)⟶𝐵)
14		fovrn 6702	. . . 4 ⊢ (((+g‘𝑈):(𝐵 × 𝐵)⟶𝐵 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → (𝑢(+g‘𝑈)𝑣) ∈ 𝐵)
15	13, 14	syl3an1 1351	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → (𝑢(+g‘𝑈)𝑣) ∈ 𝐵)
16		forn 6031	. . . . . . . . . 10 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵)
17	3, 16	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐹 = 𝐵)
18	17	eleq2d 2673	. . . . . . . 8 ⊢ (𝜑 → (𝑢 ∈ ran 𝐹 ↔ 𝑢 ∈ 𝐵))
19	17	eleq2d 2673	. . . . . . . 8 ⊢ (𝜑 → (𝑣 ∈ ran 𝐹 ↔ 𝑣 ∈ 𝐵))
20	17	eleq2d 2673	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ ran 𝐹 ↔ 𝑤 ∈ 𝐵))
21	18, 19, 20	3anbi123d 1391	. . . . . . 7 ⊢ (𝜑 → ((𝑢 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ∧ 𝑤 ∈ ran 𝐹) ↔ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)))
22		fofn 6030	. . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹 Fn 𝑉)
23	3, 22	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝑉)
24		fvelrnb 6153	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑉 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢))
25		fvelrnb 6153	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑉 → (𝑣 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣))
26		fvelrnb 6153	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑉 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤))
27	24, 25, 26	3anbi123d 1391	. . . . . . . 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 269	. . . . . 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 3087	. . . . . 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 277	. . . . 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 472	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝜑)
34	10	3adant3r3 1268	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
35		simpr3 1062	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ 𝑉)
36	3, 7, 1, 2, 4, 8, 9	imasaddval 16015	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 + 𝑦) ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘(𝑥 + 𝑦))(+g‘𝑈)(𝐹‘𝑧)) = (𝐹‘((𝑥 + 𝑦) + 𝑧)))
37	33, 34, 35, 36	syl3anc 1318	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘(𝑥 + 𝑦))(+g‘𝑈)(𝐹‘𝑧)) = (𝐹‘((𝑥 + 𝑦) + 𝑧)))
38		simpr1 1060	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑥 ∈ 𝑉)
39	12	caovclg 6724	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
40	39	3adantr1 1213	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
41	3, 7, 1, 2, 4, 8, 9	imasaddval 16015	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ (𝑦 + 𝑧) ∈ 𝑉) → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥 + (𝑦 + 𝑧))))
42	33, 38, 40, 41	syl3anc 1318	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥 + (𝑦 + 𝑧))))
43	32, 37, 42	3eqtr4d 2654	. . . . . . . . . . 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 16015	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥 + 𝑦)))
45	44	3adant3r3 1268	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥 + 𝑦)))
46	45	oveq1d 6564	. . . . . . . . . . 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 16015	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘𝑦)(+g‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦 + 𝑧)))
48	47	3adant3r1 1266	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑦)(+g‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦 + 𝑧)))
49	48	oveq2d 6565	. . . . . . . . . . 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 2654	. . . . . . . . . 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 1054	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑥) = 𝑢)
52		simp2 1055	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑦) = 𝑣)
53	51, 52	oveq12d 6567	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)) = (𝑢(+g‘𝑈)𝑣))
54		simp3 1056	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑧) = 𝑤)
55	53, 54	oveq12d 6567	. . . . . . . . . . 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 6567	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑦)(+g‘𝑈)(𝐹‘𝑧)) = (𝑣(+g‘𝑈)𝑤))
57	51, 56	oveq12d 6567	. . . . . . . . . . 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 2625	. . . . . . . . . 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 234	. . . . . . . . 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 1277	. . . . . . . 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 448	. . . . . . 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 3012	. . . . . 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 3020	. . . . 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 229	. . . 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 444	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢(+g‘𝑈)𝑣)(+g‘𝑈)𝑤) = (𝑢(+g‘𝑈)(𝑣(+g‘𝑈)𝑤)))
66		fof 6028	. . . . 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 6268	. . 3 ⊢ (𝜑 → (𝐹‘ 0 ) ∈ 𝐵)
70	23, 24	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢))
71	18, 70	bitr3d 269	. . . . 5 ⊢ (𝜑 → (𝑢 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢))
72		simpl 472	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝜑)
73	68	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ∈ 𝑉)
74		simpr 476	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝑉)
75	3, 7, 1, 2, 4, 8, 9	imasaddval 16015	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘ 0 )(+g‘𝑈)(𝐹‘𝑥)) = (𝐹‘( 0 + 𝑥)))
76	72, 73, 74, 75	syl3anc 1318	. . . . . . . 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 2644	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘ 0 )(+g‘𝑈)(𝐹‘𝑥)) = (𝐹‘𝑥))
79		oveq2 6557	. . . . . . . 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 2625	. . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑢 → (((𝐹‘ 0 )(+g‘𝑈)(𝐹‘𝑥)) = (𝐹‘𝑥) ↔ ((𝐹‘ 0 )(+g‘𝑈)𝑢) = 𝑢))
82	78, 81	syl5ibcom 234	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘𝑥) = 𝑢 → ((𝐹‘ 0 )(+g‘𝑈)𝑢) = 𝑢))
83	82	rexlimdva 3013	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢 → ((𝐹‘ 0 )(+g‘𝑈)𝑢) = 𝑢))
84	71, 83	sylbid 229	. . . 4 ⊢ (𝜑 → (𝑢 ∈ 𝐵 → ((𝐹‘ 0 )(+g‘𝑈)𝑢) = 𝑢))
85	84	imp 444	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵) → ((𝐹‘ 0 )(+g‘𝑈)𝑢) = 𝑢)
86	3, 7, 1, 2, 4, 8, 9	imasaddval 16015	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 0 ∈ 𝑉) → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘ 0 )) = (𝐹‘(𝑥 + 0 )))
87	73, 86	mpd3an3 1417	. . . . . . . 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 2644	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘ 0 )) = (𝐹‘𝑥))
90		oveq1 6556	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑢 → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘ 0 )) = (𝑢(+g‘𝑈)(𝐹‘ 0 )))
91	90, 80	eqeq12d 2625	. . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑢 → (((𝐹‘𝑥)(+g‘𝑈)(𝐹‘ 0 )) = (𝐹‘𝑥) ↔ (𝑢(+g‘𝑈)(𝐹‘ 0 )) = 𝑢))
92	89, 91	syl5ibcom 234	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘𝑥) = 𝑢 → (𝑢(+g‘𝑈)(𝐹‘ 0 )) = 𝑢))
93	92	rexlimdva 3013	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢 → (𝑢(+g‘𝑈)(𝐹‘ 0 )) = 𝑢))
94	71, 93	sylbid 229	. . . 4 ⊢ (𝜑 → (𝑢 ∈ 𝐵 → (𝑢(+g‘𝑈)(𝐹‘ 0 )) = 𝑢))
95	94	imp 444	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵) → (𝑢(+g‘𝑈)(𝐹‘ 0 )) = 𝑢)
96	5, 6, 15, 65, 69, 85, 95	ismndd 17136	. 2 ⊢ (𝜑 → 𝑈 ∈ Mnd)
97	5, 6, 69, 85, 95	grpidd 17091	. 2 ⊢ (𝜑 → (𝐹‘ 0 ) = (0g‘𝑈))
98	96, 97	jca 553	1 ⊢ (𝜑 → (𝑈 ∈ Mnd ∧ (𝐹‘ 0 ) = (0g‘𝑈)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   × cxp 5036  ran crn 5039   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +_gcplusg 15768  0_gc0g 15923   “_s cimas 15987  Mndcmnd 17117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-0g 15925  df-imas 15991  df-mgm 17065  df-sgrp 17107  df-mnd 17118

This theorem is referenced by:  imasmnd  17151