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Theorem imasmnd2 17595
Description: The image structure of a monoid is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
Hypotheses
Ref Expression
imasmnd.u (𝜑𝑈 = (𝐹s 𝑅))
imasmnd.v (𝜑𝑉 = (Base‘𝑅))
imasmnd.p + = (+g𝑅)
imasmnd.f (𝜑𝐹:𝑉onto𝐵)
imasmnd.e ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
imasmnd2.r (𝜑𝑅𝑊)
imasmnd2.1 ((𝜑𝑥𝑉𝑦𝑉) → (𝑥 + 𝑦) ∈ 𝑉)
imasmnd2.2 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘((𝑥 + 𝑦) + 𝑧)) = (𝐹‘(𝑥 + (𝑦 + 𝑧))))
imasmnd2.3 (𝜑0𝑉)
imasmnd2.4 ((𝜑𝑥𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹𝑥))
imasmnd2.5 ((𝜑𝑥𝑉) → (𝐹‘(𝑥 + 0 )) = (𝐹𝑥))
Assertion
Ref Expression
imasmnd2 (𝜑 → (𝑈 ∈ Mnd ∧ (𝐹0 ) = (0g𝑈)))
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.
StepHypRef Expression
1 imasmnd.u . . . 4 (𝜑𝑈 = (𝐹s 𝑅))
2 imasmnd.v . . . 4 (𝜑𝑉 = (Base‘𝑅))
3 imasmnd.f . . . 4 (𝜑𝐹:𝑉onto𝐵)
4 imasmnd2.r . . . 4 (𝜑𝑅𝑊)
51, 2, 3, 4imasbas 16440 . . 3 (𝜑𝐵 = (Base‘𝑈))
6 eqidd 2766 . . 3 (𝜑 → (+g𝑈) = (+g𝑈))
7 imasmnd.e . . . . 5 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
8 imasmnd.p . . . . 5 + = (+g𝑅)
9 eqid 2765 . . . . 5 (+g𝑈) = (+g𝑈)
10 imasmnd2.1 . . . . . . 7 ((𝜑𝑥𝑉𝑦𝑉) → (𝑥 + 𝑦) ∈ 𝑉)
11103expb 1149 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
1211caovclg 7024 . . . . 5 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝 + 𝑞) ∈ 𝑉)
133, 7, 1, 2, 4, 8, 9, 12imasaddf 16461 . . . 4 (𝜑 → (+g𝑈):(𝐵 × 𝐵)⟶𝐵)
14 fovrn 7002 . . . 4 (((+g𝑈):(𝐵 × 𝐵)⟶𝐵𝑢𝐵𝑣𝐵) → (𝑢(+g𝑈)𝑣) ∈ 𝐵)
1513, 14syl3an1 1202 . . 3 ((𝜑𝑢𝐵𝑣𝐵) → (𝑢(+g𝑈)𝑣) ∈ 𝐵)
16 forn 6301 . . . . . . . . . 10 (𝐹:𝑉onto𝐵 → ran 𝐹 = 𝐵)
173, 16syl 17 . . . . . . . . 9 (𝜑 → ran 𝐹 = 𝐵)
1817eleq2d 2830 . . . . . . . 8 (𝜑 → (𝑢 ∈ ran 𝐹𝑢𝐵))
1917eleq2d 2830 . . . . . . . 8 (𝜑 → (𝑣 ∈ ran 𝐹𝑣𝐵))
2017eleq2d 2830 . . . . . . . 8 (𝜑 → (𝑤 ∈ ran 𝐹𝑤𝐵))
2118, 19, 203anbi123d 1560 . . . . . . 7 (𝜑 → ((𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐹𝑤 ∈ ran 𝐹) ↔ (𝑢𝐵𝑣𝐵𝑤𝐵)))
22 fofn 6300 . . . . . . . . 9 (𝐹:𝑉onto𝐵𝐹 Fn 𝑉)
233, 22syl 17 . . . . . . . 8 (𝜑𝐹 Fn 𝑉)
24 fvelrnb 6432 . . . . . . . . 9 (𝐹 Fn 𝑉 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥𝑉 (𝐹𝑥) = 𝑢))
25 fvelrnb 6432 . . . . . . . . 9 (𝐹 Fn 𝑉 → (𝑣 ∈ ran 𝐹 ↔ ∃𝑦𝑉 (𝐹𝑦) = 𝑣))
26 fvelrnb 6432 . . . . . . . . 9 (𝐹 Fn 𝑉 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑧𝑉 (𝐹𝑧) = 𝑤))
2724, 25, 263anbi123d 1560 . . . . . . . 8 (𝐹 Fn 𝑉 → ((𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐹𝑤 ∈ ran 𝐹) ↔ (∃𝑥𝑉 (𝐹𝑥) = 𝑢 ∧ ∃𝑦𝑉 (𝐹𝑦) = 𝑣 ∧ ∃𝑧𝑉 (𝐹𝑧) = 𝑤)))
2823, 27syl 17 . . . . . . 7 (𝜑 → ((𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐹𝑤 ∈ ran 𝐹) ↔ (∃𝑥𝑉 (𝐹𝑥) = 𝑢 ∧ ∃𝑦𝑉 (𝐹𝑦) = 𝑣 ∧ ∃𝑧𝑉 (𝐹𝑧) = 𝑤)))
2921, 28bitr3d 272 . . . . . 6 (𝜑 → ((𝑢𝐵𝑣𝐵𝑤𝐵) ↔ (∃𝑥𝑉 (𝐹𝑥) = 𝑢 ∧ ∃𝑦𝑉 (𝐹𝑦) = 𝑣 ∧ ∃𝑧𝑉 (𝐹𝑧) = 𝑤)))
30 3reeanv 3255 . . . . . 6 (∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) ↔ (∃𝑥𝑉 (𝐹𝑥) = 𝑢 ∧ ∃𝑦𝑉 (𝐹𝑦) = 𝑣 ∧ ∃𝑧𝑉 (𝐹𝑧) = 𝑤))
3129, 30syl6bbr 280 . . . . 5 (𝜑 → ((𝑢𝐵𝑣𝐵𝑤𝐵) ↔ ∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤)))
32 imasmnd2.2 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘((𝑥 + 𝑦) + 𝑧)) = (𝐹‘(𝑥 + (𝑦 + 𝑧))))
33 simpl 474 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝜑)
34103adant3r3 1235 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
35 simpr3 1252 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑧𝑉)
363, 7, 1, 2, 4, 8, 9imasaddval 16460 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 + 𝑦) ∈ 𝑉𝑧𝑉) → ((𝐹‘(𝑥 + 𝑦))(+g𝑈)(𝐹𝑧)) = (𝐹‘((𝑥 + 𝑦) + 𝑧)))
3733, 34, 35, 36syl3anc 1490 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 + 𝑦))(+g𝑈)(𝐹𝑧)) = (𝐹‘((𝑥 + 𝑦) + 𝑧)))
38 simpr1 1248 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑥𝑉)
3912caovclg 7024 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝑉𝑧𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
40393adantr1 1210 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
413, 7, 1, 2, 4, 8, 9imasaddval 16460 . . . . . . . . . . . . 13 ((𝜑𝑥𝑉 ∧ (𝑦 + 𝑧) ∈ 𝑉) → ((𝐹𝑥)(+g𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥 + (𝑦 + 𝑧))))
4233, 38, 40, 41syl3anc 1490 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(+g𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥 + (𝑦 + 𝑧))))
4332, 37, 423eqtr4d 2809 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 + 𝑦))(+g𝑈)(𝐹𝑧)) = ((𝐹𝑥)(+g𝑈)(𝐹‘(𝑦 + 𝑧))))
443, 7, 1, 2, 4, 8, 9imasaddval 16460 . . . . . . . . . . . . 13 ((𝜑𝑥𝑉𝑦𝑉) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝐹‘(𝑥 + 𝑦)))
45443adant3r3 1235 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝐹‘(𝑥 + 𝑦)))
4645oveq1d 6857 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(+g𝑈)(𝐹𝑦))(+g𝑈)(𝐹𝑧)) = ((𝐹‘(𝑥 + 𝑦))(+g𝑈)(𝐹𝑧)))
473, 7, 1, 2, 4, 8, 9imasaddval 16460 . . . . . . . . . . . . 13 ((𝜑𝑦𝑉𝑧𝑉) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝐹‘(𝑦 + 𝑧)))
48473adant3r1 1233 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝐹‘(𝑦 + 𝑧)))
4948oveq2d 6858 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(+g𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))) = ((𝐹𝑥)(+g𝑈)(𝐹‘(𝑦 + 𝑧))))
5043, 46, 493eqtr4d 2809 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(+g𝑈)(𝐹𝑦))(+g𝑈)(𝐹𝑧)) = ((𝐹𝑥)(+g𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))))
51 simp1 1166 . . . . . . . . . . . . 13 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝐹𝑥) = 𝑢)
52 simp2 1167 . . . . . . . . . . . . 13 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝐹𝑦) = 𝑣)
5351, 52oveq12d 6860 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝑢(+g𝑈)𝑣))
54 simp3 1168 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝐹𝑧) = 𝑤)
5553, 54oveq12d 6860 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (((𝐹𝑥)(+g𝑈)(𝐹𝑦))(+g𝑈)(𝐹𝑧)) = ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤))
5652, 54oveq12d 6860 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝑣(+g𝑈)𝑤))
5751, 56oveq12d 6860 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(+g𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤)))
5855, 57eqeq12d 2780 . . . . . . . . . 10 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((((𝐹𝑥)(+g𝑈)(𝐹𝑦))(+g𝑈)(𝐹𝑧)) = ((𝐹𝑥)(+g𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))) ↔ ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤))))
5950, 58syl5ibcom 236 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤))))
60593exp2 1463 . . . . . . . 8 (𝜑 → (𝑥𝑉 → (𝑦𝑉 → (𝑧𝑉 → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤)))))))
6160imp32 409 . . . . . . 7 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑧𝑉 → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤)))))
6261rexlimdv 3177 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (∃𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤))))
6362rexlimdvva 3185 . . . . 5 (𝜑 → (∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤))))
6431, 63sylbid 231 . . . 4 (𝜑 → ((𝑢𝐵𝑣𝐵𝑤𝐵) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤))))
6564imp 395 . . 3 ((𝜑 ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤)))
66 fof 6298 . . . . 5 (𝐹:𝑉onto𝐵𝐹:𝑉𝐵)
673, 66syl 17 . . . 4 (𝜑𝐹:𝑉𝐵)
68 imasmnd2.3 . . . 4 (𝜑0𝑉)
6967, 68ffvelrnd 6550 . . 3 (𝜑 → (𝐹0 ) ∈ 𝐵)
7023, 24syl 17 . . . . . 6 (𝜑 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥𝑉 (𝐹𝑥) = 𝑢))
7118, 70bitr3d 272 . . . . 5 (𝜑 → (𝑢𝐵 ↔ ∃𝑥𝑉 (𝐹𝑥) = 𝑢))
72 simpl 474 . . . . . . . . 9 ((𝜑𝑥𝑉) → 𝜑)
7368adantr 472 . . . . . . . . 9 ((𝜑𝑥𝑉) → 0𝑉)
74 simpr 477 . . . . . . . . 9 ((𝜑𝑥𝑉) → 𝑥𝑉)
753, 7, 1, 2, 4, 8, 9imasaddval 16460 . . . . . . . . 9 ((𝜑0𝑉𝑥𝑉) → ((𝐹0 )(+g𝑈)(𝐹𝑥)) = (𝐹‘( 0 + 𝑥)))
7672, 73, 74, 75syl3anc 1490 . . . . . . . 8 ((𝜑𝑥𝑉) → ((𝐹0 )(+g𝑈)(𝐹𝑥)) = (𝐹‘( 0 + 𝑥)))
77 imasmnd2.4 . . . . . . . 8 ((𝜑𝑥𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹𝑥))
7876, 77eqtrd 2799 . . . . . . 7 ((𝜑𝑥𝑉) → ((𝐹0 )(+g𝑈)(𝐹𝑥)) = (𝐹𝑥))
79 oveq2 6850 . . . . . . . 8 ((𝐹𝑥) = 𝑢 → ((𝐹0 )(+g𝑈)(𝐹𝑥)) = ((𝐹0 )(+g𝑈)𝑢))
80 id 22 . . . . . . . 8 ((𝐹𝑥) = 𝑢 → (𝐹𝑥) = 𝑢)
8179, 80eqeq12d 2780 . . . . . . 7 ((𝐹𝑥) = 𝑢 → (((𝐹0 )(+g𝑈)(𝐹𝑥)) = (𝐹𝑥) ↔ ((𝐹0 )(+g𝑈)𝑢) = 𝑢))
8278, 81syl5ibcom 236 . . . . . 6 ((𝜑𝑥𝑉) → ((𝐹𝑥) = 𝑢 → ((𝐹0 )(+g𝑈)𝑢) = 𝑢))
8382rexlimdva 3178 . . . . 5 (𝜑 → (∃𝑥𝑉 (𝐹𝑥) = 𝑢 → ((𝐹0 )(+g𝑈)𝑢) = 𝑢))
8471, 83sylbid 231 . . . 4 (𝜑 → (𝑢𝐵 → ((𝐹0 )(+g𝑈)𝑢) = 𝑢))
8584imp 395 . . 3 ((𝜑𝑢𝐵) → ((𝐹0 )(+g𝑈)𝑢) = 𝑢)
863, 7, 1, 2, 4, 8, 9imasaddval 16460 . . . . . . . . 9 ((𝜑𝑥𝑉0𝑉) → ((𝐹𝑥)(+g𝑈)(𝐹0 )) = (𝐹‘(𝑥 + 0 )))
8773, 86mpd3an3 1586 . . . . . . . 8 ((𝜑𝑥𝑉) → ((𝐹𝑥)(+g𝑈)(𝐹0 )) = (𝐹‘(𝑥 + 0 )))
88 imasmnd2.5 . . . . . . . 8 ((𝜑𝑥𝑉) → (𝐹‘(𝑥 + 0 )) = (𝐹𝑥))
8987, 88eqtrd 2799 . . . . . . 7 ((𝜑𝑥𝑉) → ((𝐹𝑥)(+g𝑈)(𝐹0 )) = (𝐹𝑥))
90 oveq1 6849 . . . . . . . 8 ((𝐹𝑥) = 𝑢 → ((𝐹𝑥)(+g𝑈)(𝐹0 )) = (𝑢(+g𝑈)(𝐹0 )))
9190, 80eqeq12d 2780 . . . . . . 7 ((𝐹𝑥) = 𝑢 → (((𝐹𝑥)(+g𝑈)(𝐹0 )) = (𝐹𝑥) ↔ (𝑢(+g𝑈)(𝐹0 )) = 𝑢))
9289, 91syl5ibcom 236 . . . . . 6 ((𝜑𝑥𝑉) → ((𝐹𝑥) = 𝑢 → (𝑢(+g𝑈)(𝐹0 )) = 𝑢))
9392rexlimdva 3178 . . . . 5 (𝜑 → (∃𝑥𝑉 (𝐹𝑥) = 𝑢 → (𝑢(+g𝑈)(𝐹0 )) = 𝑢))
9471, 93sylbid 231 . . . 4 (𝜑 → (𝑢𝐵 → (𝑢(+g𝑈)(𝐹0 )) = 𝑢))
9594imp 395 . . 3 ((𝜑𝑢𝐵) → (𝑢(+g𝑈)(𝐹0 )) = 𝑢)
965, 6, 15, 65, 69, 85, 95ismndd 17581 . 2 (𝜑𝑈 ∈ Mnd)
975, 6, 69, 85, 95grpidd 17536 . 2 (𝜑 → (𝐹0 ) = (0g𝑈))
9896, 97jca 507 1 (𝜑 → (𝑈 ∈ Mnd ∧ (𝐹0 ) = (0g𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wrex 3056   × cxp 5275  ran crn 5278   Fn wfn 6063  wf 6064  ontowfo 6066  cfv 6068  (class class class)co 6842  Basecbs 16132  +gcplusg 16216  0gc0g 16368  s cimas 16432  Mndcmnd 17562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-sup 8555  df-inf 8556  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-7 11340  df-8 11341  df-9 11342  df-n0 11539  df-z 11625  df-dec 11741  df-uz 11887  df-fz 12534  df-struct 16134  df-ndx 16135  df-slot 16136  df-base 16138  df-plusg 16229  df-mulr 16230  df-sca 16232  df-vsca 16233  df-ip 16234  df-tset 16235  df-ple 16236  df-ds 16238  df-0g 16370  df-imas 16436  df-mgm 17510  df-sgrp 17552  df-mnd 17563
This theorem is referenced by:  imasmnd  17596
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