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Theorem imasgrp2 18206
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 (𝜑𝑈 = (𝐹s 𝑅))
imasgrp.v (𝜑𝑉 = (Base‘𝑅))
imasgrp.p (𝜑+ = (+g𝑅))
imasgrp.f (𝜑𝐹:𝑉onto𝐵)
imasgrp.e ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
imasgrp2.r (𝜑𝑅𝑊)
imasgrp2.1 ((𝜑𝑥𝑉𝑦𝑉) → (𝑥 + 𝑦) ∈ 𝑉)
imasgrp2.2 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘((𝑥 + 𝑦) + 𝑧)) = (𝐹‘(𝑥 + (𝑦 + 𝑧))))
imasgrp2.3 (𝜑0𝑉)
imasgrp2.4 ((𝜑𝑥𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹𝑥))
imasgrp2.5 ((𝜑𝑥𝑉) → 𝑁𝑉)
imasgrp2.6 ((𝜑𝑥𝑉) → (𝐹‘(𝑁 + 𝑥)) = (𝐹0 ))
Assertion
Ref Expression
imasgrp2 (𝜑 → (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈)))
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.
StepHypRef Expression
1 imasgrp.u . . . 4 (𝜑𝑈 = (𝐹s 𝑅))
2 imasgrp.v . . . 4 (𝜑𝑉 = (Base‘𝑅))
3 imasgrp.f . . . 4 (𝜑𝐹:𝑉onto𝐵)
4 imasgrp2.r . . . 4 (𝜑𝑅𝑊)
51, 2, 3, 4imasbas 16777 . . 3 (𝜑𝐵 = (Base‘𝑈))
6 eqidd 2799 . . 3 (𝜑 → (+g𝑈) = (+g𝑈))
7 imasgrp.e . . . . . 6 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
8 imasgrp.p . . . . . . . . . 10 (𝜑+ = (+g𝑅))
98oveqd 7152 . . . . . . . . 9 (𝜑 → (𝑎 + 𝑏) = (𝑎(+g𝑅)𝑏))
109fveq2d 6649 . . . . . . . 8 (𝜑 → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑎(+g𝑅)𝑏)))
118oveqd 7152 . . . . . . . . 9 (𝜑 → (𝑝 + 𝑞) = (𝑝(+g𝑅)𝑞))
1211fveq2d 6649 . . . . . . . 8 (𝜑 → (𝐹‘(𝑝 + 𝑞)) = (𝐹‘(𝑝(+g𝑅)𝑞)))
1310, 12eqeq12d 2814 . . . . . . 7 (𝜑 → ((𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)) ↔ (𝐹‘(𝑎(+g𝑅)𝑏)) = (𝐹‘(𝑝(+g𝑅)𝑞))))
14133ad2ant1 1130 . . . . . 6 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → ((𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)) ↔ (𝐹‘(𝑎(+g𝑅)𝑏)) = (𝐹‘(𝑝(+g𝑅)𝑞))))
157, 14sylibd 242 . . . . 5 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎(+g𝑅)𝑏)) = (𝐹‘(𝑝(+g𝑅)𝑞))))
16 eqid 2798 . . . . 5 (+g𝑅) = (+g𝑅)
17 eqid 2798 . . . . 5 (+g𝑈) = (+g𝑈)
1811adantr 484 . . . . . 6 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝 + 𝑞) = (𝑝(+g𝑅)𝑞))
19 imasgrp2.1 . . . . . . . 8 ((𝜑𝑥𝑉𝑦𝑉) → (𝑥 + 𝑦) ∈ 𝑉)
20193expb 1117 . . . . . . 7 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
2120caovclg 7320 . . . . . 6 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝 + 𝑞) ∈ 𝑉)
2218, 21eqeltrrd 2891 . . . . 5 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝(+g𝑅)𝑞) ∈ 𝑉)
233, 15, 1, 2, 4, 16, 17, 22imasaddf 16798 . . . 4 (𝜑 → (+g𝑈):(𝐵 × 𝐵)⟶𝐵)
24 fovrn 7298 . . . 4 (((+g𝑈):(𝐵 × 𝐵)⟶𝐵𝑢𝐵𝑣𝐵) → (𝑢(+g𝑈)𝑣) ∈ 𝐵)
2523, 24syl3an1 1160 . . 3 ((𝜑𝑢𝐵𝑣𝐵) → (𝑢(+g𝑈)𝑣) ∈ 𝐵)
26 forn 6568 . . . . . . . . . 10 (𝐹:𝑉onto𝐵 → ran 𝐹 = 𝐵)
273, 26syl 17 . . . . . . . . 9 (𝜑 → ran 𝐹 = 𝐵)
2827eleq2d 2875 . . . . . . . 8 (𝜑 → (𝑢 ∈ ran 𝐹𝑢𝐵))
2927eleq2d 2875 . . . . . . . 8 (𝜑 → (𝑣 ∈ ran 𝐹𝑣𝐵))
3027eleq2d 2875 . . . . . . . 8 (𝜑 → (𝑤 ∈ ran 𝐹𝑤𝐵))
3128, 29, 303anbi123d 1433 . . . . . . 7 (𝜑 → ((𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐹𝑤 ∈ ran 𝐹) ↔ (𝑢𝐵𝑣𝐵𝑤𝐵)))
32 fofn 6567 . . . . . . . . 9 (𝐹:𝑉onto𝐵𝐹 Fn 𝑉)
333, 32syl 17 . . . . . . . 8 (𝜑𝐹 Fn 𝑉)
34 fvelrnb 6701 . . . . . . . . 9 (𝐹 Fn 𝑉 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥𝑉 (𝐹𝑥) = 𝑢))
35 fvelrnb 6701 . . . . . . . . 9 (𝐹 Fn 𝑉 → (𝑣 ∈ ran 𝐹 ↔ ∃𝑦𝑉 (𝐹𝑦) = 𝑣))
36 fvelrnb 6701 . . . . . . . . 9 (𝐹 Fn 𝑉 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑧𝑉 (𝐹𝑧) = 𝑤))
3734, 35, 363anbi123d 1433 . . . . . . . 8 (𝐹 Fn 𝑉 → ((𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐹𝑤 ∈ ran 𝐹) ↔ (∃𝑥𝑉 (𝐹𝑥) = 𝑢 ∧ ∃𝑦𝑉 (𝐹𝑦) = 𝑣 ∧ ∃𝑧𝑉 (𝐹𝑧) = 𝑤)))
3833, 37syl 17 . . . . . . 7 (𝜑 → ((𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐹𝑤 ∈ ran 𝐹) ↔ (∃𝑥𝑉 (𝐹𝑥) = 𝑢 ∧ ∃𝑦𝑉 (𝐹𝑦) = 𝑣 ∧ ∃𝑧𝑉 (𝐹𝑧) = 𝑤)))
3931, 38bitr3d 284 . . . . . 6 (𝜑 → ((𝑢𝐵𝑣𝐵𝑤𝐵) ↔ (∃𝑥𝑉 (𝐹𝑥) = 𝑢 ∧ ∃𝑦𝑉 (𝐹𝑦) = 𝑣 ∧ ∃𝑧𝑉 (𝐹𝑧) = 𝑤)))
40 3reeanv 3321 . . . . . 6 (∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) ↔ (∃𝑥𝑉 (𝐹𝑥) = 𝑢 ∧ ∃𝑦𝑉 (𝐹𝑦) = 𝑣 ∧ ∃𝑧𝑉 (𝐹𝑧) = 𝑤))
4139, 40syl6bbr 292 . . . . 5 (𝜑 → ((𝑢𝐵𝑣𝐵𝑤𝐵) ↔ ∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤)))
42 imasgrp2.2 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘((𝑥 + 𝑦) + 𝑧)) = (𝐹‘(𝑥 + (𝑦 + 𝑧))))
438adantr 484 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → + = (+g𝑅))
4443oveqd 7152 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝑥 + 𝑦) + 𝑧) = ((𝑥 + 𝑦)(+g𝑅)𝑧))
4544fveq2d 6649 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘((𝑥 + 𝑦) + 𝑧)) = (𝐹‘((𝑥 + 𝑦)(+g𝑅)𝑧)))
4643oveqd 7152 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 + (𝑦 + 𝑧)) = (𝑥(+g𝑅)(𝑦 + 𝑧)))
4746fveq2d 6649 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘(𝑥 + (𝑦 + 𝑧))) = (𝐹‘(𝑥(+g𝑅)(𝑦 + 𝑧))))
4842, 45, 473eqtr3d 2841 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘((𝑥 + 𝑦)(+g𝑅)𝑧)) = (𝐹‘(𝑥(+g𝑅)(𝑦 + 𝑧))))
49 simpl 486 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝜑)
50193adant3r3 1181 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
51 simpr3 1193 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑧𝑉)
523, 15, 1, 2, 4, 16, 17imasaddval 16797 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 + 𝑦) ∈ 𝑉𝑧𝑉) → ((𝐹‘(𝑥 + 𝑦))(+g𝑈)(𝐹𝑧)) = (𝐹‘((𝑥 + 𝑦)(+g𝑅)𝑧)))
5349, 50, 51, 52syl3anc 1368 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 + 𝑦))(+g𝑈)(𝐹𝑧)) = (𝐹‘((𝑥 + 𝑦)(+g𝑅)𝑧)))
54 simpr1 1191 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑥𝑉)
5521caovclg 7320 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝑉𝑧𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
56553adantr1 1166 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
573, 15, 1, 2, 4, 16, 17imasaddval 16797 . . . . . . . . . . . . 13 ((𝜑𝑥𝑉 ∧ (𝑦 + 𝑧) ∈ 𝑉) → ((𝐹𝑥)(+g𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥(+g𝑅)(𝑦 + 𝑧))))
5849, 54, 56, 57syl3anc 1368 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(+g𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥(+g𝑅)(𝑦 + 𝑧))))
5948, 53, 583eqtr4d 2843 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 + 𝑦))(+g𝑈)(𝐹𝑧)) = ((𝐹𝑥)(+g𝑈)(𝐹‘(𝑦 + 𝑧))))
603, 15, 1, 2, 4, 16, 17imasaddval 16797 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑉𝑦𝑉) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝐹‘(𝑥(+g𝑅)𝑦)))
61603adant3r3 1181 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝐹‘(𝑥(+g𝑅)𝑦)))
6243oveqd 7152 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 + 𝑦) = (𝑥(+g𝑅)𝑦))
6362fveq2d 6649 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝑥(+g𝑅)𝑦)))
6461, 63eqtr4d 2836 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝐹‘(𝑥 + 𝑦)))
6564oveq1d 7150 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(+g𝑈)(𝐹𝑦))(+g𝑈)(𝐹𝑧)) = ((𝐹‘(𝑥 + 𝑦))(+g𝑈)(𝐹𝑧)))
663, 15, 1, 2, 4, 16, 17imasaddval 16797 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑉𝑧𝑉) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝐹‘(𝑦(+g𝑅)𝑧)))
67663adant3r1 1179 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝐹‘(𝑦(+g𝑅)𝑧)))
6843oveqd 7152 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑦 + 𝑧) = (𝑦(+g𝑅)𝑧))
6968fveq2d 6649 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘(𝑦 + 𝑧)) = (𝐹‘(𝑦(+g𝑅)𝑧)))
7067, 69eqtr4d 2836 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝐹‘(𝑦 + 𝑧)))
7170oveq2d 7151 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(+g𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))) = ((𝐹𝑥)(+g𝑈)(𝐹‘(𝑦 + 𝑧))))
7259, 65, 713eqtr4d 2843 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(+g𝑈)(𝐹𝑦))(+g𝑈)(𝐹𝑧)) = ((𝐹𝑥)(+g𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))))
73 simp1 1133 . . . . . . . . . . . . 13 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝐹𝑥) = 𝑢)
74 simp2 1134 . . . . . . . . . . . . 13 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝐹𝑦) = 𝑣)
7573, 74oveq12d 7153 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝑢(+g𝑈)𝑣))
76 simp3 1135 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝐹𝑧) = 𝑤)
7775, 76oveq12d 7153 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (((𝐹𝑥)(+g𝑈)(𝐹𝑦))(+g𝑈)(𝐹𝑧)) = ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤))
7874, 76oveq12d 7153 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝑣(+g𝑈)𝑤))
7973, 78oveq12d 7153 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(+g𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤)))
8077, 79eqeq12d 2814 . . . . . . . . . 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))))
8172, 80syl5ibcom 248 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤))))
82813exp2 1351 . . . . . . . 8 (𝜑 → (𝑥𝑉 → (𝑦𝑉 → (𝑧𝑉 → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤)))))))
8382imp32 422 . . . . . . 7 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑧𝑉 → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤)))))
8483rexlimdv 3242 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (∃𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤))))
8584rexlimdvva 3253 . . . . 5 (𝜑 → (∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤))))
8641, 85sylbid 243 . . . 4 (𝜑 → ((𝑢𝐵𝑣𝐵𝑤𝐵) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤))))
8786imp 410 . . 3 ((𝜑 ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤)))
88 fof 6565 . . . . 5 (𝐹:𝑉onto𝐵𝐹:𝑉𝐵)
893, 88syl 17 . . . 4 (𝜑𝐹:𝑉𝐵)
90 imasgrp2.3 . . . 4 (𝜑0𝑉)
9189, 90ffvelrnd 6829 . . 3 (𝜑 → (𝐹0 ) ∈ 𝐵)
9233, 34syl 17 . . . . . 6 (𝜑 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥𝑉 (𝐹𝑥) = 𝑢))
9328, 92bitr3d 284 . . . . 5 (𝜑 → (𝑢𝐵 ↔ ∃𝑥𝑉 (𝐹𝑥) = 𝑢))
94 simpl 486 . . . . . . . . 9 ((𝜑𝑥𝑉) → 𝜑)
9590adantr 484 . . . . . . . . 9 ((𝜑𝑥𝑉) → 0𝑉)
96 simpr 488 . . . . . . . . 9 ((𝜑𝑥𝑉) → 𝑥𝑉)
973, 15, 1, 2, 4, 16, 17imasaddval 16797 . . . . . . . . 9 ((𝜑0𝑉𝑥𝑉) → ((𝐹0 )(+g𝑈)(𝐹𝑥)) = (𝐹‘( 0 (+g𝑅)𝑥)))
9894, 95, 96, 97syl3anc 1368 . . . . . . . 8 ((𝜑𝑥𝑉) → ((𝐹0 )(+g𝑈)(𝐹𝑥)) = (𝐹‘( 0 (+g𝑅)𝑥)))
998adantr 484 . . . . . . . . . 10 ((𝜑𝑥𝑉) → + = (+g𝑅))
10099oveqd 7152 . . . . . . . . 9 ((𝜑𝑥𝑉) → ( 0 + 𝑥) = ( 0 (+g𝑅)𝑥))
101100fveq2d 6649 . . . . . . . 8 ((𝜑𝑥𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹‘( 0 (+g𝑅)𝑥)))
102 imasgrp2.4 . . . . . . . 8 ((𝜑𝑥𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹𝑥))
10398, 101, 1023eqtr2d 2839 . . . . . . 7 ((𝜑𝑥𝑉) → ((𝐹0 )(+g𝑈)(𝐹𝑥)) = (𝐹𝑥))
104 oveq2 7143 . . . . . . . 8 ((𝐹𝑥) = 𝑢 → ((𝐹0 )(+g𝑈)(𝐹𝑥)) = ((𝐹0 )(+g𝑈)𝑢))
105 id 22 . . . . . . . 8 ((𝐹𝑥) = 𝑢 → (𝐹𝑥) = 𝑢)
106104, 105eqeq12d 2814 . . . . . . 7 ((𝐹𝑥) = 𝑢 → (((𝐹0 )(+g𝑈)(𝐹𝑥)) = (𝐹𝑥) ↔ ((𝐹0 )(+g𝑈)𝑢) = 𝑢))
107103, 106syl5ibcom 248 . . . . . 6 ((𝜑𝑥𝑉) → ((𝐹𝑥) = 𝑢 → ((𝐹0 )(+g𝑈)𝑢) = 𝑢))
108107rexlimdva 3243 . . . . 5 (𝜑 → (∃𝑥𝑉 (𝐹𝑥) = 𝑢 → ((𝐹0 )(+g𝑈)𝑢) = 𝑢))
10993, 108sylbid 243 . . . 4 (𝜑 → (𝑢𝐵 → ((𝐹0 )(+g𝑈)𝑢) = 𝑢))
110109imp 410 . . 3 ((𝜑𝑢𝐵) → ((𝐹0 )(+g𝑈)𝑢) = 𝑢)
11189adantr 484 . . . . . . . . 9 ((𝜑𝑥𝑉) → 𝐹:𝑉𝐵)
112 imasgrp2.5 . . . . . . . . 9 ((𝜑𝑥𝑉) → 𝑁𝑉)
113111, 112ffvelrnd 6829 . . . . . . . 8 ((𝜑𝑥𝑉) → (𝐹𝑁) ∈ 𝐵)
1143, 15, 1, 2, 4, 16, 17imasaddval 16797 . . . . . . . . . 10 ((𝜑𝑁𝑉𝑥𝑉) → ((𝐹𝑁)(+g𝑈)(𝐹𝑥)) = (𝐹‘(𝑁(+g𝑅)𝑥)))
11594, 112, 96, 114syl3anc 1368 . . . . . . . . 9 ((𝜑𝑥𝑉) → ((𝐹𝑁)(+g𝑈)(𝐹𝑥)) = (𝐹‘(𝑁(+g𝑅)𝑥)))
11699oveqd 7152 . . . . . . . . . 10 ((𝜑𝑥𝑉) → (𝑁 + 𝑥) = (𝑁(+g𝑅)𝑥))
117116fveq2d 6649 . . . . . . . . 9 ((𝜑𝑥𝑉) → (𝐹‘(𝑁 + 𝑥)) = (𝐹‘(𝑁(+g𝑅)𝑥)))
118 imasgrp2.6 . . . . . . . . 9 ((𝜑𝑥𝑉) → (𝐹‘(𝑁 + 𝑥)) = (𝐹0 ))
119115, 117, 1183eqtr2d 2839 . . . . . . . 8 ((𝜑𝑥𝑉) → ((𝐹𝑁)(+g𝑈)(𝐹𝑥)) = (𝐹0 ))
120 oveq1 7142 . . . . . . . . . 10 (𝑣 = (𝐹𝑁) → (𝑣(+g𝑈)(𝐹𝑥)) = ((𝐹𝑁)(+g𝑈)(𝐹𝑥)))
121120eqeq1d 2800 . . . . . . . . 9 (𝑣 = (𝐹𝑁) → ((𝑣(+g𝑈)(𝐹𝑥)) = (𝐹0 ) ↔ ((𝐹𝑁)(+g𝑈)(𝐹𝑥)) = (𝐹0 )))
122121rspcev 3571 . . . . . . . 8 (((𝐹𝑁) ∈ 𝐵 ∧ ((𝐹𝑁)(+g𝑈)(𝐹𝑥)) = (𝐹0 )) → ∃𝑣𝐵 (𝑣(+g𝑈)(𝐹𝑥)) = (𝐹0 ))
123113, 119, 122syl2anc 587 . . . . . . 7 ((𝜑𝑥𝑉) → ∃𝑣𝐵 (𝑣(+g𝑈)(𝐹𝑥)) = (𝐹0 ))
124 oveq2 7143 . . . . . . . . 9 ((𝐹𝑥) = 𝑢 → (𝑣(+g𝑈)(𝐹𝑥)) = (𝑣(+g𝑈)𝑢))
125124eqeq1d 2800 . . . . . . . 8 ((𝐹𝑥) = 𝑢 → ((𝑣(+g𝑈)(𝐹𝑥)) = (𝐹0 ) ↔ (𝑣(+g𝑈)𝑢) = (𝐹0 )))
126125rexbidv 3256 . . . . . . 7 ((𝐹𝑥) = 𝑢 → (∃𝑣𝐵 (𝑣(+g𝑈)(𝐹𝑥)) = (𝐹0 ) ↔ ∃𝑣𝐵 (𝑣(+g𝑈)𝑢) = (𝐹0 )))
127123, 126syl5ibcom 248 . . . . . 6 ((𝜑𝑥𝑉) → ((𝐹𝑥) = 𝑢 → ∃𝑣𝐵 (𝑣(+g𝑈)𝑢) = (𝐹0 )))
128127rexlimdva 3243 . . . . 5 (𝜑 → (∃𝑥𝑉 (𝐹𝑥) = 𝑢 → ∃𝑣𝐵 (𝑣(+g𝑈)𝑢) = (𝐹0 )))
12993, 128sylbid 243 . . . 4 (𝜑 → (𝑢𝐵 → ∃𝑣𝐵 (𝑣(+g𝑈)𝑢) = (𝐹0 )))
130129imp 410 . . 3 ((𝜑𝑢𝐵) → ∃𝑣𝐵 (𝑣(+g𝑈)𝑢) = (𝐹0 ))
1315, 6, 25, 87, 91, 110, 130isgrpde 18116 . 2 (𝜑𝑈 ∈ Grp)
1325, 6, 91, 110, 131grpidd2 18133 . 2 (𝜑 → (𝐹0 ) = (0g𝑈))
133131, 132jca 515 1 (𝜑 → (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2111  wrex 3107   × cxp 5517  ran crn 5520   Fn wfn 6319  wf 6320  ontowfo 6322  cfv 6324  (class class class)co 7135  Basecbs 16475  +gcplusg 16557  0gc0g 16705  s cimas 16769  Grpcgrp 18095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-sup 8890  df-inf 8891  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-fz 12886  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-plusg 16570  df-mulr 16571  df-sca 16573  df-vsca 16574  df-ip 16575  df-tset 16576  df-ple 16577  df-ds 16579  df-0g 16707  df-imas 16773  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098
This theorem is referenced by:  imasgrp  18207  qusgrp2  18209
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