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Theorem imasgrp2 18678
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 17211 . . 3 (𝜑𝐵 = (Base‘𝑈))
6 eqidd 2739 . . 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 7285 . . . . . . . . 9 (𝜑 → (𝑎 + 𝑏) = (𝑎(+g𝑅)𝑏))
109fveq2d 6771 . . . . . . . 8 (𝜑 → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑎(+g𝑅)𝑏)))
118oveqd 7285 . . . . . . . . 9 (𝜑 → (𝑝 + 𝑞) = (𝑝(+g𝑅)𝑞))
1211fveq2d 6771 . . . . . . . 8 (𝜑 → (𝐹‘(𝑝 + 𝑞)) = (𝐹‘(𝑝(+g𝑅)𝑞)))
1310, 12eqeq12d 2754 . . . . . . 7 (𝜑 → ((𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)) ↔ (𝐹‘(𝑎(+g𝑅)𝑏)) = (𝐹‘(𝑝(+g𝑅)𝑞))))
14133ad2ant1 1132 . . . . . 6 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → ((𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)) ↔ (𝐹‘(𝑎(+g𝑅)𝑏)) = (𝐹‘(𝑝(+g𝑅)𝑞))))
157, 14sylibd 238 . . . . 5 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎(+g𝑅)𝑏)) = (𝐹‘(𝑝(+g𝑅)𝑞))))
16 eqid 2738 . . . . 5 (+g𝑅) = (+g𝑅)
17 eqid 2738 . . . . 5 (+g𝑈) = (+g𝑈)
1811adantr 481 . . . . . 6 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝 + 𝑞) = (𝑝(+g𝑅)𝑞))
19 imasgrp2.1 . . . . . . . 8 ((𝜑𝑥𝑉𝑦𝑉) → (𝑥 + 𝑦) ∈ 𝑉)
20193expb 1119 . . . . . . 7 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
2120caovclg 7455 . . . . . 6 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝 + 𝑞) ∈ 𝑉)
2218, 21eqeltrrd 2840 . . . . 5 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝(+g𝑅)𝑞) ∈ 𝑉)
233, 15, 1, 2, 4, 16, 17, 22imasaddf 17232 . . . 4 (𝜑 → (+g𝑈):(𝐵 × 𝐵)⟶𝐵)
24 fovrn 7433 . . . 4 (((+g𝑈):(𝐵 × 𝐵)⟶𝐵𝑢𝐵𝑣𝐵) → (𝑢(+g𝑈)𝑣) ∈ 𝐵)
2523, 24syl3an1 1162 . . 3 ((𝜑𝑢𝐵𝑣𝐵) → (𝑢(+g𝑈)𝑣) ∈ 𝐵)
26 forn 6684 . . . . . . . . . 10 (𝐹:𝑉onto𝐵 → ran 𝐹 = 𝐵)
273, 26syl 17 . . . . . . . . 9 (𝜑 → ran 𝐹 = 𝐵)
2827eleq2d 2824 . . . . . . . 8 (𝜑 → (𝑢 ∈ ran 𝐹𝑢𝐵))
2927eleq2d 2824 . . . . . . . 8 (𝜑 → (𝑣 ∈ ran 𝐹𝑣𝐵))
3027eleq2d 2824 . . . . . . . 8 (𝜑 → (𝑤 ∈ ran 𝐹𝑤𝐵))
3128, 29, 303anbi123d 1435 . . . . . . 7 (𝜑 → ((𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐹𝑤 ∈ ran 𝐹) ↔ (𝑢𝐵𝑣𝐵𝑤𝐵)))
32 fofn 6683 . . . . . . . . 9 (𝐹:𝑉onto𝐵𝐹 Fn 𝑉)
333, 32syl 17 . . . . . . . 8 (𝜑𝐹 Fn 𝑉)
34 fvelrnb 6823 . . . . . . . . 9 (𝐹 Fn 𝑉 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥𝑉 (𝐹𝑥) = 𝑢))
35 fvelrnb 6823 . . . . . . . . 9 (𝐹 Fn 𝑉 → (𝑣 ∈ ran 𝐹 ↔ ∃𝑦𝑉 (𝐹𝑦) = 𝑣))
36 fvelrnb 6823 . . . . . . . . 9 (𝐹 Fn 𝑉 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑧𝑉 (𝐹𝑧) = 𝑤))
3734, 35, 363anbi123d 1435 . . . . . . . 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 280 . . . . . 6 (𝜑 → ((𝑢𝐵𝑣𝐵𝑤𝐵) ↔ (∃𝑥𝑉 (𝐹𝑥) = 𝑢 ∧ ∃𝑦𝑉 (𝐹𝑦) = 𝑣 ∧ ∃𝑧𝑉 (𝐹𝑧) = 𝑤)))
40 3reeanv 3293 . . . . . 6 (∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) ↔ (∃𝑥𝑉 (𝐹𝑥) = 𝑢 ∧ ∃𝑦𝑉 (𝐹𝑦) = 𝑣 ∧ ∃𝑧𝑉 (𝐹𝑧) = 𝑤))
4139, 40bitr4di 289 . . . . 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 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → + = (+g𝑅))
4443oveqd 7285 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝑥 + 𝑦) + 𝑧) = ((𝑥 + 𝑦)(+g𝑅)𝑧))
4544fveq2d 6771 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘((𝑥 + 𝑦) + 𝑧)) = (𝐹‘((𝑥 + 𝑦)(+g𝑅)𝑧)))
4643oveqd 7285 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 + (𝑦 + 𝑧)) = (𝑥(+g𝑅)(𝑦 + 𝑧)))
4746fveq2d 6771 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘(𝑥 + (𝑦 + 𝑧))) = (𝐹‘(𝑥(+g𝑅)(𝑦 + 𝑧))))
4842, 45, 473eqtr3d 2786 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘((𝑥 + 𝑦)(+g𝑅)𝑧)) = (𝐹‘(𝑥(+g𝑅)(𝑦 + 𝑧))))
49 simpl 483 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝜑)
50193adant3r3 1183 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
51 simpr3 1195 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑧𝑉)
523, 15, 1, 2, 4, 16, 17imasaddval 17231 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 + 𝑦) ∈ 𝑉𝑧𝑉) → ((𝐹‘(𝑥 + 𝑦))(+g𝑈)(𝐹𝑧)) = (𝐹‘((𝑥 + 𝑦)(+g𝑅)𝑧)))
5349, 50, 51, 52syl3anc 1370 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 + 𝑦))(+g𝑈)(𝐹𝑧)) = (𝐹‘((𝑥 + 𝑦)(+g𝑅)𝑧)))
54 simpr1 1193 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑥𝑉)
5521caovclg 7455 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝑉𝑧𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
56553adantr1 1168 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
573, 15, 1, 2, 4, 16, 17imasaddval 17231 . . . . . . . . . . . . 13 ((𝜑𝑥𝑉 ∧ (𝑦 + 𝑧) ∈ 𝑉) → ((𝐹𝑥)(+g𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥(+g𝑅)(𝑦 + 𝑧))))
5849, 54, 56, 57syl3anc 1370 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(+g𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥(+g𝑅)(𝑦 + 𝑧))))
5948, 53, 583eqtr4d 2788 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 + 𝑦))(+g𝑈)(𝐹𝑧)) = ((𝐹𝑥)(+g𝑈)(𝐹‘(𝑦 + 𝑧))))
603, 15, 1, 2, 4, 16, 17imasaddval 17231 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑉𝑦𝑉) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝐹‘(𝑥(+g𝑅)𝑦)))
61603adant3r3 1183 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝐹‘(𝑥(+g𝑅)𝑦)))
6243oveqd 7285 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 + 𝑦) = (𝑥(+g𝑅)𝑦))
6362fveq2d 6771 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝑥(+g𝑅)𝑦)))
6461, 63eqtr4d 2781 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝐹‘(𝑥 + 𝑦)))
6564oveq1d 7283 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(+g𝑈)(𝐹𝑦))(+g𝑈)(𝐹𝑧)) = ((𝐹‘(𝑥 + 𝑦))(+g𝑈)(𝐹𝑧)))
663, 15, 1, 2, 4, 16, 17imasaddval 17231 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑉𝑧𝑉) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝐹‘(𝑦(+g𝑅)𝑧)))
67663adant3r1 1181 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝐹‘(𝑦(+g𝑅)𝑧)))
6843oveqd 7285 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑦 + 𝑧) = (𝑦(+g𝑅)𝑧))
6968fveq2d 6771 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘(𝑦 + 𝑧)) = (𝐹‘(𝑦(+g𝑅)𝑧)))
7067, 69eqtr4d 2781 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝐹‘(𝑦 + 𝑧)))
7170oveq2d 7284 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(+g𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))) = ((𝐹𝑥)(+g𝑈)(𝐹‘(𝑦 + 𝑧))))
7259, 65, 713eqtr4d 2788 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(+g𝑈)(𝐹𝑦))(+g𝑈)(𝐹𝑧)) = ((𝐹𝑥)(+g𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))))
73 simp1 1135 . . . . . . . . . . . . 13 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝐹𝑥) = 𝑢)
74 simp2 1136 . . . . . . . . . . . . 13 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝐹𝑦) = 𝑣)
7573, 74oveq12d 7286 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝑢(+g𝑈)𝑣))
76 simp3 1137 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝐹𝑧) = 𝑤)
7775, 76oveq12d 7286 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (((𝐹𝑥)(+g𝑈)(𝐹𝑦))(+g𝑈)(𝐹𝑧)) = ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤))
7874, 76oveq12d 7286 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝑣(+g𝑈)𝑤))
7973, 78oveq12d 7286 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(+g𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤)))
8077, 79eqeq12d 2754 . . . . . . . . . 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 244 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤))))
82813exp2 1353 . . . . . . . 8 (𝜑 → (𝑥𝑉 → (𝑦𝑉 → (𝑧𝑉 → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤)))))))
8382imp32 419 . . . . . . 7 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑧𝑉 → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤)))))
8483rexlimdv 3210 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (∃𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤))))
8584rexlimdvva 3221 . . . . 5 (𝜑 → (∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤))))
8641, 85sylbid 239 . . . 4 (𝜑 → ((𝑢𝐵𝑣𝐵𝑤𝐵) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤))))
8786imp 407 . . 3 ((𝜑 ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤)))
88 fof 6681 . . . . 5 (𝐹:𝑉onto𝐵𝐹:𝑉𝐵)
893, 88syl 17 . . . 4 (𝜑𝐹:𝑉𝐵)
90 imasgrp2.3 . . . 4 (𝜑0𝑉)
9189, 90ffvelrnd 6955 . . 3 (𝜑 → (𝐹0 ) ∈ 𝐵)
9233, 34syl 17 . . . . . 6 (𝜑 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥𝑉 (𝐹𝑥) = 𝑢))
9328, 92bitr3d 280 . . . . 5 (𝜑 → (𝑢𝐵 ↔ ∃𝑥𝑉 (𝐹𝑥) = 𝑢))
94 simpl 483 . . . . . . . . 9 ((𝜑𝑥𝑉) → 𝜑)
9590adantr 481 . . . . . . . . 9 ((𝜑𝑥𝑉) → 0𝑉)
96 simpr 485 . . . . . . . . 9 ((𝜑𝑥𝑉) → 𝑥𝑉)
973, 15, 1, 2, 4, 16, 17imasaddval 17231 . . . . . . . . 9 ((𝜑0𝑉𝑥𝑉) → ((𝐹0 )(+g𝑈)(𝐹𝑥)) = (𝐹‘( 0 (+g𝑅)𝑥)))
9894, 95, 96, 97syl3anc 1370 . . . . . . . 8 ((𝜑𝑥𝑉) → ((𝐹0 )(+g𝑈)(𝐹𝑥)) = (𝐹‘( 0 (+g𝑅)𝑥)))
998adantr 481 . . . . . . . . . 10 ((𝜑𝑥𝑉) → + = (+g𝑅))
10099oveqd 7285 . . . . . . . . 9 ((𝜑𝑥𝑉) → ( 0 + 𝑥) = ( 0 (+g𝑅)𝑥))
101100fveq2d 6771 . . . . . . . 8 ((𝜑𝑥𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹‘( 0 (+g𝑅)𝑥)))
102 imasgrp2.4 . . . . . . . 8 ((𝜑𝑥𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹𝑥))
10398, 101, 1023eqtr2d 2784 . . . . . . 7 ((𝜑𝑥𝑉) → ((𝐹0 )(+g𝑈)(𝐹𝑥)) = (𝐹𝑥))
104 oveq2 7276 . . . . . . . 8 ((𝐹𝑥) = 𝑢 → ((𝐹0 )(+g𝑈)(𝐹𝑥)) = ((𝐹0 )(+g𝑈)𝑢))
105 id 22 . . . . . . . 8 ((𝐹𝑥) = 𝑢 → (𝐹𝑥) = 𝑢)
106104, 105eqeq12d 2754 . . . . . . 7 ((𝐹𝑥) = 𝑢 → (((𝐹0 )(+g𝑈)(𝐹𝑥)) = (𝐹𝑥) ↔ ((𝐹0 )(+g𝑈)𝑢) = 𝑢))
107103, 106syl5ibcom 244 . . . . . 6 ((𝜑𝑥𝑉) → ((𝐹𝑥) = 𝑢 → ((𝐹0 )(+g𝑈)𝑢) = 𝑢))
108107rexlimdva 3211 . . . . 5 (𝜑 → (∃𝑥𝑉 (𝐹𝑥) = 𝑢 → ((𝐹0 )(+g𝑈)𝑢) = 𝑢))
10993, 108sylbid 239 . . . 4 (𝜑 → (𝑢𝐵 → ((𝐹0 )(+g𝑈)𝑢) = 𝑢))
110109imp 407 . . 3 ((𝜑𝑢𝐵) → ((𝐹0 )(+g𝑈)𝑢) = 𝑢)
11189adantr 481 . . . . . . . . 9 ((𝜑𝑥𝑉) → 𝐹:𝑉𝐵)
112 imasgrp2.5 . . . . . . . . 9 ((𝜑𝑥𝑉) → 𝑁𝑉)
113111, 112ffvelrnd 6955 . . . . . . . 8 ((𝜑𝑥𝑉) → (𝐹𝑁) ∈ 𝐵)
1143, 15, 1, 2, 4, 16, 17imasaddval 17231 . . . . . . . . . 10 ((𝜑𝑁𝑉𝑥𝑉) → ((𝐹𝑁)(+g𝑈)(𝐹𝑥)) = (𝐹‘(𝑁(+g𝑅)𝑥)))
11594, 112, 96, 114syl3anc 1370 . . . . . . . . 9 ((𝜑𝑥𝑉) → ((𝐹𝑁)(+g𝑈)(𝐹𝑥)) = (𝐹‘(𝑁(+g𝑅)𝑥)))
11699oveqd 7285 . . . . . . . . . 10 ((𝜑𝑥𝑉) → (𝑁 + 𝑥) = (𝑁(+g𝑅)𝑥))
117116fveq2d 6771 . . . . . . . . 9 ((𝜑𝑥𝑉) → (𝐹‘(𝑁 + 𝑥)) = (𝐹‘(𝑁(+g𝑅)𝑥)))
118 imasgrp2.6 . . . . . . . . 9 ((𝜑𝑥𝑉) → (𝐹‘(𝑁 + 𝑥)) = (𝐹0 ))
119115, 117, 1183eqtr2d 2784 . . . . . . . 8 ((𝜑𝑥𝑉) → ((𝐹𝑁)(+g𝑈)(𝐹𝑥)) = (𝐹0 ))
120 oveq1 7275 . . . . . . . . . 10 (𝑣 = (𝐹𝑁) → (𝑣(+g𝑈)(𝐹𝑥)) = ((𝐹𝑁)(+g𝑈)(𝐹𝑥)))
121120eqeq1d 2740 . . . . . . . . 9 (𝑣 = (𝐹𝑁) → ((𝑣(+g𝑈)(𝐹𝑥)) = (𝐹0 ) ↔ ((𝐹𝑁)(+g𝑈)(𝐹𝑥)) = (𝐹0 )))
122121rspcev 3560 . . . . . . . 8 (((𝐹𝑁) ∈ 𝐵 ∧ ((𝐹𝑁)(+g𝑈)(𝐹𝑥)) = (𝐹0 )) → ∃𝑣𝐵 (𝑣(+g𝑈)(𝐹𝑥)) = (𝐹0 ))
123113, 119, 122syl2anc 584 . . . . . . 7 ((𝜑𝑥𝑉) → ∃𝑣𝐵 (𝑣(+g𝑈)(𝐹𝑥)) = (𝐹0 ))
124 oveq2 7276 . . . . . . . . 9 ((𝐹𝑥) = 𝑢 → (𝑣(+g𝑈)(𝐹𝑥)) = (𝑣(+g𝑈)𝑢))
125124eqeq1d 2740 . . . . . . . 8 ((𝐹𝑥) = 𝑢 → ((𝑣(+g𝑈)(𝐹𝑥)) = (𝐹0 ) ↔ (𝑣(+g𝑈)𝑢) = (𝐹0 )))
126125rexbidv 3224 . . . . . . 7 ((𝐹𝑥) = 𝑢 → (∃𝑣𝐵 (𝑣(+g𝑈)(𝐹𝑥)) = (𝐹0 ) ↔ ∃𝑣𝐵 (𝑣(+g𝑈)𝑢) = (𝐹0 )))
127123, 126syl5ibcom 244 . . . . . 6 ((𝜑𝑥𝑉) → ((𝐹𝑥) = 𝑢 → ∃𝑣𝐵 (𝑣(+g𝑈)𝑢) = (𝐹0 )))
128127rexlimdva 3211 . . . . 5 (𝜑 → (∃𝑥𝑉 (𝐹𝑥) = 𝑢 → ∃𝑣𝐵 (𝑣(+g𝑈)𝑢) = (𝐹0 )))
12993, 128sylbid 239 . . . 4 (𝜑 → (𝑢𝐵 → ∃𝑣𝐵 (𝑣(+g𝑈)𝑢) = (𝐹0 )))
130129imp 407 . . 3 ((𝜑𝑢𝐵) → ∃𝑣𝐵 (𝑣(+g𝑈)𝑢) = (𝐹0 ))
1315, 6, 25, 87, 91, 110, 130isgrpde 18588 . 2 (𝜑𝑈 ∈ Grp)
1325, 6, 91, 110, 131grpidd2 18605 . 2 (𝜑 → (𝐹0 ) = (0g𝑈))
133131, 132jca 512 1 (𝜑 → (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wrex 3065   × cxp 5583  ran crn 5586   Fn wfn 6422  wf 6423  ontowfo 6425  cfv 6427  (class class class)co 7268  Basecbs 16900  +gcplusg 16950  0gc0g 17138  s cimas 17203  Grpcgrp 18565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5222  ax-nul 5229  ax-pow 5287  ax-pr 5351  ax-un 7579  ax-cnex 10915  ax-resscn 10916  ax-1cn 10917  ax-icn 10918  ax-addcl 10919  ax-addrcl 10920  ax-mulcl 10921  ax-mulrcl 10922  ax-mulcom 10923  ax-addass 10924  ax-mulass 10925  ax-distr 10926  ax-i2m1 10927  ax-1ne0 10928  ax-1rid 10929  ax-rnegex 10930  ax-rrecex 10931  ax-cnre 10932  ax-pre-lttri 10933  ax-pre-lttrn 10934  ax-pre-ltadd 10935  ax-pre-mulgt0 10936
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3071  df-rmo 3072  df-rab 3073  df-v 3432  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-uni 4841  df-iun 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5485  df-eprel 5491  df-po 5499  df-so 5500  df-fr 5540  df-we 5542  df-xp 5591  df-rel 5592  df-cnv 5593  df-co 5594  df-dm 5595  df-rn 5596  df-res 5597  df-ima 5598  df-pred 6196  df-ord 6263  df-on 6264  df-lim 6265  df-suc 6266  df-iota 6385  df-fun 6429  df-fn 6430  df-f 6431  df-f1 6432  df-fo 6433  df-f1o 6434  df-fv 6435  df-riota 7225  df-ov 7271  df-oprab 7272  df-mpo 7273  df-om 7704  df-1st 7821  df-2nd 7822  df-frecs 8085  df-wrecs 8116  df-recs 8190  df-rdg 8229  df-1o 8285  df-er 8486  df-en 8722  df-dom 8723  df-sdom 8724  df-fin 8725  df-sup 9189  df-inf 9190  df-pnf 10999  df-mnf 11000  df-xr 11001  df-ltxr 11002  df-le 11003  df-sub 11195  df-neg 11196  df-nn 11962  df-2 12024  df-3 12025  df-4 12026  df-5 12027  df-6 12028  df-7 12029  df-8 12030  df-9 12031  df-n0 12222  df-z 12308  df-dec 12426  df-uz 12571  df-fz 13228  df-struct 16836  df-slot 16871  df-ndx 16883  df-base 16901  df-plusg 16963  df-mulr 16964  df-sca 16966  df-vsca 16967  df-ip 16968  df-tset 16969  df-ple 16970  df-ds 16972  df-0g 17140  df-imas 17207  df-mgm 18314  df-sgrp 18363  df-mnd 18374  df-grp 18568
This theorem is referenced by:  imasgrp  18679  qusgrp2  18681
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