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Theorem imasgrp2 18862
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 17394 . . 3 (𝜑𝐵 = (Base‘𝑈))
6 eqidd 2737 . . 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 7374 . . . . . . . . 9 (𝜑 → (𝑎 + 𝑏) = (𝑎(+g𝑅)𝑏))
109fveq2d 6846 . . . . . . . 8 (𝜑 → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑎(+g𝑅)𝑏)))
118oveqd 7374 . . . . . . . . 9 (𝜑 → (𝑝 + 𝑞) = (𝑝(+g𝑅)𝑞))
1211fveq2d 6846 . . . . . . . 8 (𝜑 → (𝐹‘(𝑝 + 𝑞)) = (𝐹‘(𝑝(+g𝑅)𝑞)))
1310, 12eqeq12d 2752 . . . . . . 7 (𝜑 → ((𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)) ↔ (𝐹‘(𝑎(+g𝑅)𝑏)) = (𝐹‘(𝑝(+g𝑅)𝑞))))
14133ad2ant1 1133 . . . . . 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 2736 . . . . 5 (+g𝑅) = (+g𝑅)
17 eqid 2736 . . . . 5 (+g𝑈) = (+g𝑈)
1811adantr 481 . . . . . 6 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝 + 𝑞) = (𝑝(+g𝑅)𝑞))
19 imasgrp2.1 . . . . . . . 8 ((𝜑𝑥𝑉𝑦𝑉) → (𝑥 + 𝑦) ∈ 𝑉)
20193expb 1120 . . . . . . 7 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
2120caovclg 7546 . . . . . 6 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝 + 𝑞) ∈ 𝑉)
2218, 21eqeltrrd 2839 . . . . 5 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝(+g𝑅)𝑞) ∈ 𝑉)
233, 15, 1, 2, 4, 16, 17, 22imasaddf 17415 . . . 4 (𝜑 → (+g𝑈):(𝐵 × 𝐵)⟶𝐵)
24 fovcdm 7524 . . . 4 (((+g𝑈):(𝐵 × 𝐵)⟶𝐵𝑢𝐵𝑣𝐵) → (𝑢(+g𝑈)𝑣) ∈ 𝐵)
2523, 24syl3an1 1163 . . 3 ((𝜑𝑢𝐵𝑣𝐵) → (𝑢(+g𝑈)𝑣) ∈ 𝐵)
26 forn 6759 . . . . . . . . . 10 (𝐹:𝑉onto𝐵 → ran 𝐹 = 𝐵)
273, 26syl 17 . . . . . . . . 9 (𝜑 → ran 𝐹 = 𝐵)
2827eleq2d 2823 . . . . . . . 8 (𝜑 → (𝑢 ∈ ran 𝐹𝑢𝐵))
2927eleq2d 2823 . . . . . . . 8 (𝜑 → (𝑣 ∈ ran 𝐹𝑣𝐵))
3027eleq2d 2823 . . . . . . . 8 (𝜑 → (𝑤 ∈ ran 𝐹𝑤𝐵))
3128, 29, 303anbi123d 1436 . . . . . . 7 (𝜑 → ((𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐹𝑤 ∈ ran 𝐹) ↔ (𝑢𝐵𝑣𝐵𝑤𝐵)))
32 fofn 6758 . . . . . . . . 9 (𝐹:𝑉onto𝐵𝐹 Fn 𝑉)
333, 32syl 17 . . . . . . . 8 (𝜑𝐹 Fn 𝑉)
34 fvelrnb 6903 . . . . . . . . 9 (𝐹 Fn 𝑉 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥𝑉 (𝐹𝑥) = 𝑢))
35 fvelrnb 6903 . . . . . . . . 9 (𝐹 Fn 𝑉 → (𝑣 ∈ ran 𝐹 ↔ ∃𝑦𝑉 (𝐹𝑦) = 𝑣))
36 fvelrnb 6903 . . . . . . . . 9 (𝐹 Fn 𝑉 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑧𝑉 (𝐹𝑧) = 𝑤))
3734, 35, 363anbi123d 1436 . . . . . . . 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 3218 . . . . . 6 (∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) ↔ (∃𝑥𝑉 (𝐹𝑥) = 𝑢 ∧ ∃𝑦𝑉 (𝐹𝑦) = 𝑣 ∧ ∃𝑧𝑉 (𝐹𝑧) = 𝑤))
4139, 40bitr4di 288 . . . . 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 7374 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝑥 + 𝑦) + 𝑧) = ((𝑥 + 𝑦)(+g𝑅)𝑧))
4544fveq2d 6846 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘((𝑥 + 𝑦) + 𝑧)) = (𝐹‘((𝑥 + 𝑦)(+g𝑅)𝑧)))
4643oveqd 7374 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 + (𝑦 + 𝑧)) = (𝑥(+g𝑅)(𝑦 + 𝑧)))
4746fveq2d 6846 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘(𝑥 + (𝑦 + 𝑧))) = (𝐹‘(𝑥(+g𝑅)(𝑦 + 𝑧))))
4842, 45, 473eqtr3d 2784 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘((𝑥 + 𝑦)(+g𝑅)𝑧)) = (𝐹‘(𝑥(+g𝑅)(𝑦 + 𝑧))))
49 simpl 483 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝜑)
50193adant3r3 1184 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
51 simpr3 1196 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑧𝑉)
523, 15, 1, 2, 4, 16, 17imasaddval 17414 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 + 𝑦) ∈ 𝑉𝑧𝑉) → ((𝐹‘(𝑥 + 𝑦))(+g𝑈)(𝐹𝑧)) = (𝐹‘((𝑥 + 𝑦)(+g𝑅)𝑧)))
5349, 50, 51, 52syl3anc 1371 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 + 𝑦))(+g𝑈)(𝐹𝑧)) = (𝐹‘((𝑥 + 𝑦)(+g𝑅)𝑧)))
54 simpr1 1194 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑥𝑉)
5521caovclg 7546 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝑉𝑧𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
56553adantr1 1169 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
573, 15, 1, 2, 4, 16, 17imasaddval 17414 . . . . . . . . . . . . 13 ((𝜑𝑥𝑉 ∧ (𝑦 + 𝑧) ∈ 𝑉) → ((𝐹𝑥)(+g𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥(+g𝑅)(𝑦 + 𝑧))))
5849, 54, 56, 57syl3anc 1371 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(+g𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥(+g𝑅)(𝑦 + 𝑧))))
5948, 53, 583eqtr4d 2786 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 + 𝑦))(+g𝑈)(𝐹𝑧)) = ((𝐹𝑥)(+g𝑈)(𝐹‘(𝑦 + 𝑧))))
603, 15, 1, 2, 4, 16, 17imasaddval 17414 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑉𝑦𝑉) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝐹‘(𝑥(+g𝑅)𝑦)))
61603adant3r3 1184 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝐹‘(𝑥(+g𝑅)𝑦)))
6243oveqd 7374 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 + 𝑦) = (𝑥(+g𝑅)𝑦))
6362fveq2d 6846 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝑥(+g𝑅)𝑦)))
6461, 63eqtr4d 2779 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝐹‘(𝑥 + 𝑦)))
6564oveq1d 7372 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(+g𝑈)(𝐹𝑦))(+g𝑈)(𝐹𝑧)) = ((𝐹‘(𝑥 + 𝑦))(+g𝑈)(𝐹𝑧)))
663, 15, 1, 2, 4, 16, 17imasaddval 17414 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑉𝑧𝑉) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝐹‘(𝑦(+g𝑅)𝑧)))
67663adant3r1 1182 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝐹‘(𝑦(+g𝑅)𝑧)))
6843oveqd 7374 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑦 + 𝑧) = (𝑦(+g𝑅)𝑧))
6968fveq2d 6846 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘(𝑦 + 𝑧)) = (𝐹‘(𝑦(+g𝑅)𝑧)))
7067, 69eqtr4d 2779 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝐹‘(𝑦 + 𝑧)))
7170oveq2d 7373 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(+g𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))) = ((𝐹𝑥)(+g𝑈)(𝐹‘(𝑦 + 𝑧))))
7259, 65, 713eqtr4d 2786 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(+g𝑈)(𝐹𝑦))(+g𝑈)(𝐹𝑧)) = ((𝐹𝑥)(+g𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))))
73 simp1 1136 . . . . . . . . . . . . 13 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝐹𝑥) = 𝑢)
74 simp2 1137 . . . . . . . . . . . . 13 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝐹𝑦) = 𝑣)
7573, 74oveq12d 7375 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝑢(+g𝑈)𝑣))
76 simp3 1138 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝐹𝑧) = 𝑤)
7775, 76oveq12d 7375 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (((𝐹𝑥)(+g𝑈)(𝐹𝑦))(+g𝑈)(𝐹𝑧)) = ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤))
7874, 76oveq12d 7375 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝑣(+g𝑈)𝑤))
7973, 78oveq12d 7375 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(+g𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤)))
8077, 79eqeq12d 2752 . . . . . . . . . 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 1354 . . . . . . . 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 3150 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (∃𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤))))
8584rexlimdvva 3205 . . . . 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 6756 . . . . 5 (𝐹:𝑉onto𝐵𝐹:𝑉𝐵)
893, 88syl 17 . . . 4 (𝜑𝐹:𝑉𝐵)
90 imasgrp2.3 . . . 4 (𝜑0𝑉)
9189, 90ffvelcdmd 7036 . . 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 17414 . . . . . . . . 9 ((𝜑0𝑉𝑥𝑉) → ((𝐹0 )(+g𝑈)(𝐹𝑥)) = (𝐹‘( 0 (+g𝑅)𝑥)))
9894, 95, 96, 97syl3anc 1371 . . . . . . . 8 ((𝜑𝑥𝑉) → ((𝐹0 )(+g𝑈)(𝐹𝑥)) = (𝐹‘( 0 (+g𝑅)𝑥)))
998adantr 481 . . . . . . . . . 10 ((𝜑𝑥𝑉) → + = (+g𝑅))
10099oveqd 7374 . . . . . . . . 9 ((𝜑𝑥𝑉) → ( 0 + 𝑥) = ( 0 (+g𝑅)𝑥))
101100fveq2d 6846 . . . . . . . 8 ((𝜑𝑥𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹‘( 0 (+g𝑅)𝑥)))
102 imasgrp2.4 . . . . . . . 8 ((𝜑𝑥𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹𝑥))
10398, 101, 1023eqtr2d 2782 . . . . . . 7 ((𝜑𝑥𝑉) → ((𝐹0 )(+g𝑈)(𝐹𝑥)) = (𝐹𝑥))
104 oveq2 7365 . . . . . . . 8 ((𝐹𝑥) = 𝑢 → ((𝐹0 )(+g𝑈)(𝐹𝑥)) = ((𝐹0 )(+g𝑈)𝑢))
105 id 22 . . . . . . . 8 ((𝐹𝑥) = 𝑢 → (𝐹𝑥) = 𝑢)
106104, 105eqeq12d 2752 . . . . . . 7 ((𝐹𝑥) = 𝑢 → (((𝐹0 )(+g𝑈)(𝐹𝑥)) = (𝐹𝑥) ↔ ((𝐹0 )(+g𝑈)𝑢) = 𝑢))
107103, 106syl5ibcom 244 . . . . . 6 ((𝜑𝑥𝑉) → ((𝐹𝑥) = 𝑢 → ((𝐹0 )(+g𝑈)𝑢) = 𝑢))
108107rexlimdva 3152 . . . . 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, 112ffvelcdmd 7036 . . . . . . . 8 ((𝜑𝑥𝑉) → (𝐹𝑁) ∈ 𝐵)
1143, 15, 1, 2, 4, 16, 17imasaddval 17414 . . . . . . . . . 10 ((𝜑𝑁𝑉𝑥𝑉) → ((𝐹𝑁)(+g𝑈)(𝐹𝑥)) = (𝐹‘(𝑁(+g𝑅)𝑥)))
11594, 112, 96, 114syl3anc 1371 . . . . . . . . 9 ((𝜑𝑥𝑉) → ((𝐹𝑁)(+g𝑈)(𝐹𝑥)) = (𝐹‘(𝑁(+g𝑅)𝑥)))
11699oveqd 7374 . . . . . . . . . 10 ((𝜑𝑥𝑉) → (𝑁 + 𝑥) = (𝑁(+g𝑅)𝑥))
117116fveq2d 6846 . . . . . . . . 9 ((𝜑𝑥𝑉) → (𝐹‘(𝑁 + 𝑥)) = (𝐹‘(𝑁(+g𝑅)𝑥)))
118 imasgrp2.6 . . . . . . . . 9 ((𝜑𝑥𝑉) → (𝐹‘(𝑁 + 𝑥)) = (𝐹0 ))
119115, 117, 1183eqtr2d 2782 . . . . . . . 8 ((𝜑𝑥𝑉) → ((𝐹𝑁)(+g𝑈)(𝐹𝑥)) = (𝐹0 ))
120 oveq1 7364 . . . . . . . . . 10 (𝑣 = (𝐹𝑁) → (𝑣(+g𝑈)(𝐹𝑥)) = ((𝐹𝑁)(+g𝑈)(𝐹𝑥)))
121120eqeq1d 2738 . . . . . . . . 9 (𝑣 = (𝐹𝑁) → ((𝑣(+g𝑈)(𝐹𝑥)) = (𝐹0 ) ↔ ((𝐹𝑁)(+g𝑈)(𝐹𝑥)) = (𝐹0 )))
122121rspcev 3581 . . . . . . . 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 7365 . . . . . . . . 9 ((𝐹𝑥) = 𝑢 → (𝑣(+g𝑈)(𝐹𝑥)) = (𝑣(+g𝑈)𝑢))
125124eqeq1d 2738 . . . . . . . 8 ((𝐹𝑥) = 𝑢 → ((𝑣(+g𝑈)(𝐹𝑥)) = (𝐹0 ) ↔ (𝑣(+g𝑈)𝑢) = (𝐹0 )))
126125rexbidv 3175 . . . . . . 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 3152 . . . . 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 18771 . 2 (𝜑𝑈 ∈ Grp)
1325, 6, 91, 110, 131grpidd2 18788 . 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 1087   = wceq 1541  wcel 2106  wrex 3073   × cxp 5631  ran crn 5634   Fn wfn 6491  wf 6492  ontowfo 6494  cfv 6496  (class class class)co 7357  Basecbs 17083  +gcplusg 17133  0gc0g 17321  s cimas 17386  Grpcgrp 18748
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-struct 17019  df-slot 17054  df-ndx 17066  df-base 17084  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-0g 17323  df-imas 17390  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751
This theorem is referenced by:  imasgrp  18863  qusgrp2  18865
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