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Theorem imasgrp2 18938
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 17458 . . 3 (𝜑𝐵 = (Base‘𝑈))
6 eqidd 2734 . . 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 7426 . . . . . . . . 9 (𝜑 → (𝑎 + 𝑏) = (𝑎(+g𝑅)𝑏))
109fveq2d 6896 . . . . . . . 8 (𝜑 → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑎(+g𝑅)𝑏)))
118oveqd 7426 . . . . . . . . 9 (𝜑 → (𝑝 + 𝑞) = (𝑝(+g𝑅)𝑞))
1211fveq2d 6896 . . . . . . . 8 (𝜑 → (𝐹‘(𝑝 + 𝑞)) = (𝐹‘(𝑝(+g𝑅)𝑞)))
1310, 12eqeq12d 2749 . . . . . . 7 (𝜑 → ((𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)) ↔ (𝐹‘(𝑎(+g𝑅)𝑏)) = (𝐹‘(𝑝(+g𝑅)𝑞))))
14133ad2ant1 1134 . . . . . 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 2733 . . . . 5 (+g𝑅) = (+g𝑅)
17 eqid 2733 . . . . 5 (+g𝑈) = (+g𝑈)
1811adantr 482 . . . . . 6 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝 + 𝑞) = (𝑝(+g𝑅)𝑞))
19 imasgrp2.1 . . . . . . . 8 ((𝜑𝑥𝑉𝑦𝑉) → (𝑥 + 𝑦) ∈ 𝑉)
20193expb 1121 . . . . . . 7 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
2120caovclg 7599 . . . . . 6 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝 + 𝑞) ∈ 𝑉)
2218, 21eqeltrrd 2835 . . . . 5 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝(+g𝑅)𝑞) ∈ 𝑉)
233, 15, 1, 2, 4, 16, 17, 22imasaddf 17479 . . . 4 (𝜑 → (+g𝑈):(𝐵 × 𝐵)⟶𝐵)
24 fovcdm 7577 . . . 4 (((+g𝑈):(𝐵 × 𝐵)⟶𝐵𝑢𝐵𝑣𝐵) → (𝑢(+g𝑈)𝑣) ∈ 𝐵)
2523, 24syl3an1 1164 . . 3 ((𝜑𝑢𝐵𝑣𝐵) → (𝑢(+g𝑈)𝑣) ∈ 𝐵)
26 forn 6809 . . . . . . . . . 10 (𝐹:𝑉onto𝐵 → ran 𝐹 = 𝐵)
273, 26syl 17 . . . . . . . . 9 (𝜑 → ran 𝐹 = 𝐵)
2827eleq2d 2820 . . . . . . . 8 (𝜑 → (𝑢 ∈ ran 𝐹𝑢𝐵))
2927eleq2d 2820 . . . . . . . 8 (𝜑 → (𝑣 ∈ ran 𝐹𝑣𝐵))
3027eleq2d 2820 . . . . . . . 8 (𝜑 → (𝑤 ∈ ran 𝐹𝑤𝐵))
3128, 29, 303anbi123d 1437 . . . . . . 7 (𝜑 → ((𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐹𝑤 ∈ ran 𝐹) ↔ (𝑢𝐵𝑣𝐵𝑤𝐵)))
32 fofn 6808 . . . . . . . . 9 (𝐹:𝑉onto𝐵𝐹 Fn 𝑉)
333, 32syl 17 . . . . . . . 8 (𝜑𝐹 Fn 𝑉)
34 fvelrnb 6953 . . . . . . . . 9 (𝐹 Fn 𝑉 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥𝑉 (𝐹𝑥) = 𝑢))
35 fvelrnb 6953 . . . . . . . . 9 (𝐹 Fn 𝑉 → (𝑣 ∈ ran 𝐹 ↔ ∃𝑦𝑉 (𝐹𝑦) = 𝑣))
36 fvelrnb 6953 . . . . . . . . 9 (𝐹 Fn 𝑉 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑧𝑉 (𝐹𝑧) = 𝑤))
3734, 35, 363anbi123d 1437 . . . . . . . 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 281 . . . . . 6 (𝜑 → ((𝑢𝐵𝑣𝐵𝑤𝐵) ↔ (∃𝑥𝑉 (𝐹𝑥) = 𝑢 ∧ ∃𝑦𝑉 (𝐹𝑦) = 𝑣 ∧ ∃𝑧𝑉 (𝐹𝑧) = 𝑤)))
40 3reeanv 3228 . . . . . 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 482 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → + = (+g𝑅))
4443oveqd 7426 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝑥 + 𝑦) + 𝑧) = ((𝑥 + 𝑦)(+g𝑅)𝑧))
4544fveq2d 6896 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘((𝑥 + 𝑦) + 𝑧)) = (𝐹‘((𝑥 + 𝑦)(+g𝑅)𝑧)))
4643oveqd 7426 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 + (𝑦 + 𝑧)) = (𝑥(+g𝑅)(𝑦 + 𝑧)))
4746fveq2d 6896 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘(𝑥 + (𝑦 + 𝑧))) = (𝐹‘(𝑥(+g𝑅)(𝑦 + 𝑧))))
4842, 45, 473eqtr3d 2781 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘((𝑥 + 𝑦)(+g𝑅)𝑧)) = (𝐹‘(𝑥(+g𝑅)(𝑦 + 𝑧))))
49 simpl 484 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝜑)
50193adant3r3 1185 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
51 simpr3 1197 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑧𝑉)
523, 15, 1, 2, 4, 16, 17imasaddval 17478 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 + 𝑦) ∈ 𝑉𝑧𝑉) → ((𝐹‘(𝑥 + 𝑦))(+g𝑈)(𝐹𝑧)) = (𝐹‘((𝑥 + 𝑦)(+g𝑅)𝑧)))
5349, 50, 51, 52syl3anc 1372 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 + 𝑦))(+g𝑈)(𝐹𝑧)) = (𝐹‘((𝑥 + 𝑦)(+g𝑅)𝑧)))
54 simpr1 1195 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑥𝑉)
5521caovclg 7599 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝑉𝑧𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
56553adantr1 1170 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
573, 15, 1, 2, 4, 16, 17imasaddval 17478 . . . . . . . . . . . . 13 ((𝜑𝑥𝑉 ∧ (𝑦 + 𝑧) ∈ 𝑉) → ((𝐹𝑥)(+g𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥(+g𝑅)(𝑦 + 𝑧))))
5849, 54, 56, 57syl3anc 1372 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(+g𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥(+g𝑅)(𝑦 + 𝑧))))
5948, 53, 583eqtr4d 2783 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 + 𝑦))(+g𝑈)(𝐹𝑧)) = ((𝐹𝑥)(+g𝑈)(𝐹‘(𝑦 + 𝑧))))
603, 15, 1, 2, 4, 16, 17imasaddval 17478 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑉𝑦𝑉) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝐹‘(𝑥(+g𝑅)𝑦)))
61603adant3r3 1185 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝐹‘(𝑥(+g𝑅)𝑦)))
6243oveqd 7426 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 + 𝑦) = (𝑥(+g𝑅)𝑦))
6362fveq2d 6896 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝑥(+g𝑅)𝑦)))
6461, 63eqtr4d 2776 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝐹‘(𝑥 + 𝑦)))
6564oveq1d 7424 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(+g𝑈)(𝐹𝑦))(+g𝑈)(𝐹𝑧)) = ((𝐹‘(𝑥 + 𝑦))(+g𝑈)(𝐹𝑧)))
663, 15, 1, 2, 4, 16, 17imasaddval 17478 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑉𝑧𝑉) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝐹‘(𝑦(+g𝑅)𝑧)))
67663adant3r1 1183 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝐹‘(𝑦(+g𝑅)𝑧)))
6843oveqd 7426 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑦 + 𝑧) = (𝑦(+g𝑅)𝑧))
6968fveq2d 6896 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘(𝑦 + 𝑧)) = (𝐹‘(𝑦(+g𝑅)𝑧)))
7067, 69eqtr4d 2776 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝐹‘(𝑦 + 𝑧)))
7170oveq2d 7425 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(+g𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))) = ((𝐹𝑥)(+g𝑈)(𝐹‘(𝑦 + 𝑧))))
7259, 65, 713eqtr4d 2783 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(+g𝑈)(𝐹𝑦))(+g𝑈)(𝐹𝑧)) = ((𝐹𝑥)(+g𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))))
73 simp1 1137 . . . . . . . . . . . . 13 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝐹𝑥) = 𝑢)
74 simp2 1138 . . . . . . . . . . . . 13 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝐹𝑦) = 𝑣)
7573, 74oveq12d 7427 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝑢(+g𝑈)𝑣))
76 simp3 1139 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝐹𝑧) = 𝑤)
7775, 76oveq12d 7427 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (((𝐹𝑥)(+g𝑈)(𝐹𝑦))(+g𝑈)(𝐹𝑧)) = ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤))
7874, 76oveq12d 7427 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝑣(+g𝑈)𝑤))
7973, 78oveq12d 7427 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(+g𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤)))
8077, 79eqeq12d 2749 . . . . . . . . . 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 1355 . . . . . . . 8 (𝜑 → (𝑥𝑉 → (𝑦𝑉 → (𝑧𝑉 → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤)))))))
8382imp32 420 . . . . . . 7 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑧𝑉 → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤)))))
8483rexlimdv 3154 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (∃𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤))))
8584rexlimdvva 3212 . . . . 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 408 . . 3 ((𝜑 ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤)))
88 fof 6806 . . . . 5 (𝐹:𝑉onto𝐵𝐹:𝑉𝐵)
893, 88syl 17 . . . 4 (𝜑𝐹:𝑉𝐵)
90 imasgrp2.3 . . . 4 (𝜑0𝑉)
9189, 90ffvelcdmd 7088 . . 3 (𝜑 → (𝐹0 ) ∈ 𝐵)
9233, 34syl 17 . . . . . 6 (𝜑 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥𝑉 (𝐹𝑥) = 𝑢))
9328, 92bitr3d 281 . . . . 5 (𝜑 → (𝑢𝐵 ↔ ∃𝑥𝑉 (𝐹𝑥) = 𝑢))
94 simpl 484 . . . . . . . . 9 ((𝜑𝑥𝑉) → 𝜑)
9590adantr 482 . . . . . . . . 9 ((𝜑𝑥𝑉) → 0𝑉)
96 simpr 486 . . . . . . . . 9 ((𝜑𝑥𝑉) → 𝑥𝑉)
973, 15, 1, 2, 4, 16, 17imasaddval 17478 . . . . . . . . 9 ((𝜑0𝑉𝑥𝑉) → ((𝐹0 )(+g𝑈)(𝐹𝑥)) = (𝐹‘( 0 (+g𝑅)𝑥)))
9894, 95, 96, 97syl3anc 1372 . . . . . . . 8 ((𝜑𝑥𝑉) → ((𝐹0 )(+g𝑈)(𝐹𝑥)) = (𝐹‘( 0 (+g𝑅)𝑥)))
998adantr 482 . . . . . . . . . 10 ((𝜑𝑥𝑉) → + = (+g𝑅))
10099oveqd 7426 . . . . . . . . 9 ((𝜑𝑥𝑉) → ( 0 + 𝑥) = ( 0 (+g𝑅)𝑥))
101100fveq2d 6896 . . . . . . . 8 ((𝜑𝑥𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹‘( 0 (+g𝑅)𝑥)))
102 imasgrp2.4 . . . . . . . 8 ((𝜑𝑥𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹𝑥))
10398, 101, 1023eqtr2d 2779 . . . . . . 7 ((𝜑𝑥𝑉) → ((𝐹0 )(+g𝑈)(𝐹𝑥)) = (𝐹𝑥))
104 oveq2 7417 . . . . . . . 8 ((𝐹𝑥) = 𝑢 → ((𝐹0 )(+g𝑈)(𝐹𝑥)) = ((𝐹0 )(+g𝑈)𝑢))
105 id 22 . . . . . . . 8 ((𝐹𝑥) = 𝑢 → (𝐹𝑥) = 𝑢)
106104, 105eqeq12d 2749 . . . . . . 7 ((𝐹𝑥) = 𝑢 → (((𝐹0 )(+g𝑈)(𝐹𝑥)) = (𝐹𝑥) ↔ ((𝐹0 )(+g𝑈)𝑢) = 𝑢))
107103, 106syl5ibcom 244 . . . . . 6 ((𝜑𝑥𝑉) → ((𝐹𝑥) = 𝑢 → ((𝐹0 )(+g𝑈)𝑢) = 𝑢))
108107rexlimdva 3156 . . . . 5 (𝜑 → (∃𝑥𝑉 (𝐹𝑥) = 𝑢 → ((𝐹0 )(+g𝑈)𝑢) = 𝑢))
10993, 108sylbid 239 . . . 4 (𝜑 → (𝑢𝐵 → ((𝐹0 )(+g𝑈)𝑢) = 𝑢))
110109imp 408 . . 3 ((𝜑𝑢𝐵) → ((𝐹0 )(+g𝑈)𝑢) = 𝑢)
11189adantr 482 . . . . . . . . 9 ((𝜑𝑥𝑉) → 𝐹:𝑉𝐵)
112 imasgrp2.5 . . . . . . . . 9 ((𝜑𝑥𝑉) → 𝑁𝑉)
113111, 112ffvelcdmd 7088 . . . . . . . 8 ((𝜑𝑥𝑉) → (𝐹𝑁) ∈ 𝐵)
1143, 15, 1, 2, 4, 16, 17imasaddval 17478 . . . . . . . . . 10 ((𝜑𝑁𝑉𝑥𝑉) → ((𝐹𝑁)(+g𝑈)(𝐹𝑥)) = (𝐹‘(𝑁(+g𝑅)𝑥)))
11594, 112, 96, 114syl3anc 1372 . . . . . . . . 9 ((𝜑𝑥𝑉) → ((𝐹𝑁)(+g𝑈)(𝐹𝑥)) = (𝐹‘(𝑁(+g𝑅)𝑥)))
11699oveqd 7426 . . . . . . . . . 10 ((𝜑𝑥𝑉) → (𝑁 + 𝑥) = (𝑁(+g𝑅)𝑥))
117116fveq2d 6896 . . . . . . . . 9 ((𝜑𝑥𝑉) → (𝐹‘(𝑁 + 𝑥)) = (𝐹‘(𝑁(+g𝑅)𝑥)))
118 imasgrp2.6 . . . . . . . . 9 ((𝜑𝑥𝑉) → (𝐹‘(𝑁 + 𝑥)) = (𝐹0 ))
119115, 117, 1183eqtr2d 2779 . . . . . . . 8 ((𝜑𝑥𝑉) → ((𝐹𝑁)(+g𝑈)(𝐹𝑥)) = (𝐹0 ))
120 oveq1 7416 . . . . . . . . . 10 (𝑣 = (𝐹𝑁) → (𝑣(+g𝑈)(𝐹𝑥)) = ((𝐹𝑁)(+g𝑈)(𝐹𝑥)))
121120eqeq1d 2735 . . . . . . . . 9 (𝑣 = (𝐹𝑁) → ((𝑣(+g𝑈)(𝐹𝑥)) = (𝐹0 ) ↔ ((𝐹𝑁)(+g𝑈)(𝐹𝑥)) = (𝐹0 )))
122121rspcev 3613 . . . . . . . 8 (((𝐹𝑁) ∈ 𝐵 ∧ ((𝐹𝑁)(+g𝑈)(𝐹𝑥)) = (𝐹0 )) → ∃𝑣𝐵 (𝑣(+g𝑈)(𝐹𝑥)) = (𝐹0 ))
123113, 119, 122syl2anc 585 . . . . . . 7 ((𝜑𝑥𝑉) → ∃𝑣𝐵 (𝑣(+g𝑈)(𝐹𝑥)) = (𝐹0 ))
124 oveq2 7417 . . . . . . . . 9 ((𝐹𝑥) = 𝑢 → (𝑣(+g𝑈)(𝐹𝑥)) = (𝑣(+g𝑈)𝑢))
125124eqeq1d 2735 . . . . . . . 8 ((𝐹𝑥) = 𝑢 → ((𝑣(+g𝑈)(𝐹𝑥)) = (𝐹0 ) ↔ (𝑣(+g𝑈)𝑢) = (𝐹0 )))
126125rexbidv 3179 . . . . . . 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 3156 . . . . 5 (𝜑 → (∃𝑥𝑉 (𝐹𝑥) = 𝑢 → ∃𝑣𝐵 (𝑣(+g𝑈)𝑢) = (𝐹0 )))
12993, 128sylbid 239 . . . 4 (𝜑 → (𝑢𝐵 → ∃𝑣𝐵 (𝑣(+g𝑈)𝑢) = (𝐹0 )))
130129imp 408 . . 3 ((𝜑𝑢𝐵) → ∃𝑣𝐵 (𝑣(+g𝑈)𝑢) = (𝐹0 ))
1315, 6, 25, 87, 91, 110, 130isgrpde 18843 . 2 (𝜑𝑈 ∈ Grp)
1325, 6, 91, 110, 131grpidd2 18862 . 2 (𝜑 → (𝐹0 ) = (0g𝑈))
133131, 132jca 513 1 (𝜑 → (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wrex 3071   × cxp 5675  ran crn 5678   Fn wfn 6539  wf 6540  ontowfo 6542  cfv 6544  (class class class)co 7409  Basecbs 17144  +gcplusg 17197  0gc0g 17385  s cimas 17450  Grpcgrp 18819
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9437  df-inf 9438  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-uz 12823  df-fz 13485  df-struct 17080  df-slot 17115  df-ndx 17127  df-base 17145  df-plusg 17210  df-mulr 17211  df-sca 17213  df-vsca 17214  df-ip 17215  df-tset 17216  df-ple 17217  df-ds 17219  df-0g 17387  df-imas 17454  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822
This theorem is referenced by:  imasgrp  18939  qusgrp2  18941
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