MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  imasgrp2 Structured version   Visualization version   GIF version

Theorem imasgrp2 17302
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 15944 . . 3 (𝜑𝐵 = (Base‘𝑈))
6 eqidd 2611 . . 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 6544 . . . . . . . . 9 (𝜑 → (𝑎 + 𝑏) = (𝑎(+g𝑅)𝑏))
109fveq2d 6092 . . . . . . . 8 (𝜑 → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑎(+g𝑅)𝑏)))
118oveqd 6544 . . . . . . . . 9 (𝜑 → (𝑝 + 𝑞) = (𝑝(+g𝑅)𝑞))
1211fveq2d 6092 . . . . . . . 8 (𝜑 → (𝐹‘(𝑝 + 𝑞)) = (𝐹‘(𝑝(+g𝑅)𝑞)))
1310, 12eqeq12d 2625 . . . . . . 7 (𝜑 → ((𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)) ↔ (𝐹‘(𝑎(+g𝑅)𝑏)) = (𝐹‘(𝑝(+g𝑅)𝑞))))
14133ad2ant1 1075 . . . . . 6 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → ((𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)) ↔ (𝐹‘(𝑎(+g𝑅)𝑏)) = (𝐹‘(𝑝(+g𝑅)𝑞))))
157, 14sylibd 228 . . . . 5 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎(+g𝑅)𝑏)) = (𝐹‘(𝑝(+g𝑅)𝑞))))
16 eqid 2610 . . . . 5 (+g𝑅) = (+g𝑅)
17 eqid 2610 . . . . 5 (+g𝑈) = (+g𝑈)
1811adantr 480 . . . . . 6 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝 + 𝑞) = (𝑝(+g𝑅)𝑞))
19 imasgrp2.1 . . . . . . . 8 ((𝜑𝑥𝑉𝑦𝑉) → (𝑥 + 𝑦) ∈ 𝑉)
20193expb 1258 . . . . . . 7 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
2120caovclg 6702 . . . . . 6 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝 + 𝑞) ∈ 𝑉)
2218, 21eqeltrrd 2689 . . . . 5 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝(+g𝑅)𝑞) ∈ 𝑉)
233, 15, 1, 2, 4, 16, 17, 22imasaddf 15965 . . . 4 (𝜑 → (+g𝑈):(𝐵 × 𝐵)⟶𝐵)
24 fovrn 6680 . . . 4 (((+g𝑈):(𝐵 × 𝐵)⟶𝐵𝑢𝐵𝑣𝐵) → (𝑢(+g𝑈)𝑣) ∈ 𝐵)
2523, 24syl3an1 1351 . . 3 ((𝜑𝑢𝐵𝑣𝐵) → (𝑢(+g𝑈)𝑣) ∈ 𝐵)
26 forn 6016 . . . . . . . . . 10 (𝐹:𝑉onto𝐵 → ran 𝐹 = 𝐵)
273, 26syl 17 . . . . . . . . 9 (𝜑 → ran 𝐹 = 𝐵)
2827eleq2d 2673 . . . . . . . 8 (𝜑 → (𝑢 ∈ ran 𝐹𝑢𝐵))
2927eleq2d 2673 . . . . . . . 8 (𝜑 → (𝑣 ∈ ran 𝐹𝑣𝐵))
3027eleq2d 2673 . . . . . . . 8 (𝜑 → (𝑤 ∈ ran 𝐹𝑤𝐵))
3128, 29, 303anbi123d 1391 . . . . . . 7 (𝜑 → ((𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐹𝑤 ∈ ran 𝐹) ↔ (𝑢𝐵𝑣𝐵𝑤𝐵)))
32 fofn 6015 . . . . . . . . 9 (𝐹:𝑉onto𝐵𝐹 Fn 𝑉)
333, 32syl 17 . . . . . . . 8 (𝜑𝐹 Fn 𝑉)
34 fvelrnb 6138 . . . . . . . . 9 (𝐹 Fn 𝑉 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥𝑉 (𝐹𝑥) = 𝑢))
35 fvelrnb 6138 . . . . . . . . 9 (𝐹 Fn 𝑉 → (𝑣 ∈ ran 𝐹 ↔ ∃𝑦𝑉 (𝐹𝑦) = 𝑣))
36 fvelrnb 6138 . . . . . . . . 9 (𝐹 Fn 𝑉 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑧𝑉 (𝐹𝑧) = 𝑤))
3734, 35, 363anbi123d 1391 . . . . . . . 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 269 . . . . . 6 (𝜑 → ((𝑢𝐵𝑣𝐵𝑤𝐵) ↔ (∃𝑥𝑉 (𝐹𝑥) = 𝑢 ∧ ∃𝑦𝑉 (𝐹𝑦) = 𝑣 ∧ ∃𝑧𝑉 (𝐹𝑧) = 𝑤)))
40 3reeanv 3087 . . . . . 6 (∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) ↔ (∃𝑥𝑉 (𝐹𝑥) = 𝑢 ∧ ∃𝑦𝑉 (𝐹𝑦) = 𝑣 ∧ ∃𝑧𝑉 (𝐹𝑧) = 𝑤))
4139, 40syl6bbr 277 . . . . 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 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → + = (+g𝑅))
4443oveqd 6544 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝑥 + 𝑦) + 𝑧) = ((𝑥 + 𝑦)(+g𝑅)𝑧))
4544fveq2d 6092 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘((𝑥 + 𝑦) + 𝑧)) = (𝐹‘((𝑥 + 𝑦)(+g𝑅)𝑧)))
4643oveqd 6544 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 + (𝑦 + 𝑧)) = (𝑥(+g𝑅)(𝑦 + 𝑧)))
4746fveq2d 6092 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘(𝑥 + (𝑦 + 𝑧))) = (𝐹‘(𝑥(+g𝑅)(𝑦 + 𝑧))))
4842, 45, 473eqtr3d 2652 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘((𝑥 + 𝑦)(+g𝑅)𝑧)) = (𝐹‘(𝑥(+g𝑅)(𝑦 + 𝑧))))
49 simpl 472 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝜑)
50193adant3r3 1268 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
51 simpr3 1062 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑧𝑉)
523, 15, 1, 2, 4, 16, 17imasaddval 15964 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 + 𝑦) ∈ 𝑉𝑧𝑉) → ((𝐹‘(𝑥 + 𝑦))(+g𝑈)(𝐹𝑧)) = (𝐹‘((𝑥 + 𝑦)(+g𝑅)𝑧)))
5349, 50, 51, 52syl3anc 1318 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 + 𝑦))(+g𝑈)(𝐹𝑧)) = (𝐹‘((𝑥 + 𝑦)(+g𝑅)𝑧)))
54 simpr1 1060 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑥𝑉)
5521caovclg 6702 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝑉𝑧𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
56553adantr1 1213 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
573, 15, 1, 2, 4, 16, 17imasaddval 15964 . . . . . . . . . . . . 13 ((𝜑𝑥𝑉 ∧ (𝑦 + 𝑧) ∈ 𝑉) → ((𝐹𝑥)(+g𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥(+g𝑅)(𝑦 + 𝑧))))
5849, 54, 56, 57syl3anc 1318 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(+g𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥(+g𝑅)(𝑦 + 𝑧))))
5948, 53, 583eqtr4d 2654 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 + 𝑦))(+g𝑈)(𝐹𝑧)) = ((𝐹𝑥)(+g𝑈)(𝐹‘(𝑦 + 𝑧))))
603, 15, 1, 2, 4, 16, 17imasaddval 15964 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑉𝑦𝑉) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝐹‘(𝑥(+g𝑅)𝑦)))
61603adant3r3 1268 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝐹‘(𝑥(+g𝑅)𝑦)))
6243oveqd 6544 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 + 𝑦) = (𝑥(+g𝑅)𝑦))
6362fveq2d 6092 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝑥(+g𝑅)𝑦)))
6461, 63eqtr4d 2647 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝐹‘(𝑥 + 𝑦)))
6564oveq1d 6542 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(+g𝑈)(𝐹𝑦))(+g𝑈)(𝐹𝑧)) = ((𝐹‘(𝑥 + 𝑦))(+g𝑈)(𝐹𝑧)))
663, 15, 1, 2, 4, 16, 17imasaddval 15964 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑉𝑧𝑉) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝐹‘(𝑦(+g𝑅)𝑧)))
67663adant3r1 1266 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝐹‘(𝑦(+g𝑅)𝑧)))
6843oveqd 6544 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑦 + 𝑧) = (𝑦(+g𝑅)𝑧))
6968fveq2d 6092 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘(𝑦 + 𝑧)) = (𝐹‘(𝑦(+g𝑅)𝑧)))
7067, 69eqtr4d 2647 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝐹‘(𝑦 + 𝑧)))
7170oveq2d 6543 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(+g𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))) = ((𝐹𝑥)(+g𝑈)(𝐹‘(𝑦 + 𝑧))))
7259, 65, 713eqtr4d 2654 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(+g𝑈)(𝐹𝑦))(+g𝑈)(𝐹𝑧)) = ((𝐹𝑥)(+g𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))))
73 simp1 1054 . . . . . . . . . . . . 13 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝐹𝑥) = 𝑢)
74 simp2 1055 . . . . . . . . . . . . 13 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝐹𝑦) = 𝑣)
7573, 74oveq12d 6545 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝑢(+g𝑈)𝑣))
76 simp3 1056 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝐹𝑧) = 𝑤)
7775, 76oveq12d 6545 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (((𝐹𝑥)(+g𝑈)(𝐹𝑦))(+g𝑈)(𝐹𝑧)) = ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤))
7874, 76oveq12d 6545 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝑣(+g𝑈)𝑤))
7973, 78oveq12d 6545 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(+g𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤)))
8077, 79eqeq12d 2625 . . . . . . . . . 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 234 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤))))
82813exp2 1277 . . . . . . . 8 (𝜑 → (𝑥𝑉 → (𝑦𝑉 → (𝑧𝑉 → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤)))))))
8382imp32 448 . . . . . . 7 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑧𝑉 → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤)))))
8483rexlimdv 3012 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (∃𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤))))
8584rexlimdvva 3020 . . . . 5 (𝜑 → (∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤))))
8641, 85sylbid 229 . . . 4 (𝜑 → ((𝑢𝐵𝑣𝐵𝑤𝐵) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤))))
8786imp 444 . . 3 ((𝜑 ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢(+g𝑈)𝑣)(+g𝑈)𝑤) = (𝑢(+g𝑈)(𝑣(+g𝑈)𝑤)))
88 fof 6013 . . . . 5 (𝐹:𝑉onto𝐵𝐹:𝑉𝐵)
893, 88syl 17 . . . 4 (𝜑𝐹:𝑉𝐵)
90 imasgrp2.3 . . . 4 (𝜑0𝑉)
9189, 90ffvelrnd 6253 . . 3 (𝜑 → (𝐹0 ) ∈ 𝐵)
9233, 34syl 17 . . . . . 6 (𝜑 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥𝑉 (𝐹𝑥) = 𝑢))
9328, 92bitr3d 269 . . . . 5 (𝜑 → (𝑢𝐵 ↔ ∃𝑥𝑉 (𝐹𝑥) = 𝑢))
94 simpl 472 . . . . . . . . 9 ((𝜑𝑥𝑉) → 𝜑)
9590adantr 480 . . . . . . . . 9 ((𝜑𝑥𝑉) → 0𝑉)
96 simpr 476 . . . . . . . . 9 ((𝜑𝑥𝑉) → 𝑥𝑉)
973, 15, 1, 2, 4, 16, 17imasaddval 15964 . . . . . . . . 9 ((𝜑0𝑉𝑥𝑉) → ((𝐹0 )(+g𝑈)(𝐹𝑥)) = (𝐹‘( 0 (+g𝑅)𝑥)))
9894, 95, 96, 97syl3anc 1318 . . . . . . . 8 ((𝜑𝑥𝑉) → ((𝐹0 )(+g𝑈)(𝐹𝑥)) = (𝐹‘( 0 (+g𝑅)𝑥)))
998adantr 480 . . . . . . . . . 10 ((𝜑𝑥𝑉) → + = (+g𝑅))
10099oveqd 6544 . . . . . . . . 9 ((𝜑𝑥𝑉) → ( 0 + 𝑥) = ( 0 (+g𝑅)𝑥))
101100fveq2d 6092 . . . . . . . 8 ((𝜑𝑥𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹‘( 0 (+g𝑅)𝑥)))
102 imasgrp2.4 . . . . . . . 8 ((𝜑𝑥𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹𝑥))
10398, 101, 1023eqtr2d 2650 . . . . . . 7 ((𝜑𝑥𝑉) → ((𝐹0 )(+g𝑈)(𝐹𝑥)) = (𝐹𝑥))
104 oveq2 6535 . . . . . . . 8 ((𝐹𝑥) = 𝑢 → ((𝐹0 )(+g𝑈)(𝐹𝑥)) = ((𝐹0 )(+g𝑈)𝑢))
105 id 22 . . . . . . . 8 ((𝐹𝑥) = 𝑢 → (𝐹𝑥) = 𝑢)
106104, 105eqeq12d 2625 . . . . . . 7 ((𝐹𝑥) = 𝑢 → (((𝐹0 )(+g𝑈)(𝐹𝑥)) = (𝐹𝑥) ↔ ((𝐹0 )(+g𝑈)𝑢) = 𝑢))
107103, 106syl5ibcom 234 . . . . . 6 ((𝜑𝑥𝑉) → ((𝐹𝑥) = 𝑢 → ((𝐹0 )(+g𝑈)𝑢) = 𝑢))
108107rexlimdva 3013 . . . . 5 (𝜑 → (∃𝑥𝑉 (𝐹𝑥) = 𝑢 → ((𝐹0 )(+g𝑈)𝑢) = 𝑢))
10993, 108sylbid 229 . . . 4 (𝜑 → (𝑢𝐵 → ((𝐹0 )(+g𝑈)𝑢) = 𝑢))
110109imp 444 . . 3 ((𝜑𝑢𝐵) → ((𝐹0 )(+g𝑈)𝑢) = 𝑢)
11189adantr 480 . . . . . . . . 9 ((𝜑𝑥𝑉) → 𝐹:𝑉𝐵)
112 imasgrp2.5 . . . . . . . . 9 ((𝜑𝑥𝑉) → 𝑁𝑉)
113111, 112ffvelrnd 6253 . . . . . . . 8 ((𝜑𝑥𝑉) → (𝐹𝑁) ∈ 𝐵)
1143, 15, 1, 2, 4, 16, 17imasaddval 15964 . . . . . . . . . 10 ((𝜑𝑁𝑉𝑥𝑉) → ((𝐹𝑁)(+g𝑈)(𝐹𝑥)) = (𝐹‘(𝑁(+g𝑅)𝑥)))
11594, 112, 96, 114syl3anc 1318 . . . . . . . . 9 ((𝜑𝑥𝑉) → ((𝐹𝑁)(+g𝑈)(𝐹𝑥)) = (𝐹‘(𝑁(+g𝑅)𝑥)))
11699oveqd 6544 . . . . . . . . . 10 ((𝜑𝑥𝑉) → (𝑁 + 𝑥) = (𝑁(+g𝑅)𝑥))
117116fveq2d 6092 . . . . . . . . 9 ((𝜑𝑥𝑉) → (𝐹‘(𝑁 + 𝑥)) = (𝐹‘(𝑁(+g𝑅)𝑥)))
118 imasgrp2.6 . . . . . . . . 9 ((𝜑𝑥𝑉) → (𝐹‘(𝑁 + 𝑥)) = (𝐹0 ))
119115, 117, 1183eqtr2d 2650 . . . . . . . 8 ((𝜑𝑥𝑉) → ((𝐹𝑁)(+g𝑈)(𝐹𝑥)) = (𝐹0 ))
120 oveq1 6534 . . . . . . . . . 10 (𝑣 = (𝐹𝑁) → (𝑣(+g𝑈)(𝐹𝑥)) = ((𝐹𝑁)(+g𝑈)(𝐹𝑥)))
121120eqeq1d 2612 . . . . . . . . 9 (𝑣 = (𝐹𝑁) → ((𝑣(+g𝑈)(𝐹𝑥)) = (𝐹0 ) ↔ ((𝐹𝑁)(+g𝑈)(𝐹𝑥)) = (𝐹0 )))
122121rspcev 3282 . . . . . . . 8 (((𝐹𝑁) ∈ 𝐵 ∧ ((𝐹𝑁)(+g𝑈)(𝐹𝑥)) = (𝐹0 )) → ∃𝑣𝐵 (𝑣(+g𝑈)(𝐹𝑥)) = (𝐹0 ))
123113, 119, 122syl2anc 691 . . . . . . 7 ((𝜑𝑥𝑉) → ∃𝑣𝐵 (𝑣(+g𝑈)(𝐹𝑥)) = (𝐹0 ))
124 oveq2 6535 . . . . . . . . 9 ((𝐹𝑥) = 𝑢 → (𝑣(+g𝑈)(𝐹𝑥)) = (𝑣(+g𝑈)𝑢))
125124eqeq1d 2612 . . . . . . . 8 ((𝐹𝑥) = 𝑢 → ((𝑣(+g𝑈)(𝐹𝑥)) = (𝐹0 ) ↔ (𝑣(+g𝑈)𝑢) = (𝐹0 )))
126125rexbidv 3034 . . . . . . 7 ((𝐹𝑥) = 𝑢 → (∃𝑣𝐵 (𝑣(+g𝑈)(𝐹𝑥)) = (𝐹0 ) ↔ ∃𝑣𝐵 (𝑣(+g𝑈)𝑢) = (𝐹0 )))
127123, 126syl5ibcom 234 . . . . . 6 ((𝜑𝑥𝑉) → ((𝐹𝑥) = 𝑢 → ∃𝑣𝐵 (𝑣(+g𝑈)𝑢) = (𝐹0 )))
128127rexlimdva 3013 . . . . 5 (𝜑 → (∃𝑥𝑉 (𝐹𝑥) = 𝑢 → ∃𝑣𝐵 (𝑣(+g𝑈)𝑢) = (𝐹0 )))
12993, 128sylbid 229 . . . 4 (𝜑 → (𝑢𝐵 → ∃𝑣𝐵 (𝑣(+g𝑈)𝑢) = (𝐹0 )))
130129imp 444 . . 3 ((𝜑𝑢𝐵) → ∃𝑣𝐵 (𝑣(+g𝑈)𝑢) = (𝐹0 ))
1315, 6, 25, 87, 91, 110, 130isgrpde 17215 . 2 (𝜑𝑈 ∈ Grp)
1325, 6, 91, 110, 131grpidd2 17231 . 2 (𝜑 → (𝐹0 ) = (0g𝑈))
133131, 132jca 553 1 (𝜑 → (𝑈 ∈ Grp ∧ (𝐹0 ) = (0g𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wrex 2897   × cxp 5026  ran crn 5029   Fn wfn 5785  wf 5786  ontowfo 5788  cfv 5790  (class class class)co 6527  Basecbs 15644  +gcplusg 15717  0gc0g 15872  s cimas 15936  Grpcgrp 17194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4368  df-int 4406  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-tr 4676  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6936  df-1st 7037  df-2nd 7038  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-1o 7425  df-oadd 7429  df-er 7607  df-en 7820  df-dom 7821  df-sdom 7822  df-fin 7823  df-sup 8209  df-inf 8210  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-nn 10871  df-2 10929  df-3 10930  df-4 10931  df-5 10932  df-6 10933  df-7 10934  df-8 10935  df-9 10936  df-n0 11143  df-z 11214  df-dec 11329  df-uz 11523  df-fz 12156  df-struct 15646  df-ndx 15647  df-slot 15648  df-base 15649  df-plusg 15730  df-mulr 15731  df-sca 15733  df-vsca 15734  df-ip 15735  df-tset 15736  df-ple 15737  df-ds 15740  df-0g 15874  df-imas 15940  df-mgm 17014  df-sgrp 17056  df-mnd 17067  df-grp 17197
This theorem is referenced by:  imasgrp  17303  qusgrp2  17305
  Copyright terms: Public domain W3C validator