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Theorem imasring 20344
Description: The image structure of a ring is a ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
Hypotheses
Ref Expression
imasring.u (𝜑𝑈 = (𝐹s 𝑅))
imasring.v (𝜑𝑉 = (Base‘𝑅))
imasring.p + = (+g𝑅)
imasring.t · = (.r𝑅)
imasring.o 1 = (1r𝑅)
imasring.f (𝜑𝐹:𝑉onto𝐵)
imasring.e1 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
imasring.e2 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))
imasring.r (𝜑𝑅 ∈ Ring)
Assertion
Ref Expression
imasring (𝜑 → (𝑈 ∈ Ring ∧ (𝐹1 ) = (1r𝑈)))
Distinct variable groups:   𝑞,𝑝, +   𝑎,𝑏,𝑝,𝑞,𝜑   𝑈,𝑎,𝑏,𝑝,𝑞   1 ,𝑝,𝑞   𝐵,𝑝,𝑞   𝐹,𝑎,𝑏,𝑝,𝑞   𝑅,𝑝,𝑞   𝑉,𝑎,𝑏,𝑝,𝑞   · ,𝑝,𝑞
Allowed substitution hints:   𝐵(𝑎,𝑏)   + (𝑎,𝑏)   𝑅(𝑎,𝑏)   · (𝑎,𝑏)   1 (𝑎,𝑏)

Proof of Theorem imasring
Dummy variables 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imasring.u . . . 4 (𝜑𝑈 = (𝐹s 𝑅))
2 imasring.v . . . 4 (𝜑𝑉 = (Base‘𝑅))
3 imasring.f . . . 4 (𝜑𝐹:𝑉onto𝐵)
4 imasring.r . . . 4 (𝜑𝑅 ∈ Ring)
51, 2, 3, 4imasbas 17559 . . 3 (𝜑𝐵 = (Base‘𝑈))
6 eqidd 2736 . . 3 (𝜑 → (+g𝑈) = (+g𝑈))
7 eqidd 2736 . . 3 (𝜑 → (.r𝑈) = (.r𝑈))
8 imasring.p . . . . . 6 + = (+g𝑅)
98a1i 11 . . . . 5 (𝜑+ = (+g𝑅))
10 imasring.e1 . . . . 5 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
11 ringgrp 20256 . . . . . 6 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
124, 11syl 17 . . . . 5 (𝜑𝑅 ∈ Grp)
13 eqid 2735 . . . . 5 (0g𝑅) = (0g𝑅)
141, 2, 9, 3, 10, 12, 13imasgrp 19087 . . . 4 (𝜑 → (𝑈 ∈ Grp ∧ (𝐹‘(0g𝑅)) = (0g𝑈)))
1514simpld 494 . . 3 (𝜑𝑈 ∈ Grp)
16 imasring.e2 . . . . 5 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))
17 imasring.t . . . . 5 · = (.r𝑅)
18 eqid 2735 . . . . 5 (.r𝑈) = (.r𝑈)
194adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → 𝑅 ∈ Ring)
20 simprl 771 . . . . . . . . 9 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → 𝑢𝑉)
212adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → 𝑉 = (Base‘𝑅))
2220, 21eleqtrd 2841 . . . . . . . 8 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → 𝑢 ∈ (Base‘𝑅))
23 simprr 773 . . . . . . . . 9 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → 𝑣𝑉)
2423, 21eleqtrd 2841 . . . . . . . 8 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → 𝑣 ∈ (Base‘𝑅))
25 eqid 2735 . . . . . . . . 9 (Base‘𝑅) = (Base‘𝑅)
2625, 17ringcl 20268 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ (Base‘𝑅)) → (𝑢 · 𝑣) ∈ (Base‘𝑅))
2719, 22, 24, 26syl3anc 1370 . . . . . . 7 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → (𝑢 · 𝑣) ∈ (Base‘𝑅))
2827, 21eleqtrrd 2842 . . . . . 6 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → (𝑢 · 𝑣) ∈ 𝑉)
2928caovclg 7625 . . . . 5 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)
303, 16, 1, 2, 4, 17, 18, 29imasmulf 17583 . . . 4 (𝜑 → (.r𝑈):(𝐵 × 𝐵)⟶𝐵)
31 fovcdm 7603 . . . 4 (((.r𝑈):(𝐵 × 𝐵)⟶𝐵𝑢𝐵𝑣𝐵) → (𝑢(.r𝑈)𝑣) ∈ 𝐵)
3230, 31syl3an1 1162 . . 3 ((𝜑𝑢𝐵𝑣𝐵) → (𝑢(.r𝑈)𝑣) ∈ 𝐵)
33 forn 6824 . . . . . . . . . 10 (𝐹:𝑉onto𝐵 → ran 𝐹 = 𝐵)
343, 33syl 17 . . . . . . . . 9 (𝜑 → ran 𝐹 = 𝐵)
3534eleq2d 2825 . . . . . . . 8 (𝜑 → (𝑢 ∈ ran 𝐹𝑢𝐵))
3634eleq2d 2825 . . . . . . . 8 (𝜑 → (𝑣 ∈ ran 𝐹𝑣𝐵))
3734eleq2d 2825 . . . . . . . 8 (𝜑 → (𝑤 ∈ ran 𝐹𝑤𝐵))
3835, 36, 373anbi123d 1435 . . . . . . 7 (𝜑 → ((𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐹𝑤 ∈ ran 𝐹) ↔ (𝑢𝐵𝑣𝐵𝑤𝐵)))
39 fofn 6823 . . . . . . . . 9 (𝐹:𝑉onto𝐵𝐹 Fn 𝑉)
403, 39syl 17 . . . . . . . 8 (𝜑𝐹 Fn 𝑉)
41 fvelrnb 6969 . . . . . . . . 9 (𝐹 Fn 𝑉 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥𝑉 (𝐹𝑥) = 𝑢))
42 fvelrnb 6969 . . . . . . . . 9 (𝐹 Fn 𝑉 → (𝑣 ∈ ran 𝐹 ↔ ∃𝑦𝑉 (𝐹𝑦) = 𝑣))
43 fvelrnb 6969 . . . . . . . . 9 (𝐹 Fn 𝑉 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑧𝑉 (𝐹𝑧) = 𝑤))
4441, 42, 433anbi123d 1435 . . . . . . . 8 (𝐹 Fn 𝑉 → ((𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐹𝑤 ∈ ran 𝐹) ↔ (∃𝑥𝑉 (𝐹𝑥) = 𝑢 ∧ ∃𝑦𝑉 (𝐹𝑦) = 𝑣 ∧ ∃𝑧𝑉 (𝐹𝑧) = 𝑤)))
4540, 44syl 17 . . . . . . 7 (𝜑 → ((𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐹𝑤 ∈ ran 𝐹) ↔ (∃𝑥𝑉 (𝐹𝑥) = 𝑢 ∧ ∃𝑦𝑉 (𝐹𝑦) = 𝑣 ∧ ∃𝑧𝑉 (𝐹𝑧) = 𝑤)))
4638, 45bitr3d 281 . . . . . 6 (𝜑 → ((𝑢𝐵𝑣𝐵𝑤𝐵) ↔ (∃𝑥𝑉 (𝐹𝑥) = 𝑢 ∧ ∃𝑦𝑉 (𝐹𝑦) = 𝑣 ∧ ∃𝑧𝑉 (𝐹𝑧) = 𝑤)))
47 3reeanv 3228 . . . . . 6 (∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) ↔ (∃𝑥𝑉 (𝐹𝑥) = 𝑢 ∧ ∃𝑦𝑉 (𝐹𝑦) = 𝑣 ∧ ∃𝑧𝑉 (𝐹𝑧) = 𝑤))
4846, 47bitr4di 289 . . . . 5 (𝜑 → ((𝑢𝐵𝑣𝐵𝑤𝐵) ↔ ∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤)))
494adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑅 ∈ Ring)
50 simp2 1136 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝑉𝑦𝑉) → 𝑥𝑉)
5123ad2ant1 1132 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝑉𝑦𝑉) → 𝑉 = (Base‘𝑅))
5250, 51eleqtrd 2841 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑉𝑦𝑉) → 𝑥 ∈ (Base‘𝑅))
53523adant3r3 1183 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑥 ∈ (Base‘𝑅))
54 simp3 1137 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝑉𝑦𝑉) → 𝑦𝑉)
5554, 51eleqtrd 2841 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑉𝑦𝑉) → 𝑦 ∈ (Base‘𝑅))
56553adant3r3 1183 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑦 ∈ (Base‘𝑅))
57 simpr3 1195 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑧𝑉)
582adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑉 = (Base‘𝑅))
5957, 58eleqtrd 2841 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑧 ∈ (Base‘𝑅))
6025, 17ringass 20271 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)))
6149, 53, 56, 59, 60syl13anc 1371 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)))
6261fveq2d 6911 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘((𝑥 · 𝑦) · 𝑧)) = (𝐹‘(𝑥 · (𝑦 · 𝑧))))
63 simpl 482 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝜑)
6428caovclg 7625 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥 · 𝑦) ∈ 𝑉)
65643adantr3 1170 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 · 𝑦) ∈ 𝑉)
663, 16, 1, 2, 4, 17, 18imasmulval 17582 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 · 𝑦) ∈ 𝑉𝑧𝑉) → ((𝐹‘(𝑥 · 𝑦))(.r𝑈)(𝐹𝑧)) = (𝐹‘((𝑥 · 𝑦) · 𝑧)))
6763, 65, 57, 66syl3anc 1370 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 · 𝑦))(.r𝑈)(𝐹𝑧)) = (𝐹‘((𝑥 · 𝑦) · 𝑧)))
68 simpr1 1193 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑥𝑉)
6928caovclg 7625 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝑉𝑧𝑉)) → (𝑦 · 𝑧) ∈ 𝑉)
70693adantr1 1168 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑦 · 𝑧) ∈ 𝑉)
713, 16, 1, 2, 4, 17, 18imasmulval 17582 . . . . . . . . . . . . 13 ((𝜑𝑥𝑉 ∧ (𝑦 · 𝑧) ∈ 𝑉) → ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘(𝑥 · (𝑦 · 𝑧))))
7263, 68, 70, 71syl3anc 1370 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘(𝑥 · (𝑦 · 𝑧))))
7362, 67, 723eqtr4d 2785 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 · 𝑦))(.r𝑈)(𝐹𝑧)) = ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 · 𝑧))))
743, 16, 1, 2, 4, 17, 18imasmulval 17582 . . . . . . . . . . . . 13 ((𝜑𝑥𝑉𝑦𝑉) → ((𝐹𝑥)(.r𝑈)(𝐹𝑦)) = (𝐹‘(𝑥 · 𝑦)))
75743adant3r3 1183 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)(𝐹𝑦)) = (𝐹‘(𝑥 · 𝑦)))
7675oveq1d 7446 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(.r𝑈)(𝐹𝑦))(.r𝑈)(𝐹𝑧)) = ((𝐹‘(𝑥 · 𝑦))(.r𝑈)(𝐹𝑧)))
773, 16, 1, 2, 4, 17, 18imasmulval 17582 . . . . . . . . . . . . 13 ((𝜑𝑦𝑉𝑧𝑉) → ((𝐹𝑦)(.r𝑈)(𝐹𝑧)) = (𝐹‘(𝑦 · 𝑧)))
78773adant3r1 1181 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑦)(.r𝑈)(𝐹𝑧)) = (𝐹‘(𝑦 · 𝑧)))
7978oveq2d 7447 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)((𝐹𝑦)(.r𝑈)(𝐹𝑧))) = ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 · 𝑧))))
8073, 76, 793eqtr4d 2785 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(.r𝑈)(𝐹𝑦))(.r𝑈)(𝐹𝑧)) = ((𝐹𝑥)(.r𝑈)((𝐹𝑦)(.r𝑈)(𝐹𝑧))))
81 simp1 1135 . . . . . . . . . . . . 13 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝐹𝑥) = 𝑢)
82 simp2 1136 . . . . . . . . . . . . 13 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝐹𝑦) = 𝑣)
8381, 82oveq12d 7449 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(.r𝑈)(𝐹𝑦)) = (𝑢(.r𝑈)𝑣))
84 simp3 1137 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝐹𝑧) = 𝑤)
8583, 84oveq12d 7449 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (((𝐹𝑥)(.r𝑈)(𝐹𝑦))(.r𝑈)(𝐹𝑧)) = ((𝑢(.r𝑈)𝑣)(.r𝑈)𝑤))
8682, 84oveq12d 7449 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑦)(.r𝑈)(𝐹𝑧)) = (𝑣(.r𝑈)𝑤))
8781, 86oveq12d 7449 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(.r𝑈)((𝐹𝑦)(.r𝑈)(𝐹𝑧))) = (𝑢(.r𝑈)(𝑣(.r𝑈)𝑤)))
8885, 87eqeq12d 2751 . . . . . . . . . 10 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((((𝐹𝑥)(.r𝑈)(𝐹𝑦))(.r𝑈)(𝐹𝑧)) = ((𝐹𝑥)(.r𝑈)((𝐹𝑦)(.r𝑈)(𝐹𝑧))) ↔ ((𝑢(.r𝑈)𝑣)(.r𝑈)𝑤) = (𝑢(.r𝑈)(𝑣(.r𝑈)𝑤))))
8980, 88syl5ibcom 245 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(.r𝑈)𝑣)(.r𝑈)𝑤) = (𝑢(.r𝑈)(𝑣(.r𝑈)𝑤))))
90893exp2 1353 . . . . . . . 8 (𝜑 → (𝑥𝑉 → (𝑦𝑉 → (𝑧𝑉 → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(.r𝑈)𝑣)(.r𝑈)𝑤) = (𝑢(.r𝑈)(𝑣(.r𝑈)𝑤)))))))
9190imp32 418 . . . . . . 7 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑧𝑉 → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(.r𝑈)𝑣)(.r𝑈)𝑤) = (𝑢(.r𝑈)(𝑣(.r𝑈)𝑤)))))
9291rexlimdv 3151 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (∃𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(.r𝑈)𝑣)(.r𝑈)𝑤) = (𝑢(.r𝑈)(𝑣(.r𝑈)𝑤))))
9392rexlimdvva 3211 . . . . 5 (𝜑 → (∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(.r𝑈)𝑣)(.r𝑈)𝑤) = (𝑢(.r𝑈)(𝑣(.r𝑈)𝑤))))
9448, 93sylbid 240 . . . 4 (𝜑 → ((𝑢𝐵𝑣𝐵𝑤𝐵) → ((𝑢(.r𝑈)𝑣)(.r𝑈)𝑤) = (𝑢(.r𝑈)(𝑣(.r𝑈)𝑤))))
9594imp 406 . . 3 ((𝜑 ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢(.r𝑈)𝑣)(.r𝑈)𝑤) = (𝑢(.r𝑈)(𝑣(.r𝑈)𝑤)))
9625, 8, 17ringdi 20278 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
9749, 53, 56, 59, 96syl13anc 1371 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
9897fveq2d 6911 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘(𝑥 · (𝑦 + 𝑧))) = (𝐹‘((𝑥 · 𝑦) + (𝑥 · 𝑧))))
9925, 8ringacl 20292 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Ring ∧ 𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ (Base‘𝑅)) → (𝑢 + 𝑣) ∈ (Base‘𝑅))
10019, 22, 24, 99syl3anc 1370 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → (𝑢 + 𝑣) ∈ (Base‘𝑅))
101100, 21eleqtrrd 2842 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → (𝑢 + 𝑣) ∈ 𝑉)
102101caovclg 7625 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝑉𝑧𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
1031023adantr1 1168 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
1043, 16, 1, 2, 4, 17, 18imasmulval 17582 . . . . . . . . . . . . 13 ((𝜑𝑥𝑉 ∧ (𝑦 + 𝑧) ∈ 𝑉) → ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥 · (𝑦 + 𝑧))))
10563, 68, 103, 104syl3anc 1370 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥 · (𝑦 + 𝑧))))
10628caovclg 7625 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑧𝑉)) → (𝑥 · 𝑧) ∈ 𝑉)
1071063adantr2 1169 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 · 𝑧) ∈ 𝑉)
108 eqid 2735 . . . . . . . . . . . . . 14 (+g𝑈) = (+g𝑈)
1093, 10, 1, 2, 4, 8, 108imasaddval 17579 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 · 𝑦) ∈ 𝑉 ∧ (𝑥 · 𝑧) ∈ 𝑉) → ((𝐹‘(𝑥 · 𝑦))(+g𝑈)(𝐹‘(𝑥 · 𝑧))) = (𝐹‘((𝑥 · 𝑦) + (𝑥 · 𝑧))))
11063, 65, 107, 109syl3anc 1370 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 · 𝑦))(+g𝑈)(𝐹‘(𝑥 · 𝑧))) = (𝐹‘((𝑥 · 𝑦) + (𝑥 · 𝑧))))
11198, 105, 1103eqtr4d 2785 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 + 𝑧))) = ((𝐹‘(𝑥 · 𝑦))(+g𝑈)(𝐹‘(𝑥 · 𝑧))))
1123, 10, 1, 2, 4, 8, 108imasaddval 17579 . . . . . . . . . . . . 13 ((𝜑𝑦𝑉𝑧𝑉) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝐹‘(𝑦 + 𝑧)))
1131123adant3r1 1181 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝐹‘(𝑦 + 𝑧)))
114113oveq2d 7447 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))) = ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 + 𝑧))))
1153, 16, 1, 2, 4, 17, 18imasmulval 17582 . . . . . . . . . . . . 13 ((𝜑𝑥𝑉𝑧𝑉) → ((𝐹𝑥)(.r𝑈)(𝐹𝑧)) = (𝐹‘(𝑥 · 𝑧)))
1161153adant3r2 1182 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)(𝐹𝑧)) = (𝐹‘(𝑥 · 𝑧)))
11775, 116oveq12d 7449 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(.r𝑈)(𝐹𝑦))(+g𝑈)((𝐹𝑥)(.r𝑈)(𝐹𝑧))) = ((𝐹‘(𝑥 · 𝑦))(+g𝑈)(𝐹‘(𝑥 · 𝑧))))
118111, 114, 1173eqtr4d 2785 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))) = (((𝐹𝑥)(.r𝑈)(𝐹𝑦))(+g𝑈)((𝐹𝑥)(.r𝑈)(𝐹𝑧))))
11982, 84oveq12d 7449 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝑣(+g𝑈)𝑤))
12081, 119oveq12d 7449 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(.r𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))) = (𝑢(.r𝑈)(𝑣(+g𝑈)𝑤)))
12181, 84oveq12d 7449 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(.r𝑈)(𝐹𝑧)) = (𝑢(.r𝑈)𝑤))
12283, 121oveq12d 7449 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (((𝐹𝑥)(.r𝑈)(𝐹𝑦))(+g𝑈)((𝐹𝑥)(.r𝑈)(𝐹𝑧))) = ((𝑢(.r𝑈)𝑣)(+g𝑈)(𝑢(.r𝑈)𝑤)))
123120, 122eqeq12d 2751 . . . . . . . . . 10 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (((𝐹𝑥)(.r𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))) = (((𝐹𝑥)(.r𝑈)(𝐹𝑦))(+g𝑈)((𝐹𝑥)(.r𝑈)(𝐹𝑧))) ↔ (𝑢(.r𝑈)(𝑣(+g𝑈)𝑤)) = ((𝑢(.r𝑈)𝑣)(+g𝑈)(𝑢(.r𝑈)𝑤))))
124118, 123syl5ibcom 245 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝑢(.r𝑈)(𝑣(+g𝑈)𝑤)) = ((𝑢(.r𝑈)𝑣)(+g𝑈)(𝑢(.r𝑈)𝑤))))
1251243exp2 1353 . . . . . . . 8 (𝜑 → (𝑥𝑉 → (𝑦𝑉 → (𝑧𝑉 → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝑢(.r𝑈)(𝑣(+g𝑈)𝑤)) = ((𝑢(.r𝑈)𝑣)(+g𝑈)(𝑢(.r𝑈)𝑤)))))))
126125imp32 418 . . . . . . 7 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑧𝑉 → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝑢(.r𝑈)(𝑣(+g𝑈)𝑤)) = ((𝑢(.r𝑈)𝑣)(+g𝑈)(𝑢(.r𝑈)𝑤)))))
127126rexlimdv 3151 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (∃𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝑢(.r𝑈)(𝑣(+g𝑈)𝑤)) = ((𝑢(.r𝑈)𝑣)(+g𝑈)(𝑢(.r𝑈)𝑤))))
128127rexlimdvva 3211 . . . . 5 (𝜑 → (∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝑢(.r𝑈)(𝑣(+g𝑈)𝑤)) = ((𝑢(.r𝑈)𝑣)(+g𝑈)(𝑢(.r𝑈)𝑤))))
12948, 128sylbid 240 . . . 4 (𝜑 → ((𝑢𝐵𝑣𝐵𝑤𝐵) → (𝑢(.r𝑈)(𝑣(+g𝑈)𝑤)) = ((𝑢(.r𝑈)𝑣)(+g𝑈)(𝑢(.r𝑈)𝑤))))
130129imp 406 . . 3 ((𝜑 ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → (𝑢(.r𝑈)(𝑣(+g𝑈)𝑤)) = ((𝑢(.r𝑈)𝑣)(+g𝑈)(𝑢(.r𝑈)𝑤)))
13125, 8, 17ringdir 20279 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
13249, 53, 56, 59, 131syl13anc 1371 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
133132fveq2d 6911 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘((𝑥 + 𝑦) · 𝑧)) = (𝐹‘((𝑥 · 𝑧) + (𝑦 · 𝑧))))
134101caovclg 7625 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
1351343adantr3 1170 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
1363, 16, 1, 2, 4, 17, 18imasmulval 17582 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 + 𝑦) ∈ 𝑉𝑧𝑉) → ((𝐹‘(𝑥 + 𝑦))(.r𝑈)(𝐹𝑧)) = (𝐹‘((𝑥 + 𝑦) · 𝑧)))
13763, 135, 57, 136syl3anc 1370 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 + 𝑦))(.r𝑈)(𝐹𝑧)) = (𝐹‘((𝑥 + 𝑦) · 𝑧)))
1383, 10, 1, 2, 4, 8, 108imasaddval 17579 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 · 𝑧) ∈ 𝑉 ∧ (𝑦 · 𝑧) ∈ 𝑉) → ((𝐹‘(𝑥 · 𝑧))(+g𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘((𝑥 · 𝑧) + (𝑦 · 𝑧))))
13963, 107, 70, 138syl3anc 1370 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 · 𝑧))(+g𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘((𝑥 · 𝑧) + (𝑦 · 𝑧))))
140133, 137, 1393eqtr4d 2785 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 + 𝑦))(.r𝑈)(𝐹𝑧)) = ((𝐹‘(𝑥 · 𝑧))(+g𝑈)(𝐹‘(𝑦 · 𝑧))))
1413, 10, 1, 2, 4, 8, 108imasaddval 17579 . . . . . . . . . . . . 13 ((𝜑𝑥𝑉𝑦𝑉) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝐹‘(𝑥 + 𝑦)))
1421413adant3r3 1183 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝐹‘(𝑥 + 𝑦)))
143142oveq1d 7446 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(+g𝑈)(𝐹𝑦))(.r𝑈)(𝐹𝑧)) = ((𝐹‘(𝑥 + 𝑦))(.r𝑈)(𝐹𝑧)))
144116, 78oveq12d 7449 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(.r𝑈)(𝐹𝑧))(+g𝑈)((𝐹𝑦)(.r𝑈)(𝐹𝑧))) = ((𝐹‘(𝑥 · 𝑧))(+g𝑈)(𝐹‘(𝑦 · 𝑧))))
145140, 143, 1443eqtr4d 2785 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(+g𝑈)(𝐹𝑦))(.r𝑈)(𝐹𝑧)) = (((𝐹𝑥)(.r𝑈)(𝐹𝑧))(+g𝑈)((𝐹𝑦)(.r𝑈)(𝐹𝑧))))
14681, 82oveq12d 7449 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝑢(+g𝑈)𝑣))
147146, 84oveq12d 7449 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (((𝐹𝑥)(+g𝑈)(𝐹𝑦))(.r𝑈)(𝐹𝑧)) = ((𝑢(+g𝑈)𝑣)(.r𝑈)𝑤))
148121, 86oveq12d 7449 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (((𝐹𝑥)(.r𝑈)(𝐹𝑧))(+g𝑈)((𝐹𝑦)(.r𝑈)(𝐹𝑧))) = ((𝑢(.r𝑈)𝑤)(+g𝑈)(𝑣(.r𝑈)𝑤)))
149147, 148eqeq12d 2751 . . . . . . . . . 10 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((((𝐹𝑥)(+g𝑈)(𝐹𝑦))(.r𝑈)(𝐹𝑧)) = (((𝐹𝑥)(.r𝑈)(𝐹𝑧))(+g𝑈)((𝐹𝑦)(.r𝑈)(𝐹𝑧))) ↔ ((𝑢(+g𝑈)𝑣)(.r𝑈)𝑤) = ((𝑢(.r𝑈)𝑤)(+g𝑈)(𝑣(.r𝑈)𝑤))))
150145, 149syl5ibcom 245 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(.r𝑈)𝑤) = ((𝑢(.r𝑈)𝑤)(+g𝑈)(𝑣(.r𝑈)𝑤))))
1511503exp2 1353 . . . . . . . 8 (𝜑 → (𝑥𝑉 → (𝑦𝑉 → (𝑧𝑉 → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(.r𝑈)𝑤) = ((𝑢(.r𝑈)𝑤)(+g𝑈)(𝑣(.r𝑈)𝑤)))))))
152151imp32 418 . . . . . . 7 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑧𝑉 → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(.r𝑈)𝑤) = ((𝑢(.r𝑈)𝑤)(+g𝑈)(𝑣(.r𝑈)𝑤)))))
153152rexlimdv 3151 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (∃𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(.r𝑈)𝑤) = ((𝑢(.r𝑈)𝑤)(+g𝑈)(𝑣(.r𝑈)𝑤))))
154153rexlimdvva 3211 . . . . 5 (𝜑 → (∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(.r𝑈)𝑤) = ((𝑢(.r𝑈)𝑤)(+g𝑈)(𝑣(.r𝑈)𝑤))))
15548, 154sylbid 240 . . . 4 (𝜑 → ((𝑢𝐵𝑣𝐵𝑤𝐵) → ((𝑢(+g𝑈)𝑣)(.r𝑈)𝑤) = ((𝑢(.r𝑈)𝑤)(+g𝑈)(𝑣(.r𝑈)𝑤))))
156155imp 406 . . 3 ((𝜑 ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢(+g𝑈)𝑣)(.r𝑈)𝑤) = ((𝑢(.r𝑈)𝑤)(+g𝑈)(𝑣(.r𝑈)𝑤)))
157 fof 6821 . . . . 5 (𝐹:𝑉onto𝐵𝐹:𝑉𝐵)
1583, 157syl 17 . . . 4 (𝜑𝐹:𝑉𝐵)
159 imasring.o . . . . . . 7 1 = (1r𝑅)
16025, 159ringidcl 20280 . . . . . 6 (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
1614, 160syl 17 . . . . 5 (𝜑1 ∈ (Base‘𝑅))
162161, 2eleqtrrd 2842 . . . 4 (𝜑1𝑉)
163158, 162ffvelcdmd 7105 . . 3 (𝜑 → (𝐹1 ) ∈ 𝐵)
16440, 41syl 17 . . . . . 6 (𝜑 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥𝑉 (𝐹𝑥) = 𝑢))
16535, 164bitr3d 281 . . . . 5 (𝜑 → (𝑢𝐵 ↔ ∃𝑥𝑉 (𝐹𝑥) = 𝑢))
166 simpl 482 . . . . . . . . 9 ((𝜑𝑥𝑉) → 𝜑)
167162adantr 480 . . . . . . . . 9 ((𝜑𝑥𝑉) → 1𝑉)
168 simpr 484 . . . . . . . . 9 ((𝜑𝑥𝑉) → 𝑥𝑉)
1693, 16, 1, 2, 4, 17, 18imasmulval 17582 . . . . . . . . 9 ((𝜑1𝑉𝑥𝑉) → ((𝐹1 )(.r𝑈)(𝐹𝑥)) = (𝐹‘( 1 · 𝑥)))
170166, 167, 168, 169syl3anc 1370 . . . . . . . 8 ((𝜑𝑥𝑉) → ((𝐹1 )(.r𝑈)(𝐹𝑥)) = (𝐹‘( 1 · 𝑥)))
1712eleq2d 2825 . . . . . . . . . . 11 (𝜑 → (𝑥𝑉𝑥 ∈ (Base‘𝑅)))
172171biimpa 476 . . . . . . . . . 10 ((𝜑𝑥𝑉) → 𝑥 ∈ (Base‘𝑅))
17325, 17, 159ringlidm 20283 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → ( 1 · 𝑥) = 𝑥)
1744, 172, 173syl2an2r 685 . . . . . . . . 9 ((𝜑𝑥𝑉) → ( 1 · 𝑥) = 𝑥)
175174fveq2d 6911 . . . . . . . 8 ((𝜑𝑥𝑉) → (𝐹‘( 1 · 𝑥)) = (𝐹𝑥))
176170, 175eqtrd 2775 . . . . . . 7 ((𝜑𝑥𝑉) → ((𝐹1 )(.r𝑈)(𝐹𝑥)) = (𝐹𝑥))
177 oveq2 7439 . . . . . . . 8 ((𝐹𝑥) = 𝑢 → ((𝐹1 )(.r𝑈)(𝐹𝑥)) = ((𝐹1 )(.r𝑈)𝑢))
178 id 22 . . . . . . . 8 ((𝐹𝑥) = 𝑢 → (𝐹𝑥) = 𝑢)
179177, 178eqeq12d 2751 . . . . . . 7 ((𝐹𝑥) = 𝑢 → (((𝐹1 )(.r𝑈)(𝐹𝑥)) = (𝐹𝑥) ↔ ((𝐹1 )(.r𝑈)𝑢) = 𝑢))
180176, 179syl5ibcom 245 . . . . . 6 ((𝜑𝑥𝑉) → ((𝐹𝑥) = 𝑢 → ((𝐹1 )(.r𝑈)𝑢) = 𝑢))
181180rexlimdva 3153 . . . . 5 (𝜑 → (∃𝑥𝑉 (𝐹𝑥) = 𝑢 → ((𝐹1 )(.r𝑈)𝑢) = 𝑢))
182165, 181sylbid 240 . . . 4 (𝜑 → (𝑢𝐵 → ((𝐹1 )(.r𝑈)𝑢) = 𝑢))
183182imp 406 . . 3 ((𝜑𝑢𝐵) → ((𝐹1 )(.r𝑈)𝑢) = 𝑢)
1843, 16, 1, 2, 4, 17, 18imasmulval 17582 . . . . . . . . 9 ((𝜑𝑥𝑉1𝑉) → ((𝐹𝑥)(.r𝑈)(𝐹1 )) = (𝐹‘(𝑥 · 1 )))
185167, 184mpd3an3 1461 . . . . . . . 8 ((𝜑𝑥𝑉) → ((𝐹𝑥)(.r𝑈)(𝐹1 )) = (𝐹‘(𝑥 · 1 )))
18625, 17, 159ringridm 20284 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥 · 1 ) = 𝑥)
1874, 172, 186syl2an2r 685 . . . . . . . . 9 ((𝜑𝑥𝑉) → (𝑥 · 1 ) = 𝑥)
188187fveq2d 6911 . . . . . . . 8 ((𝜑𝑥𝑉) → (𝐹‘(𝑥 · 1 )) = (𝐹𝑥))
189185, 188eqtrd 2775 . . . . . . 7 ((𝜑𝑥𝑉) → ((𝐹𝑥)(.r𝑈)(𝐹1 )) = (𝐹𝑥))
190 oveq1 7438 . . . . . . . 8 ((𝐹𝑥) = 𝑢 → ((𝐹𝑥)(.r𝑈)(𝐹1 )) = (𝑢(.r𝑈)(𝐹1 )))
191190, 178eqeq12d 2751 . . . . . . 7 ((𝐹𝑥) = 𝑢 → (((𝐹𝑥)(.r𝑈)(𝐹1 )) = (𝐹𝑥) ↔ (𝑢(.r𝑈)(𝐹1 )) = 𝑢))
192189, 191syl5ibcom 245 . . . . . 6 ((𝜑𝑥𝑉) → ((𝐹𝑥) = 𝑢 → (𝑢(.r𝑈)(𝐹1 )) = 𝑢))
193192rexlimdva 3153 . . . . 5 (𝜑 → (∃𝑥𝑉 (𝐹𝑥) = 𝑢 → (𝑢(.r𝑈)(𝐹1 )) = 𝑢))
194165, 193sylbid 240 . . . 4 (𝜑 → (𝑢𝐵 → (𝑢(.r𝑈)(𝐹1 )) = 𝑢))
195194imp 406 . . 3 ((𝜑𝑢𝐵) → (𝑢(.r𝑈)(𝐹1 )) = 𝑢)
1965, 6, 7, 15, 32, 95, 130, 156, 163, 183, 195isringd 20305 . 2 (𝜑𝑈 ∈ Ring)
197163, 5eleqtrd 2841 . . . 4 (𝜑 → (𝐹1 ) ∈ (Base‘𝑈))
1985eleq2d 2825 . . . . . 6 (𝜑 → (𝑢𝐵𝑢 ∈ (Base‘𝑈)))
199182, 194jcad 512 . . . . . 6 (𝜑 → (𝑢𝐵 → (((𝐹1 )(.r𝑈)𝑢) = 𝑢 ∧ (𝑢(.r𝑈)(𝐹1 )) = 𝑢)))
200198, 199sylbird 260 . . . . 5 (𝜑 → (𝑢 ∈ (Base‘𝑈) → (((𝐹1 )(.r𝑈)𝑢) = 𝑢 ∧ (𝑢(.r𝑈)(𝐹1 )) = 𝑢)))
201200ralrimiv 3143 . . . 4 (𝜑 → ∀𝑢 ∈ (Base‘𝑈)(((𝐹1 )(.r𝑈)𝑢) = 𝑢 ∧ (𝑢(.r𝑈)(𝐹1 )) = 𝑢))
202 eqid 2735 . . . . . 6 (Base‘𝑈) = (Base‘𝑈)
203 eqid 2735 . . . . . 6 (1r𝑈) = (1r𝑈)
204202, 18, 203isringid 20285 . . . . 5 (𝑈 ∈ Ring → (((𝐹1 ) ∈ (Base‘𝑈) ∧ ∀𝑢 ∈ (Base‘𝑈)(((𝐹1 )(.r𝑈)𝑢) = 𝑢 ∧ (𝑢(.r𝑈)(𝐹1 )) = 𝑢)) ↔ (1r𝑈) = (𝐹1 )))
205196, 204syl 17 . . . 4 (𝜑 → (((𝐹1 ) ∈ (Base‘𝑈) ∧ ∀𝑢 ∈ (Base‘𝑈)(((𝐹1 )(.r𝑈)𝑢) = 𝑢 ∧ (𝑢(.r𝑈)(𝐹1 )) = 𝑢)) ↔ (1r𝑈) = (𝐹1 )))
206197, 201, 205mpbi2and 712 . . 3 (𝜑 → (1r𝑈) = (𝐹1 ))
207206eqcomd 2741 . 2 (𝜑 → (𝐹1 ) = (1r𝑈))
208196, 207jca 511 1 (𝜑 → (𝑈 ∈ Ring ∧ (𝐹1 ) = (1r𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068   × cxp 5687  ran crn 5690   Fn wfn 6558  wf 6559  ontowfo 6561  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  .rcmulr 17299  0gc0g 17486  s cimas 17551  Grpcgrp 18964  1rcur 20199  Ringcrg 20251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-0g 17488  df-imas 17555  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-mgp 20153  df-ur 20200  df-ring 20253
This theorem is referenced by:  imasringf1  20345  qusring2  20348  imasrhm  33364
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