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Theorem imasring 20045
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 17394 . . 3 (𝜑𝐵 = (Base‘𝑈))
6 eqidd 2737 . . 3 (𝜑 → (+g𝑈) = (+g𝑈))
7 eqidd 2737 . . 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 19969 . . . . . 6 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
124, 11syl 17 . . . . 5 (𝜑𝑅 ∈ Grp)
13 eqid 2736 . . . . 5 (0g𝑅) = (0g𝑅)
141, 2, 9, 3, 10, 12, 13imasgrp 18863 . . . 4 (𝜑 → (𝑈 ∈ Grp ∧ (𝐹‘(0g𝑅)) = (0g𝑈)))
1514simpld 495 . . 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 2736 . . . . 5 (.r𝑈) = (.r𝑈)
194adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → 𝑅 ∈ Ring)
20 simprl 769 . . . . . . . . 9 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → 𝑢𝑉)
212adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → 𝑉 = (Base‘𝑅))
2220, 21eleqtrd 2840 . . . . . . . 8 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → 𝑢 ∈ (Base‘𝑅))
23 simprr 771 . . . . . . . . 9 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → 𝑣𝑉)
2423, 21eleqtrd 2840 . . . . . . . 8 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → 𝑣 ∈ (Base‘𝑅))
25 eqid 2736 . . . . . . . . 9 (Base‘𝑅) = (Base‘𝑅)
2625, 17ringcl 19981 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ (Base‘𝑅)) → (𝑢 · 𝑣) ∈ (Base‘𝑅))
2719, 22, 24, 26syl3anc 1371 . . . . . . 7 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → (𝑢 · 𝑣) ∈ (Base‘𝑅))
2827, 21eleqtrrd 2841 . . . . . 6 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → (𝑢 · 𝑣) ∈ 𝑉)
2928caovclg 7546 . . . . 5 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)
303, 16, 1, 2, 4, 17, 18, 29imasmulf 17418 . . . 4 (𝜑 → (.r𝑈):(𝐵 × 𝐵)⟶𝐵)
31 fovcdm 7524 . . . 4 (((.r𝑈):(𝐵 × 𝐵)⟶𝐵𝑢𝐵𝑣𝐵) → (𝑢(.r𝑈)𝑣) ∈ 𝐵)
3230, 31syl3an1 1163 . . 3 ((𝜑𝑢𝐵𝑣𝐵) → (𝑢(.r𝑈)𝑣) ∈ 𝐵)
33 forn 6759 . . . . . . . . . 10 (𝐹:𝑉onto𝐵 → ran 𝐹 = 𝐵)
343, 33syl 17 . . . . . . . . 9 (𝜑 → ran 𝐹 = 𝐵)
3534eleq2d 2823 . . . . . . . 8 (𝜑 → (𝑢 ∈ ran 𝐹𝑢𝐵))
3634eleq2d 2823 . . . . . . . 8 (𝜑 → (𝑣 ∈ ran 𝐹𝑣𝐵))
3734eleq2d 2823 . . . . . . . 8 (𝜑 → (𝑤 ∈ ran 𝐹𝑤𝐵))
3835, 36, 373anbi123d 1436 . . . . . . 7 (𝜑 → ((𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐹𝑤 ∈ ran 𝐹) ↔ (𝑢𝐵𝑣𝐵𝑤𝐵)))
39 fofn 6758 . . . . . . . . 9 (𝐹:𝑉onto𝐵𝐹 Fn 𝑉)
403, 39syl 17 . . . . . . . 8 (𝜑𝐹 Fn 𝑉)
41 fvelrnb 6903 . . . . . . . . 9 (𝐹 Fn 𝑉 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥𝑉 (𝐹𝑥) = 𝑢))
42 fvelrnb 6903 . . . . . . . . 9 (𝐹 Fn 𝑉 → (𝑣 ∈ ran 𝐹 ↔ ∃𝑦𝑉 (𝐹𝑦) = 𝑣))
43 fvelrnb 6903 . . . . . . . . 9 (𝐹 Fn 𝑉 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑧𝑉 (𝐹𝑧) = 𝑤))
4441, 42, 433anbi123d 1436 . . . . . . . 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 280 . . . . . 6 (𝜑 → ((𝑢𝐵𝑣𝐵𝑤𝐵) ↔ (∃𝑥𝑉 (𝐹𝑥) = 𝑢 ∧ ∃𝑦𝑉 (𝐹𝑦) = 𝑣 ∧ ∃𝑧𝑉 (𝐹𝑧) = 𝑤)))
47 3reeanv 3218 . . . . . 6 (∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) ↔ (∃𝑥𝑉 (𝐹𝑥) = 𝑢 ∧ ∃𝑦𝑉 (𝐹𝑦) = 𝑣 ∧ ∃𝑧𝑉 (𝐹𝑧) = 𝑤))
4846, 47bitr4di 288 . . . . 5 (𝜑 → ((𝑢𝐵𝑣𝐵𝑤𝐵) ↔ ∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤)))
494adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑅 ∈ Ring)
50 simp2 1137 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝑉𝑦𝑉) → 𝑥𝑉)
5123ad2ant1 1133 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝑉𝑦𝑉) → 𝑉 = (Base‘𝑅))
5250, 51eleqtrd 2840 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑉𝑦𝑉) → 𝑥 ∈ (Base‘𝑅))
53523adant3r3 1184 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑥 ∈ (Base‘𝑅))
54 simp3 1138 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝑉𝑦𝑉) → 𝑦𝑉)
5554, 51eleqtrd 2840 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑉𝑦𝑉) → 𝑦 ∈ (Base‘𝑅))
56553adant3r3 1184 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑦 ∈ (Base‘𝑅))
57 simpr3 1196 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑧𝑉)
582adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑉 = (Base‘𝑅))
5957, 58eleqtrd 2840 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑧 ∈ (Base‘𝑅))
6025, 17ringass 19984 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)))
6149, 53, 56, 59, 60syl13anc 1372 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)))
6261fveq2d 6846 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘((𝑥 · 𝑦) · 𝑧)) = (𝐹‘(𝑥 · (𝑦 · 𝑧))))
63 simpl 483 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝜑)
6428caovclg 7546 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥 · 𝑦) ∈ 𝑉)
65643adantr3 1171 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 · 𝑦) ∈ 𝑉)
663, 16, 1, 2, 4, 17, 18imasmulval 17417 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 · 𝑦) ∈ 𝑉𝑧𝑉) → ((𝐹‘(𝑥 · 𝑦))(.r𝑈)(𝐹𝑧)) = (𝐹‘((𝑥 · 𝑦) · 𝑧)))
6763, 65, 57, 66syl3anc 1371 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 · 𝑦))(.r𝑈)(𝐹𝑧)) = (𝐹‘((𝑥 · 𝑦) · 𝑧)))
68 simpr1 1194 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑥𝑉)
6928caovclg 7546 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝑉𝑧𝑉)) → (𝑦 · 𝑧) ∈ 𝑉)
70693adantr1 1169 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑦 · 𝑧) ∈ 𝑉)
713, 16, 1, 2, 4, 17, 18imasmulval 17417 . . . . . . . . . . . . 13 ((𝜑𝑥𝑉 ∧ (𝑦 · 𝑧) ∈ 𝑉) → ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘(𝑥 · (𝑦 · 𝑧))))
7263, 68, 70, 71syl3anc 1371 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘(𝑥 · (𝑦 · 𝑧))))
7362, 67, 723eqtr4d 2786 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 · 𝑦))(.r𝑈)(𝐹𝑧)) = ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 · 𝑧))))
743, 16, 1, 2, 4, 17, 18imasmulval 17417 . . . . . . . . . . . . 13 ((𝜑𝑥𝑉𝑦𝑉) → ((𝐹𝑥)(.r𝑈)(𝐹𝑦)) = (𝐹‘(𝑥 · 𝑦)))
75743adant3r3 1184 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)(𝐹𝑦)) = (𝐹‘(𝑥 · 𝑦)))
7675oveq1d 7372 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(.r𝑈)(𝐹𝑦))(.r𝑈)(𝐹𝑧)) = ((𝐹‘(𝑥 · 𝑦))(.r𝑈)(𝐹𝑧)))
773, 16, 1, 2, 4, 17, 18imasmulval 17417 . . . . . . . . . . . . 13 ((𝜑𝑦𝑉𝑧𝑉) → ((𝐹𝑦)(.r𝑈)(𝐹𝑧)) = (𝐹‘(𝑦 · 𝑧)))
78773adant3r1 1182 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑦)(.r𝑈)(𝐹𝑧)) = (𝐹‘(𝑦 · 𝑧)))
7978oveq2d 7373 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)((𝐹𝑦)(.r𝑈)(𝐹𝑧))) = ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 · 𝑧))))
8073, 76, 793eqtr4d 2786 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(.r𝑈)(𝐹𝑦))(.r𝑈)(𝐹𝑧)) = ((𝐹𝑥)(.r𝑈)((𝐹𝑦)(.r𝑈)(𝐹𝑧))))
81 simp1 1136 . . . . . . . . . . . . 13 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝐹𝑥) = 𝑢)
82 simp2 1137 . . . . . . . . . . . . 13 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝐹𝑦) = 𝑣)
8381, 82oveq12d 7375 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(.r𝑈)(𝐹𝑦)) = (𝑢(.r𝑈)𝑣))
84 simp3 1138 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝐹𝑧) = 𝑤)
8583, 84oveq12d 7375 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (((𝐹𝑥)(.r𝑈)(𝐹𝑦))(.r𝑈)(𝐹𝑧)) = ((𝑢(.r𝑈)𝑣)(.r𝑈)𝑤))
8682, 84oveq12d 7375 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑦)(.r𝑈)(𝐹𝑧)) = (𝑣(.r𝑈)𝑤))
8781, 86oveq12d 7375 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(.r𝑈)((𝐹𝑦)(.r𝑈)(𝐹𝑧))) = (𝑢(.r𝑈)(𝑣(.r𝑈)𝑤)))
8885, 87eqeq12d 2752 . . . . . . . . . 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 244 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(.r𝑈)𝑣)(.r𝑈)𝑤) = (𝑢(.r𝑈)(𝑣(.r𝑈)𝑤))))
90893exp2 1354 . . . . . . . 8 (𝜑 → (𝑥𝑉 → (𝑦𝑉 → (𝑧𝑉 → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(.r𝑈)𝑣)(.r𝑈)𝑤) = (𝑢(.r𝑈)(𝑣(.r𝑈)𝑤)))))))
9190imp32 419 . . . . . . 7 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑧𝑉 → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(.r𝑈)𝑣)(.r𝑈)𝑤) = (𝑢(.r𝑈)(𝑣(.r𝑈)𝑤)))))
9291rexlimdv 3150 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (∃𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(.r𝑈)𝑣)(.r𝑈)𝑤) = (𝑢(.r𝑈)(𝑣(.r𝑈)𝑤))))
9392rexlimdvva 3205 . . . . 5 (𝜑 → (∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(.r𝑈)𝑣)(.r𝑈)𝑤) = (𝑢(.r𝑈)(𝑣(.r𝑈)𝑤))))
9448, 93sylbid 239 . . . 4 (𝜑 → ((𝑢𝐵𝑣𝐵𝑤𝐵) → ((𝑢(.r𝑈)𝑣)(.r𝑈)𝑤) = (𝑢(.r𝑈)(𝑣(.r𝑈)𝑤))))
9594imp 407 . . 3 ((𝜑 ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢(.r𝑈)𝑣)(.r𝑈)𝑤) = (𝑢(.r𝑈)(𝑣(.r𝑈)𝑤)))
9625, 8, 17ringdi 19987 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
9749, 53, 56, 59, 96syl13anc 1372 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
9897fveq2d 6846 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘(𝑥 · (𝑦 + 𝑧))) = (𝐹‘((𝑥 · 𝑦) + (𝑥 · 𝑧))))
9925, 8ringacl 19999 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Ring ∧ 𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ (Base‘𝑅)) → (𝑢 + 𝑣) ∈ (Base‘𝑅))
10019, 22, 24, 99syl3anc 1371 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → (𝑢 + 𝑣) ∈ (Base‘𝑅))
101100, 21eleqtrrd 2841 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → (𝑢 + 𝑣) ∈ 𝑉)
102101caovclg 7546 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝑉𝑧𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
1031023adantr1 1169 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
1043, 16, 1, 2, 4, 17, 18imasmulval 17417 . . . . . . . . . . . . 13 ((𝜑𝑥𝑉 ∧ (𝑦 + 𝑧) ∈ 𝑉) → ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥 · (𝑦 + 𝑧))))
10563, 68, 103, 104syl3anc 1371 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥 · (𝑦 + 𝑧))))
10628caovclg 7546 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑧𝑉)) → (𝑥 · 𝑧) ∈ 𝑉)
1071063adantr2 1170 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 · 𝑧) ∈ 𝑉)
108 eqid 2736 . . . . . . . . . . . . . 14 (+g𝑈) = (+g𝑈)
1093, 10, 1, 2, 4, 8, 108imasaddval 17414 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 · 𝑦) ∈ 𝑉 ∧ (𝑥 · 𝑧) ∈ 𝑉) → ((𝐹‘(𝑥 · 𝑦))(+g𝑈)(𝐹‘(𝑥 · 𝑧))) = (𝐹‘((𝑥 · 𝑦) + (𝑥 · 𝑧))))
11063, 65, 107, 109syl3anc 1371 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 · 𝑦))(+g𝑈)(𝐹‘(𝑥 · 𝑧))) = (𝐹‘((𝑥 · 𝑦) + (𝑥 · 𝑧))))
11198, 105, 1103eqtr4d 2786 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 + 𝑧))) = ((𝐹‘(𝑥 · 𝑦))(+g𝑈)(𝐹‘(𝑥 · 𝑧))))
1123, 10, 1, 2, 4, 8, 108imasaddval 17414 . . . . . . . . . . . . 13 ((𝜑𝑦𝑉𝑧𝑉) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝐹‘(𝑦 + 𝑧)))
1131123adant3r1 1182 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝐹‘(𝑦 + 𝑧)))
114113oveq2d 7373 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))) = ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 + 𝑧))))
1153, 16, 1, 2, 4, 17, 18imasmulval 17417 . . . . . . . . . . . . 13 ((𝜑𝑥𝑉𝑧𝑉) → ((𝐹𝑥)(.r𝑈)(𝐹𝑧)) = (𝐹‘(𝑥 · 𝑧)))
1161153adant3r2 1183 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)(𝐹𝑧)) = (𝐹‘(𝑥 · 𝑧)))
11775, 116oveq12d 7375 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(.r𝑈)(𝐹𝑦))(+g𝑈)((𝐹𝑥)(.r𝑈)(𝐹𝑧))) = ((𝐹‘(𝑥 · 𝑦))(+g𝑈)(𝐹‘(𝑥 · 𝑧))))
118111, 114, 1173eqtr4d 2786 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))) = (((𝐹𝑥)(.r𝑈)(𝐹𝑦))(+g𝑈)((𝐹𝑥)(.r𝑈)(𝐹𝑧))))
11982, 84oveq12d 7375 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝑣(+g𝑈)𝑤))
12081, 119oveq12d 7375 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(.r𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))) = (𝑢(.r𝑈)(𝑣(+g𝑈)𝑤)))
12181, 84oveq12d 7375 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(.r𝑈)(𝐹𝑧)) = (𝑢(.r𝑈)𝑤))
12283, 121oveq12d 7375 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (((𝐹𝑥)(.r𝑈)(𝐹𝑦))(+g𝑈)((𝐹𝑥)(.r𝑈)(𝐹𝑧))) = ((𝑢(.r𝑈)𝑣)(+g𝑈)(𝑢(.r𝑈)𝑤)))
123120, 122eqeq12d 2752 . . . . . . . . . 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 244 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝑢(.r𝑈)(𝑣(+g𝑈)𝑤)) = ((𝑢(.r𝑈)𝑣)(+g𝑈)(𝑢(.r𝑈)𝑤))))
1251243exp2 1354 . . . . . . . 8 (𝜑 → (𝑥𝑉 → (𝑦𝑉 → (𝑧𝑉 → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝑢(.r𝑈)(𝑣(+g𝑈)𝑤)) = ((𝑢(.r𝑈)𝑣)(+g𝑈)(𝑢(.r𝑈)𝑤)))))))
126125imp32 419 . . . . . . 7 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑧𝑉 → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝑢(.r𝑈)(𝑣(+g𝑈)𝑤)) = ((𝑢(.r𝑈)𝑣)(+g𝑈)(𝑢(.r𝑈)𝑤)))))
127126rexlimdv 3150 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (∃𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝑢(.r𝑈)(𝑣(+g𝑈)𝑤)) = ((𝑢(.r𝑈)𝑣)(+g𝑈)(𝑢(.r𝑈)𝑤))))
128127rexlimdvva 3205 . . . . 5 (𝜑 → (∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝑢(.r𝑈)(𝑣(+g𝑈)𝑤)) = ((𝑢(.r𝑈)𝑣)(+g𝑈)(𝑢(.r𝑈)𝑤))))
12948, 128sylbid 239 . . . 4 (𝜑 → ((𝑢𝐵𝑣𝐵𝑤𝐵) → (𝑢(.r𝑈)(𝑣(+g𝑈)𝑤)) = ((𝑢(.r𝑈)𝑣)(+g𝑈)(𝑢(.r𝑈)𝑤))))
130129imp 407 . . 3 ((𝜑 ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → (𝑢(.r𝑈)(𝑣(+g𝑈)𝑤)) = ((𝑢(.r𝑈)𝑣)(+g𝑈)(𝑢(.r𝑈)𝑤)))
13125, 8, 17ringdir 19988 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
13249, 53, 56, 59, 131syl13anc 1372 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
133132fveq2d 6846 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘((𝑥 + 𝑦) · 𝑧)) = (𝐹‘((𝑥 · 𝑧) + (𝑦 · 𝑧))))
134101caovclg 7546 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
1351343adantr3 1171 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
1363, 16, 1, 2, 4, 17, 18imasmulval 17417 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 + 𝑦) ∈ 𝑉𝑧𝑉) → ((𝐹‘(𝑥 + 𝑦))(.r𝑈)(𝐹𝑧)) = (𝐹‘((𝑥 + 𝑦) · 𝑧)))
13763, 135, 57, 136syl3anc 1371 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 + 𝑦))(.r𝑈)(𝐹𝑧)) = (𝐹‘((𝑥 + 𝑦) · 𝑧)))
1383, 10, 1, 2, 4, 8, 108imasaddval 17414 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 · 𝑧) ∈ 𝑉 ∧ (𝑦 · 𝑧) ∈ 𝑉) → ((𝐹‘(𝑥 · 𝑧))(+g𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘((𝑥 · 𝑧) + (𝑦 · 𝑧))))
13963, 107, 70, 138syl3anc 1371 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 · 𝑧))(+g𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘((𝑥 · 𝑧) + (𝑦 · 𝑧))))
140133, 137, 1393eqtr4d 2786 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 + 𝑦))(.r𝑈)(𝐹𝑧)) = ((𝐹‘(𝑥 · 𝑧))(+g𝑈)(𝐹‘(𝑦 · 𝑧))))
1413, 10, 1, 2, 4, 8, 108imasaddval 17414 . . . . . . . . . . . . 13 ((𝜑𝑥𝑉𝑦𝑉) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝐹‘(𝑥 + 𝑦)))
1421413adant3r3 1184 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝐹‘(𝑥 + 𝑦)))
143142oveq1d 7372 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(+g𝑈)(𝐹𝑦))(.r𝑈)(𝐹𝑧)) = ((𝐹‘(𝑥 + 𝑦))(.r𝑈)(𝐹𝑧)))
144116, 78oveq12d 7375 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(.r𝑈)(𝐹𝑧))(+g𝑈)((𝐹𝑦)(.r𝑈)(𝐹𝑧))) = ((𝐹‘(𝑥 · 𝑧))(+g𝑈)(𝐹‘(𝑦 · 𝑧))))
145140, 143, 1443eqtr4d 2786 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(+g𝑈)(𝐹𝑦))(.r𝑈)(𝐹𝑧)) = (((𝐹𝑥)(.r𝑈)(𝐹𝑧))(+g𝑈)((𝐹𝑦)(.r𝑈)(𝐹𝑧))))
14681, 82oveq12d 7375 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝑢(+g𝑈)𝑣))
147146, 84oveq12d 7375 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (((𝐹𝑥)(+g𝑈)(𝐹𝑦))(.r𝑈)(𝐹𝑧)) = ((𝑢(+g𝑈)𝑣)(.r𝑈)𝑤))
148121, 86oveq12d 7375 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (((𝐹𝑥)(.r𝑈)(𝐹𝑧))(+g𝑈)((𝐹𝑦)(.r𝑈)(𝐹𝑧))) = ((𝑢(.r𝑈)𝑤)(+g𝑈)(𝑣(.r𝑈)𝑤)))
149147, 148eqeq12d 2752 . . . . . . . . . 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 244 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(.r𝑈)𝑤) = ((𝑢(.r𝑈)𝑤)(+g𝑈)(𝑣(.r𝑈)𝑤))))
1511503exp2 1354 . . . . . . . 8 (𝜑 → (𝑥𝑉 → (𝑦𝑉 → (𝑧𝑉 → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(.r𝑈)𝑤) = ((𝑢(.r𝑈)𝑤)(+g𝑈)(𝑣(.r𝑈)𝑤)))))))
152151imp32 419 . . . . . . 7 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑧𝑉 → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(.r𝑈)𝑤) = ((𝑢(.r𝑈)𝑤)(+g𝑈)(𝑣(.r𝑈)𝑤)))))
153152rexlimdv 3150 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (∃𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(.r𝑈)𝑤) = ((𝑢(.r𝑈)𝑤)(+g𝑈)(𝑣(.r𝑈)𝑤))))
154153rexlimdvva 3205 . . . . 5 (𝜑 → (∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(.r𝑈)𝑤) = ((𝑢(.r𝑈)𝑤)(+g𝑈)(𝑣(.r𝑈)𝑤))))
15548, 154sylbid 239 . . . 4 (𝜑 → ((𝑢𝐵𝑣𝐵𝑤𝐵) → ((𝑢(+g𝑈)𝑣)(.r𝑈)𝑤) = ((𝑢(.r𝑈)𝑤)(+g𝑈)(𝑣(.r𝑈)𝑤))))
156155imp 407 . . 3 ((𝜑 ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢(+g𝑈)𝑣)(.r𝑈)𝑤) = ((𝑢(.r𝑈)𝑤)(+g𝑈)(𝑣(.r𝑈)𝑤)))
157 fof 6756 . . . . 5 (𝐹:𝑉onto𝐵𝐹:𝑉𝐵)
1583, 157syl 17 . . . 4 (𝜑𝐹:𝑉𝐵)
159 imasring.o . . . . . . 7 1 = (1r𝑅)
16025, 159ringidcl 19989 . . . . . 6 (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
1614, 160syl 17 . . . . 5 (𝜑1 ∈ (Base‘𝑅))
162161, 2eleqtrrd 2841 . . . 4 (𝜑1𝑉)
163158, 162ffvelcdmd 7036 . . 3 (𝜑 → (𝐹1 ) ∈ 𝐵)
16440, 41syl 17 . . . . . 6 (𝜑 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥𝑉 (𝐹𝑥) = 𝑢))
16535, 164bitr3d 280 . . . . 5 (𝜑 → (𝑢𝐵 ↔ ∃𝑥𝑉 (𝐹𝑥) = 𝑢))
166 simpl 483 . . . . . . . . 9 ((𝜑𝑥𝑉) → 𝜑)
167162adantr 481 . . . . . . . . 9 ((𝜑𝑥𝑉) → 1𝑉)
168 simpr 485 . . . . . . . . 9 ((𝜑𝑥𝑉) → 𝑥𝑉)
1693, 16, 1, 2, 4, 17, 18imasmulval 17417 . . . . . . . . 9 ((𝜑1𝑉𝑥𝑉) → ((𝐹1 )(.r𝑈)(𝐹𝑥)) = (𝐹‘( 1 · 𝑥)))
170166, 167, 168, 169syl3anc 1371 . . . . . . . 8 ((𝜑𝑥𝑉) → ((𝐹1 )(.r𝑈)(𝐹𝑥)) = (𝐹‘( 1 · 𝑥)))
1712eleq2d 2823 . . . . . . . . . . 11 (𝜑 → (𝑥𝑉𝑥 ∈ (Base‘𝑅)))
172171biimpa 477 . . . . . . . . . 10 ((𝜑𝑥𝑉) → 𝑥 ∈ (Base‘𝑅))
17325, 17, 159ringlidm 19992 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → ( 1 · 𝑥) = 𝑥)
1744, 172, 173syl2an2r 683 . . . . . . . . 9 ((𝜑𝑥𝑉) → ( 1 · 𝑥) = 𝑥)
175174fveq2d 6846 . . . . . . . 8 ((𝜑𝑥𝑉) → (𝐹‘( 1 · 𝑥)) = (𝐹𝑥))
176170, 175eqtrd 2776 . . . . . . 7 ((𝜑𝑥𝑉) → ((𝐹1 )(.r𝑈)(𝐹𝑥)) = (𝐹𝑥))
177 oveq2 7365 . . . . . . . 8 ((𝐹𝑥) = 𝑢 → ((𝐹1 )(.r𝑈)(𝐹𝑥)) = ((𝐹1 )(.r𝑈)𝑢))
178 id 22 . . . . . . . 8 ((𝐹𝑥) = 𝑢 → (𝐹𝑥) = 𝑢)
179177, 178eqeq12d 2752 . . . . . . 7 ((𝐹𝑥) = 𝑢 → (((𝐹1 )(.r𝑈)(𝐹𝑥)) = (𝐹𝑥) ↔ ((𝐹1 )(.r𝑈)𝑢) = 𝑢))
180176, 179syl5ibcom 244 . . . . . 6 ((𝜑𝑥𝑉) → ((𝐹𝑥) = 𝑢 → ((𝐹1 )(.r𝑈)𝑢) = 𝑢))
181180rexlimdva 3152 . . . . 5 (𝜑 → (∃𝑥𝑉 (𝐹𝑥) = 𝑢 → ((𝐹1 )(.r𝑈)𝑢) = 𝑢))
182165, 181sylbid 239 . . . 4 (𝜑 → (𝑢𝐵 → ((𝐹1 )(.r𝑈)𝑢) = 𝑢))
183182imp 407 . . 3 ((𝜑𝑢𝐵) → ((𝐹1 )(.r𝑈)𝑢) = 𝑢)
1843, 16, 1, 2, 4, 17, 18imasmulval 17417 . . . . . . . . 9 ((𝜑𝑥𝑉1𝑉) → ((𝐹𝑥)(.r𝑈)(𝐹1 )) = (𝐹‘(𝑥 · 1 )))
185167, 184mpd3an3 1462 . . . . . . . 8 ((𝜑𝑥𝑉) → ((𝐹𝑥)(.r𝑈)(𝐹1 )) = (𝐹‘(𝑥 · 1 )))
18625, 17, 159ringridm 19993 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥 · 1 ) = 𝑥)
1874, 172, 186syl2an2r 683 . . . . . . . . 9 ((𝜑𝑥𝑉) → (𝑥 · 1 ) = 𝑥)
188187fveq2d 6846 . . . . . . . 8 ((𝜑𝑥𝑉) → (𝐹‘(𝑥 · 1 )) = (𝐹𝑥))
189185, 188eqtrd 2776 . . . . . . 7 ((𝜑𝑥𝑉) → ((𝐹𝑥)(.r𝑈)(𝐹1 )) = (𝐹𝑥))
190 oveq1 7364 . . . . . . . 8 ((𝐹𝑥) = 𝑢 → ((𝐹𝑥)(.r𝑈)(𝐹1 )) = (𝑢(.r𝑈)(𝐹1 )))
191190, 178eqeq12d 2752 . . . . . . 7 ((𝐹𝑥) = 𝑢 → (((𝐹𝑥)(.r𝑈)(𝐹1 )) = (𝐹𝑥) ↔ (𝑢(.r𝑈)(𝐹1 )) = 𝑢))
192189, 191syl5ibcom 244 . . . . . 6 ((𝜑𝑥𝑉) → ((𝐹𝑥) = 𝑢 → (𝑢(.r𝑈)(𝐹1 )) = 𝑢))
193192rexlimdva 3152 . . . . 5 (𝜑 → (∃𝑥𝑉 (𝐹𝑥) = 𝑢 → (𝑢(.r𝑈)(𝐹1 )) = 𝑢))
194165, 193sylbid 239 . . . 4 (𝜑 → (𝑢𝐵 → (𝑢(.r𝑈)(𝐹1 )) = 𝑢))
195194imp 407 . . 3 ((𝜑𝑢𝐵) → (𝑢(.r𝑈)(𝐹1 )) = 𝑢)
1965, 6, 7, 15, 32, 95, 130, 156, 163, 183, 195isringd 20009 . 2 (𝜑𝑈 ∈ Ring)
197163, 5eleqtrd 2840 . . . 4 (𝜑 → (𝐹1 ) ∈ (Base‘𝑈))
1985eleq2d 2823 . . . . . 6 (𝜑 → (𝑢𝐵𝑢 ∈ (Base‘𝑈)))
199182, 194jcad 513 . . . . . 6 (𝜑 → (𝑢𝐵 → (((𝐹1 )(.r𝑈)𝑢) = 𝑢 ∧ (𝑢(.r𝑈)(𝐹1 )) = 𝑢)))
200198, 199sylbird 259 . . . . 5 (𝜑 → (𝑢 ∈ (Base‘𝑈) → (((𝐹1 )(.r𝑈)𝑢) = 𝑢 ∧ (𝑢(.r𝑈)(𝐹1 )) = 𝑢)))
201200ralrimiv 3142 . . . 4 (𝜑 → ∀𝑢 ∈ (Base‘𝑈)(((𝐹1 )(.r𝑈)𝑢) = 𝑢 ∧ (𝑢(.r𝑈)(𝐹1 )) = 𝑢))
202 eqid 2736 . . . . . 6 (Base‘𝑈) = (Base‘𝑈)
203 eqid 2736 . . . . . 6 (1r𝑈) = (1r𝑈)
204202, 18, 203isringid 19994 . . . . 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 710 . . 3 (𝜑 → (1r𝑈) = (𝐹1 ))
207206eqcomd 2742 . 2 (𝜑 → (𝐹1 ) = (1r𝑈))
208196, 207jca 512 1 (𝜑 → (𝑈 ∈ Ring ∧ (𝐹1 ) = (1r𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  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  .rcmulr 17134  0gc0g 17321  s cimas 17386  Grpcgrp 18748  1rcur 19913  Ringcrg 19964
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-sets 17036  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  df-minusg 18752  df-mgp 19897  df-ur 19914  df-ring 19966
This theorem is referenced by:  qusring2  20046
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