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Theorem imasring 19289
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 16775 . . 3 (𝜑𝐵 = (Base‘𝑈))
6 eqidd 2827 . . 3 (𝜑 → (+g𝑈) = (+g𝑈))
7 eqidd 2827 . . 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 19222 . . . . . 6 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
124, 11syl 17 . . . . 5 (𝜑𝑅 ∈ Grp)
13 eqid 2826 . . . . 5 (0g𝑅) = (0g𝑅)
141, 2, 9, 3, 10, 12, 13imasgrp 18145 . . . 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 2826 . . . . 5 (.r𝑈) = (.r𝑈)
194adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → 𝑅 ∈ Ring)
20 simprl 767 . . . . . . . . 9 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → 𝑢𝑉)
212adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → 𝑉 = (Base‘𝑅))
2220, 21eleqtrd 2920 . . . . . . . 8 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → 𝑢 ∈ (Base‘𝑅))
23 simprr 769 . . . . . . . . 9 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → 𝑣𝑉)
2423, 21eleqtrd 2920 . . . . . . . 8 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → 𝑣 ∈ (Base‘𝑅))
25 eqid 2826 . . . . . . . . 9 (Base‘𝑅) = (Base‘𝑅)
2625, 17ringcl 19231 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ (Base‘𝑅)) → (𝑢 · 𝑣) ∈ (Base‘𝑅))
2719, 22, 24, 26syl3anc 1365 . . . . . . 7 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → (𝑢 · 𝑣) ∈ (Base‘𝑅))
2827, 21eleqtrrd 2921 . . . . . 6 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → (𝑢 · 𝑣) ∈ 𝑉)
2928caovclg 7330 . . . . 5 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)
303, 16, 1, 2, 4, 17, 18, 29imasmulf 16799 . . . 4 (𝜑 → (.r𝑈):(𝐵 × 𝐵)⟶𝐵)
31 fovrn 7308 . . . 4 (((.r𝑈):(𝐵 × 𝐵)⟶𝐵𝑢𝐵𝑣𝐵) → (𝑢(.r𝑈)𝑣) ∈ 𝐵)
3230, 31syl3an1 1157 . . 3 ((𝜑𝑢𝐵𝑣𝐵) → (𝑢(.r𝑈)𝑣) ∈ 𝐵)
33 forn 6590 . . . . . . . . . 10 (𝐹:𝑉onto𝐵 → ran 𝐹 = 𝐵)
343, 33syl 17 . . . . . . . . 9 (𝜑 → ran 𝐹 = 𝐵)
3534eleq2d 2903 . . . . . . . 8 (𝜑 → (𝑢 ∈ ran 𝐹𝑢𝐵))
3634eleq2d 2903 . . . . . . . 8 (𝜑 → (𝑣 ∈ ran 𝐹𝑣𝐵))
3734eleq2d 2903 . . . . . . . 8 (𝜑 → (𝑤 ∈ ran 𝐹𝑤𝐵))
3835, 36, 373anbi123d 1429 . . . . . . 7 (𝜑 → ((𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐹𝑤 ∈ ran 𝐹) ↔ (𝑢𝐵𝑣𝐵𝑤𝐵)))
39 fofn 6589 . . . . . . . . 9 (𝐹:𝑉onto𝐵𝐹 Fn 𝑉)
403, 39syl 17 . . . . . . . 8 (𝜑𝐹 Fn 𝑉)
41 fvelrnb 6723 . . . . . . . . 9 (𝐹 Fn 𝑉 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥𝑉 (𝐹𝑥) = 𝑢))
42 fvelrnb 6723 . . . . . . . . 9 (𝐹 Fn 𝑉 → (𝑣 ∈ ran 𝐹 ↔ ∃𝑦𝑉 (𝐹𝑦) = 𝑣))
43 fvelrnb 6723 . . . . . . . . 9 (𝐹 Fn 𝑉 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑧𝑉 (𝐹𝑧) = 𝑤))
4441, 42, 433anbi123d 1429 . . . . . . . 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 282 . . . . . 6 (𝜑 → ((𝑢𝐵𝑣𝐵𝑤𝐵) ↔ (∃𝑥𝑉 (𝐹𝑥) = 𝑢 ∧ ∃𝑦𝑉 (𝐹𝑦) = 𝑣 ∧ ∃𝑧𝑉 (𝐹𝑧) = 𝑤)))
47 3reeanv 3374 . . . . . 6 (∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) ↔ (∃𝑥𝑉 (𝐹𝑥) = 𝑢 ∧ ∃𝑦𝑉 (𝐹𝑦) = 𝑣 ∧ ∃𝑧𝑉 (𝐹𝑧) = 𝑤))
4846, 47syl6bbr 290 . . . . 5 (𝜑 → ((𝑢𝐵𝑣𝐵𝑤𝐵) ↔ ∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤)))
494adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑅 ∈ Ring)
50 simp2 1131 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝑉𝑦𝑉) → 𝑥𝑉)
5123ad2ant1 1127 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝑉𝑦𝑉) → 𝑉 = (Base‘𝑅))
5250, 51eleqtrd 2920 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑉𝑦𝑉) → 𝑥 ∈ (Base‘𝑅))
53523adant3r3 1178 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑥 ∈ (Base‘𝑅))
54 simp3 1132 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝑉𝑦𝑉) → 𝑦𝑉)
5554, 51eleqtrd 2920 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑉𝑦𝑉) → 𝑦 ∈ (Base‘𝑅))
56553adant3r3 1178 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑦 ∈ (Base‘𝑅))
57 simpr3 1190 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑧𝑉)
582adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑉 = (Base‘𝑅))
5957, 58eleqtrd 2920 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑧 ∈ (Base‘𝑅))
6025, 17ringass 19234 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)))
6149, 53, 56, 59, 60syl13anc 1366 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)))
6261fveq2d 6671 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘((𝑥 · 𝑦) · 𝑧)) = (𝐹‘(𝑥 · (𝑦 · 𝑧))))
63 simpl 483 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝜑)
6428caovclg 7330 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥 · 𝑦) ∈ 𝑉)
65643adantr3 1165 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 · 𝑦) ∈ 𝑉)
663, 16, 1, 2, 4, 17, 18imasmulval 16798 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 · 𝑦) ∈ 𝑉𝑧𝑉) → ((𝐹‘(𝑥 · 𝑦))(.r𝑈)(𝐹𝑧)) = (𝐹‘((𝑥 · 𝑦) · 𝑧)))
6763, 65, 57, 66syl3anc 1365 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 · 𝑦))(.r𝑈)(𝐹𝑧)) = (𝐹‘((𝑥 · 𝑦) · 𝑧)))
68 simpr1 1188 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑥𝑉)
6928caovclg 7330 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝑉𝑧𝑉)) → (𝑦 · 𝑧) ∈ 𝑉)
70693adantr1 1163 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑦 · 𝑧) ∈ 𝑉)
713, 16, 1, 2, 4, 17, 18imasmulval 16798 . . . . . . . . . . . . 13 ((𝜑𝑥𝑉 ∧ (𝑦 · 𝑧) ∈ 𝑉) → ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘(𝑥 · (𝑦 · 𝑧))))
7263, 68, 70, 71syl3anc 1365 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘(𝑥 · (𝑦 · 𝑧))))
7362, 67, 723eqtr4d 2871 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 · 𝑦))(.r𝑈)(𝐹𝑧)) = ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 · 𝑧))))
743, 16, 1, 2, 4, 17, 18imasmulval 16798 . . . . . . . . . . . . 13 ((𝜑𝑥𝑉𝑦𝑉) → ((𝐹𝑥)(.r𝑈)(𝐹𝑦)) = (𝐹‘(𝑥 · 𝑦)))
75743adant3r3 1178 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)(𝐹𝑦)) = (𝐹‘(𝑥 · 𝑦)))
7675oveq1d 7163 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(.r𝑈)(𝐹𝑦))(.r𝑈)(𝐹𝑧)) = ((𝐹‘(𝑥 · 𝑦))(.r𝑈)(𝐹𝑧)))
773, 16, 1, 2, 4, 17, 18imasmulval 16798 . . . . . . . . . . . . 13 ((𝜑𝑦𝑉𝑧𝑉) → ((𝐹𝑦)(.r𝑈)(𝐹𝑧)) = (𝐹‘(𝑦 · 𝑧)))
78773adant3r1 1176 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑦)(.r𝑈)(𝐹𝑧)) = (𝐹‘(𝑦 · 𝑧)))
7978oveq2d 7164 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)((𝐹𝑦)(.r𝑈)(𝐹𝑧))) = ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 · 𝑧))))
8073, 76, 793eqtr4d 2871 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(.r𝑈)(𝐹𝑦))(.r𝑈)(𝐹𝑧)) = ((𝐹𝑥)(.r𝑈)((𝐹𝑦)(.r𝑈)(𝐹𝑧))))
81 simp1 1130 . . . . . . . . . . . . 13 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝐹𝑥) = 𝑢)
82 simp2 1131 . . . . . . . . . . . . 13 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝐹𝑦) = 𝑣)
8381, 82oveq12d 7166 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(.r𝑈)(𝐹𝑦)) = (𝑢(.r𝑈)𝑣))
84 simp3 1132 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝐹𝑧) = 𝑤)
8583, 84oveq12d 7166 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (((𝐹𝑥)(.r𝑈)(𝐹𝑦))(.r𝑈)(𝐹𝑧)) = ((𝑢(.r𝑈)𝑣)(.r𝑈)𝑤))
8682, 84oveq12d 7166 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑦)(.r𝑈)(𝐹𝑧)) = (𝑣(.r𝑈)𝑤))
8781, 86oveq12d 7166 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(.r𝑈)((𝐹𝑦)(.r𝑈)(𝐹𝑧))) = (𝑢(.r𝑈)(𝑣(.r𝑈)𝑤)))
8885, 87eqeq12d 2842 . . . . . . . . . 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 246 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(.r𝑈)𝑣)(.r𝑈)𝑤) = (𝑢(.r𝑈)(𝑣(.r𝑈)𝑤))))
90893exp2 1348 . . . . . . . 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 3288 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (∃𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(.r𝑈)𝑣)(.r𝑈)𝑤) = (𝑢(.r𝑈)(𝑣(.r𝑈)𝑤))))
9392rexlimdvva 3299 . . . . 5 (𝜑 → (∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(.r𝑈)𝑣)(.r𝑈)𝑤) = (𝑢(.r𝑈)(𝑣(.r𝑈)𝑤))))
9448, 93sylbid 241 . . . 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 19236 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
9749, 53, 56, 59, 96syl13anc 1366 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
9897fveq2d 6671 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘(𝑥 · (𝑦 + 𝑧))) = (𝐹‘((𝑥 · 𝑦) + (𝑥 · 𝑧))))
9925, 8ringacl 19248 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Ring ∧ 𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ (Base‘𝑅)) → (𝑢 + 𝑣) ∈ (Base‘𝑅))
10019, 22, 24, 99syl3anc 1365 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → (𝑢 + 𝑣) ∈ (Base‘𝑅))
101100, 21eleqtrrd 2921 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → (𝑢 + 𝑣) ∈ 𝑉)
102101caovclg 7330 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝑉𝑧𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
1031023adantr1 1163 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
1043, 16, 1, 2, 4, 17, 18imasmulval 16798 . . . . . . . . . . . . 13 ((𝜑𝑥𝑉 ∧ (𝑦 + 𝑧) ∈ 𝑉) → ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥 · (𝑦 + 𝑧))))
10563, 68, 103, 104syl3anc 1365 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥 · (𝑦 + 𝑧))))
10628caovclg 7330 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑧𝑉)) → (𝑥 · 𝑧) ∈ 𝑉)
1071063adantr2 1164 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 · 𝑧) ∈ 𝑉)
108 eqid 2826 . . . . . . . . . . . . . 14 (+g𝑈) = (+g𝑈)
1093, 10, 1, 2, 4, 8, 108imasaddval 16795 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 · 𝑦) ∈ 𝑉 ∧ (𝑥 · 𝑧) ∈ 𝑉) → ((𝐹‘(𝑥 · 𝑦))(+g𝑈)(𝐹‘(𝑥 · 𝑧))) = (𝐹‘((𝑥 · 𝑦) + (𝑥 · 𝑧))))
11063, 65, 107, 109syl3anc 1365 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 · 𝑦))(+g𝑈)(𝐹‘(𝑥 · 𝑧))) = (𝐹‘((𝑥 · 𝑦) + (𝑥 · 𝑧))))
11198, 105, 1103eqtr4d 2871 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 + 𝑧))) = ((𝐹‘(𝑥 · 𝑦))(+g𝑈)(𝐹‘(𝑥 · 𝑧))))
1123, 10, 1, 2, 4, 8, 108imasaddval 16795 . . . . . . . . . . . . 13 ((𝜑𝑦𝑉𝑧𝑉) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝐹‘(𝑦 + 𝑧)))
1131123adant3r1 1176 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝐹‘(𝑦 + 𝑧)))
114113oveq2d 7164 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))) = ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 + 𝑧))))
1153, 16, 1, 2, 4, 17, 18imasmulval 16798 . . . . . . . . . . . . 13 ((𝜑𝑥𝑉𝑧𝑉) → ((𝐹𝑥)(.r𝑈)(𝐹𝑧)) = (𝐹‘(𝑥 · 𝑧)))
1161153adant3r2 1177 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)(𝐹𝑧)) = (𝐹‘(𝑥 · 𝑧)))
11775, 116oveq12d 7166 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(.r𝑈)(𝐹𝑦))(+g𝑈)((𝐹𝑥)(.r𝑈)(𝐹𝑧))) = ((𝐹‘(𝑥 · 𝑦))(+g𝑈)(𝐹‘(𝑥 · 𝑧))))
118111, 114, 1173eqtr4d 2871 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))) = (((𝐹𝑥)(.r𝑈)(𝐹𝑦))(+g𝑈)((𝐹𝑥)(.r𝑈)(𝐹𝑧))))
11982, 84oveq12d 7166 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝑣(+g𝑈)𝑤))
12081, 119oveq12d 7166 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(.r𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))) = (𝑢(.r𝑈)(𝑣(+g𝑈)𝑤)))
12181, 84oveq12d 7166 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(.r𝑈)(𝐹𝑧)) = (𝑢(.r𝑈)𝑤))
12283, 121oveq12d 7166 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (((𝐹𝑥)(.r𝑈)(𝐹𝑦))(+g𝑈)((𝐹𝑥)(.r𝑈)(𝐹𝑧))) = ((𝑢(.r𝑈)𝑣)(+g𝑈)(𝑢(.r𝑈)𝑤)))
123120, 122eqeq12d 2842 . . . . . . . . . 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 246 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝑢(.r𝑈)(𝑣(+g𝑈)𝑤)) = ((𝑢(.r𝑈)𝑣)(+g𝑈)(𝑢(.r𝑈)𝑤))))
1251243exp2 1348 . . . . . . . 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 3288 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (∃𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝑢(.r𝑈)(𝑣(+g𝑈)𝑤)) = ((𝑢(.r𝑈)𝑣)(+g𝑈)(𝑢(.r𝑈)𝑤))))
128127rexlimdvva 3299 . . . . 5 (𝜑 → (∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝑢(.r𝑈)(𝑣(+g𝑈)𝑤)) = ((𝑢(.r𝑈)𝑣)(+g𝑈)(𝑢(.r𝑈)𝑤))))
12948, 128sylbid 241 . . . 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 19237 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
13249, 53, 56, 59, 131syl13anc 1366 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
133132fveq2d 6671 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘((𝑥 + 𝑦) · 𝑧)) = (𝐹‘((𝑥 · 𝑧) + (𝑦 · 𝑧))))
134101caovclg 7330 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
1351343adantr3 1165 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
1363, 16, 1, 2, 4, 17, 18imasmulval 16798 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 + 𝑦) ∈ 𝑉𝑧𝑉) → ((𝐹‘(𝑥 + 𝑦))(.r𝑈)(𝐹𝑧)) = (𝐹‘((𝑥 + 𝑦) · 𝑧)))
13763, 135, 57, 136syl3anc 1365 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 + 𝑦))(.r𝑈)(𝐹𝑧)) = (𝐹‘((𝑥 + 𝑦) · 𝑧)))
1383, 10, 1, 2, 4, 8, 108imasaddval 16795 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 · 𝑧) ∈ 𝑉 ∧ (𝑦 · 𝑧) ∈ 𝑉) → ((𝐹‘(𝑥 · 𝑧))(+g𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘((𝑥 · 𝑧) + (𝑦 · 𝑧))))
13963, 107, 70, 138syl3anc 1365 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 · 𝑧))(+g𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘((𝑥 · 𝑧) + (𝑦 · 𝑧))))
140133, 137, 1393eqtr4d 2871 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 + 𝑦))(.r𝑈)(𝐹𝑧)) = ((𝐹‘(𝑥 · 𝑧))(+g𝑈)(𝐹‘(𝑦 · 𝑧))))
1413, 10, 1, 2, 4, 8, 108imasaddval 16795 . . . . . . . . . . . . 13 ((𝜑𝑥𝑉𝑦𝑉) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝐹‘(𝑥 + 𝑦)))
1421413adant3r3 1178 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝐹‘(𝑥 + 𝑦)))
143142oveq1d 7163 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(+g𝑈)(𝐹𝑦))(.r𝑈)(𝐹𝑧)) = ((𝐹‘(𝑥 + 𝑦))(.r𝑈)(𝐹𝑧)))
144116, 78oveq12d 7166 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(.r𝑈)(𝐹𝑧))(+g𝑈)((𝐹𝑦)(.r𝑈)(𝐹𝑧))) = ((𝐹‘(𝑥 · 𝑧))(+g𝑈)(𝐹‘(𝑦 · 𝑧))))
145140, 143, 1443eqtr4d 2871 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(+g𝑈)(𝐹𝑦))(.r𝑈)(𝐹𝑧)) = (((𝐹𝑥)(.r𝑈)(𝐹𝑧))(+g𝑈)((𝐹𝑦)(.r𝑈)(𝐹𝑧))))
14681, 82oveq12d 7166 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝑢(+g𝑈)𝑣))
147146, 84oveq12d 7166 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (((𝐹𝑥)(+g𝑈)(𝐹𝑦))(.r𝑈)(𝐹𝑧)) = ((𝑢(+g𝑈)𝑣)(.r𝑈)𝑤))
148121, 86oveq12d 7166 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (((𝐹𝑥)(.r𝑈)(𝐹𝑧))(+g𝑈)((𝐹𝑦)(.r𝑈)(𝐹𝑧))) = ((𝑢(.r𝑈)𝑤)(+g𝑈)(𝑣(.r𝑈)𝑤)))
149147, 148eqeq12d 2842 . . . . . . . . . 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 246 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(.r𝑈)𝑤) = ((𝑢(.r𝑈)𝑤)(+g𝑈)(𝑣(.r𝑈)𝑤))))
1511503exp2 1348 . . . . . . . 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 3288 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (∃𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(.r𝑈)𝑤) = ((𝑢(.r𝑈)𝑤)(+g𝑈)(𝑣(.r𝑈)𝑤))))
154153rexlimdvva 3299 . . . . 5 (𝜑 → (∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(.r𝑈)𝑤) = ((𝑢(.r𝑈)𝑤)(+g𝑈)(𝑣(.r𝑈)𝑤))))
15548, 154sylbid 241 . . . 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 6587 . . . . 5 (𝐹:𝑉onto𝐵𝐹:𝑉𝐵)
1583, 157syl 17 . . . 4 (𝜑𝐹:𝑉𝐵)
159 imasring.o . . . . . . 7 1 = (1r𝑅)
16025, 159ringidcl 19238 . . . . . 6 (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
1614, 160syl 17 . . . . 5 (𝜑1 ∈ (Base‘𝑅))
162161, 2eleqtrrd 2921 . . . 4 (𝜑1𝑉)
163158, 162ffvelrnd 6848 . . 3 (𝜑 → (𝐹1 ) ∈ 𝐵)
16440, 41syl 17 . . . . . 6 (𝜑 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥𝑉 (𝐹𝑥) = 𝑢))
16535, 164bitr3d 282 . . . . 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 16798 . . . . . . . . 9 ((𝜑1𝑉𝑥𝑉) → ((𝐹1 )(.r𝑈)(𝐹𝑥)) = (𝐹‘( 1 · 𝑥)))
170166, 167, 168, 169syl3anc 1365 . . . . . . . 8 ((𝜑𝑥𝑉) → ((𝐹1 )(.r𝑈)(𝐹𝑥)) = (𝐹‘( 1 · 𝑥)))
1712eleq2d 2903 . . . . . . . . . . 11 (𝜑 → (𝑥𝑉𝑥 ∈ (Base‘𝑅)))
172171biimpa 477 . . . . . . . . . 10 ((𝜑𝑥𝑉) → 𝑥 ∈ (Base‘𝑅))
17325, 17, 159ringlidm 19241 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → ( 1 · 𝑥) = 𝑥)
1744, 172, 173syl2an2r 681 . . . . . . . . 9 ((𝜑𝑥𝑉) → ( 1 · 𝑥) = 𝑥)
175174fveq2d 6671 . . . . . . . 8 ((𝜑𝑥𝑉) → (𝐹‘( 1 · 𝑥)) = (𝐹𝑥))
176170, 175eqtrd 2861 . . . . . . 7 ((𝜑𝑥𝑉) → ((𝐹1 )(.r𝑈)(𝐹𝑥)) = (𝐹𝑥))
177 oveq2 7156 . . . . . . . 8 ((𝐹𝑥) = 𝑢 → ((𝐹1 )(.r𝑈)(𝐹𝑥)) = ((𝐹1 )(.r𝑈)𝑢))
178 id 22 . . . . . . . 8 ((𝐹𝑥) = 𝑢 → (𝐹𝑥) = 𝑢)
179177, 178eqeq12d 2842 . . . . . . 7 ((𝐹𝑥) = 𝑢 → (((𝐹1 )(.r𝑈)(𝐹𝑥)) = (𝐹𝑥) ↔ ((𝐹1 )(.r𝑈)𝑢) = 𝑢))
180176, 179syl5ibcom 246 . . . . . 6 ((𝜑𝑥𝑉) → ((𝐹𝑥) = 𝑢 → ((𝐹1 )(.r𝑈)𝑢) = 𝑢))
181180rexlimdva 3289 . . . . 5 (𝜑 → (∃𝑥𝑉 (𝐹𝑥) = 𝑢 → ((𝐹1 )(.r𝑈)𝑢) = 𝑢))
182165, 181sylbid 241 . . . 4 (𝜑 → (𝑢𝐵 → ((𝐹1 )(.r𝑈)𝑢) = 𝑢))
183182imp 407 . . 3 ((𝜑𝑢𝐵) → ((𝐹1 )(.r𝑈)𝑢) = 𝑢)
1843, 16, 1, 2, 4, 17, 18imasmulval 16798 . . . . . . . . 9 ((𝜑𝑥𝑉1𝑉) → ((𝐹𝑥)(.r𝑈)(𝐹1 )) = (𝐹‘(𝑥 · 1 )))
185167, 184mpd3an3 1455 . . . . . . . 8 ((𝜑𝑥𝑉) → ((𝐹𝑥)(.r𝑈)(𝐹1 )) = (𝐹‘(𝑥 · 1 )))
18625, 17, 159ringridm 19242 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥 · 1 ) = 𝑥)
1874, 172, 186syl2an2r 681 . . . . . . . . 9 ((𝜑𝑥𝑉) → (𝑥 · 1 ) = 𝑥)
188187fveq2d 6671 . . . . . . . 8 ((𝜑𝑥𝑉) → (𝐹‘(𝑥 · 1 )) = (𝐹𝑥))
189185, 188eqtrd 2861 . . . . . . 7 ((𝜑𝑥𝑉) → ((𝐹𝑥)(.r𝑈)(𝐹1 )) = (𝐹𝑥))
190 oveq1 7155 . . . . . . . 8 ((𝐹𝑥) = 𝑢 → ((𝐹𝑥)(.r𝑈)(𝐹1 )) = (𝑢(.r𝑈)(𝐹1 )))
191190, 178eqeq12d 2842 . . . . . . 7 ((𝐹𝑥) = 𝑢 → (((𝐹𝑥)(.r𝑈)(𝐹1 )) = (𝐹𝑥) ↔ (𝑢(.r𝑈)(𝐹1 )) = 𝑢))
192189, 191syl5ibcom 246 . . . . . 6 ((𝜑𝑥𝑉) → ((𝐹𝑥) = 𝑢 → (𝑢(.r𝑈)(𝐹1 )) = 𝑢))
193192rexlimdva 3289 . . . . 5 (𝜑 → (∃𝑥𝑉 (𝐹𝑥) = 𝑢 → (𝑢(.r𝑈)(𝐹1 )) = 𝑢))
194165, 193sylbid 241 . . . 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 19255 . 2 (𝜑𝑈 ∈ Ring)
197163, 5eleqtrd 2920 . . . 4 (𝜑 → (𝐹1 ) ∈ (Base‘𝑈))
1985eleq2d 2903 . . . . . 6 (𝜑 → (𝑢𝐵𝑢 ∈ (Base‘𝑈)))
199182, 194jcad 513 . . . . . 6 (𝜑 → (𝑢𝐵 → (((𝐹1 )(.r𝑈)𝑢) = 𝑢 ∧ (𝑢(.r𝑈)(𝐹1 )) = 𝑢)))
200198, 199sylbird 261 . . . . 5 (𝜑 → (𝑢 ∈ (Base‘𝑈) → (((𝐹1 )(.r𝑈)𝑢) = 𝑢 ∧ (𝑢(.r𝑈)(𝐹1 )) = 𝑢)))
201200ralrimiv 3186 . . . 4 (𝜑 → ∀𝑢 ∈ (Base‘𝑈)(((𝐹1 )(.r𝑈)𝑢) = 𝑢 ∧ (𝑢(.r𝑈)(𝐹1 )) = 𝑢))
202 eqid 2826 . . . . . 6 (Base‘𝑈) = (Base‘𝑈)
203 eqid 2826 . . . . . 6 (1r𝑈) = (1r𝑈)
204202, 18, 203isringid 19243 . . . . 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 708 . . 3 (𝜑 → (1r𝑈) = (𝐹1 ))
207206eqcomd 2832 . 2 (𝜑 → (𝐹1 ) = (1r𝑈))
208196, 207jca 512 1 (𝜑 → (𝑈 ∈ Ring ∧ (𝐹1 ) = (1r𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wral 3143  wrex 3144   × cxp 5552  ran crn 5555   Fn wfn 6347  wf 6348  ontowfo 6350  cfv 6352  (class class class)co 7148  Basecbs 16473  +gcplusg 16555  .rcmulr 16556  0gc0g 16703  s cimas 16767  Grpcgrp 18033  1rcur 19171  Ringcrg 19217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-sup 8895  df-inf 8896  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-3 11690  df-4 11691  df-5 11692  df-6 11693  df-7 11694  df-8 11695  df-9 11696  df-n0 11887  df-z 11971  df-dec 12088  df-uz 12233  df-fz 12883  df-struct 16475  df-ndx 16476  df-slot 16477  df-base 16479  df-sets 16480  df-plusg 16568  df-mulr 16569  df-sca 16571  df-vsca 16572  df-ip 16573  df-tset 16574  df-ple 16575  df-ds 16577  df-0g 16705  df-imas 16771  df-mgm 17842  df-sgrp 17890  df-mnd 17901  df-grp 18036  df-minusg 18037  df-mgp 19160  df-ur 19172  df-ring 19219
This theorem is referenced by:  qusring2  19290
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