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Theorem unitgrp 13238
Description: The group of units is a group under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.)
Hypotheses
Ref Expression
unitgrp.1 𝑈 = (Unit‘𝑅)
unitgrp.2 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)
Assertion
Ref Expression
unitgrp (𝑅 ∈ Ring → 𝐺 ∈ Grp)

Proof of Theorem unitgrp
Dummy variables 𝑥 𝑦 𝑧 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unitgrp.1 . . . 4 𝑈 = (Unit‘𝑅)
21a1i 9 . . 3 (𝑅 ∈ Ring → 𝑈 = (Unit‘𝑅))
3 unitgrp.2 . . . 4 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)
43a1i 9 . . 3 (𝑅 ∈ Ring → 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈))
5 ringsrg 13177 . . 3 (𝑅 ∈ Ring → 𝑅 ∈ SRing)
62, 4, 5unitgrpbasd 13237 . 2 (𝑅 ∈ Ring → 𝑈 = (Base‘𝐺))
7 eqid 2177 . . . 4 (mulGrp‘𝑅) = (mulGrp‘𝑅)
8 eqid 2177 . . . 4 (.r𝑅) = (.r𝑅)
97, 8mgpplusgg 13087 . . 3 (𝑅 ∈ Ring → (.r𝑅) = (+g‘(mulGrp‘𝑅)))
10 basfn 12514 . . . . 5 Base Fn V
11 elex 2748 . . . . 5 (𝑅 ∈ Ring → 𝑅 ∈ V)
12 funfvex 5532 . . . . . 6 ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
1312funfni 5316 . . . . 5 ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
1410, 11, 13sylancr 414 . . . 4 (𝑅 ∈ Ring → (Base‘𝑅) ∈ V)
15 eqidd 2178 . . . . 5 (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘𝑅))
1615, 2, 5unitssd 13231 . . . 4 (𝑅 ∈ Ring → 𝑈 ⊆ (Base‘𝑅))
1714, 16ssexd 4143 . . 3 (𝑅 ∈ Ring → 𝑈 ∈ V)
187mgpex 13088 . . 3 (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ V)
194, 9, 17, 18ressplusgd 12581 . 2 (𝑅 ∈ Ring → (.r𝑅) = (+g𝐺))
201, 8unitmulcl 13235 . 2 ((𝑅 ∈ Ring ∧ 𝑥𝑈𝑦𝑈) → (𝑥(.r𝑅)𝑦) ∈ 𝑈)
21 eqidd 2178 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑥𝑈𝑦𝑈𝑧𝑈)) → (Base‘𝑅) = (Base‘𝑅))
221a1i 9 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑥𝑈𝑦𝑈𝑧𝑈)) → 𝑈 = (Unit‘𝑅))
235adantr 276 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑥𝑈𝑦𝑈𝑧𝑈)) → 𝑅 ∈ SRing)
24 simpr1 1003 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑥𝑈𝑦𝑈𝑧𝑈)) → 𝑥𝑈)
2521, 22, 23, 24unitcld 13230 . . . 4 ((𝑅 ∈ Ring ∧ (𝑥𝑈𝑦𝑈𝑧𝑈)) → 𝑥 ∈ (Base‘𝑅))
26 simpr2 1004 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑥𝑈𝑦𝑈𝑧𝑈)) → 𝑦𝑈)
2721, 22, 23, 26unitcld 13230 . . . 4 ((𝑅 ∈ Ring ∧ (𝑥𝑈𝑦𝑈𝑧𝑈)) → 𝑦 ∈ (Base‘𝑅))
28 simpr3 1005 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑥𝑈𝑦𝑈𝑧𝑈)) → 𝑧𝑈)
2921, 22, 23, 28unitcld 13230 . . . 4 ((𝑅 ∈ Ring ∧ (𝑥𝑈𝑦𝑈𝑧𝑈)) → 𝑧 ∈ (Base‘𝑅))
3025, 27, 293jca 1177 . . 3 ((𝑅 ∈ Ring ∧ (𝑥𝑈𝑦𝑈𝑧𝑈)) → (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅)))
31 eqid 2177 . . . 4 (Base‘𝑅) = (Base‘𝑅)
3231, 8ringass 13152 . . 3 ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥(.r𝑅)𝑦)(.r𝑅)𝑧) = (𝑥(.r𝑅)(𝑦(.r𝑅)𝑧)))
3330, 32syldan 282 . 2 ((𝑅 ∈ Ring ∧ (𝑥𝑈𝑦𝑈𝑧𝑈)) → ((𝑥(.r𝑅)𝑦)(.r𝑅)𝑧) = (𝑥(.r𝑅)(𝑦(.r𝑅)𝑧)))
34 eqid 2177 . . 3 (1r𝑅) = (1r𝑅)
351, 341unit 13229 . 2 (𝑅 ∈ Ring → (1r𝑅) ∈ 𝑈)
36 eqidd 2178 . . . 4 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → (Base‘𝑅) = (Base‘𝑅))
371a1i 9 . . . 4 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → 𝑈 = (Unit‘𝑅))
385adantr 276 . . . 4 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → 𝑅 ∈ SRing)
39 simpr 110 . . . 4 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → 𝑥𝑈)
4036, 37, 38, 39unitcld 13230 . . 3 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → 𝑥 ∈ (Base‘𝑅))
4131, 8, 34ringlidm 13159 . . 3 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → ((1r𝑅)(.r𝑅)𝑥) = 𝑥)
4240, 41syldan 282 . 2 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → ((1r𝑅)(.r𝑅)𝑥) = 𝑥)
43 eqidd 2178 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → (1r𝑅) = (1r𝑅))
44 eqidd 2178 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → (∥r𝑅) = (∥r𝑅))
45 eqidd 2178 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → (oppr𝑅) = (oppr𝑅))
46 eqidd 2178 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → (∥r‘(oppr𝑅)) = (∥r‘(oppr𝑅)))
4737, 43, 44, 45, 46, 38isunitd 13228 . . . 4 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → (𝑥𝑈 ↔ (𝑥(∥r𝑅)(1r𝑅) ∧ 𝑥(∥r‘(oppr𝑅))(1r𝑅))))
4839, 47mpbid 147 . . 3 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → (𝑥(∥r𝑅)(1r𝑅) ∧ 𝑥(∥r‘(oppr𝑅))(1r𝑅)))
49 eqidd 2178 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → (.r𝑅) = (.r𝑅))
5036, 44, 38, 49, 40dvdsr2d 13217 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → (𝑥(∥r𝑅)(1r𝑅) ↔ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r𝑅)𝑥) = (1r𝑅)))
51 eqid 2177 . . . . . . . 8 (oppr𝑅) = (oppr𝑅)
5251, 31opprbasg 13200 . . . . . . 7 (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘(oppr𝑅)))
5352adantr 276 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → (Base‘𝑅) = (Base‘(oppr𝑅)))
5451opprring 13202 . . . . . . . 8 (𝑅 ∈ Ring → (oppr𝑅) ∈ Ring)
55 ringsrg 13177 . . . . . . . 8 ((oppr𝑅) ∈ Ring → (oppr𝑅) ∈ SRing)
5654, 55syl 14 . . . . . . 7 (𝑅 ∈ Ring → (oppr𝑅) ∈ SRing)
5756adantr 276 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → (oppr𝑅) ∈ SRing)
58 eqidd 2178 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → (.r‘(oppr𝑅)) = (.r‘(oppr𝑅)))
5953, 46, 57, 58, 40dvdsr2d 13217 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → (𝑥(∥r‘(oppr𝑅))(1r𝑅) ↔ ∃𝑚 ∈ (Base‘𝑅)(𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))
6050, 59anbi12d 473 . . . 4 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → ((𝑥(∥r𝑅)(1r𝑅) ∧ 𝑥(∥r‘(oppr𝑅))(1r𝑅)) ↔ (∃𝑦 ∈ (Base‘𝑅)(𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ ∃𝑚 ∈ (Base‘𝑅)(𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅))))
61 reeanv 2646 . . . . 5 (∃𝑦 ∈ (Base‘𝑅)∃𝑚 ∈ (Base‘𝑅)((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)) ↔ (∃𝑦 ∈ (Base‘𝑅)(𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ ∃𝑚 ∈ (Base‘𝑅)(𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))
62 eqidd 2178 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (Base‘𝑅) = (Base‘𝑅))
63 eqidd 2178 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (∥r𝑅) = (∥r𝑅))
6438ad2antrr 488 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑅 ∈ SRing)
65 eqidd 2178 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (.r𝑅) = (.r𝑅))
66 simprl 529 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑚 ∈ (Base‘𝑅))
6740ad2antrr 488 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑥 ∈ (Base‘𝑅))
6862, 63, 64, 65, 66, 67dvdsrmuld 13218 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑚(∥r𝑅)(𝑥(.r𝑅)𝑚))
69 simplll 533 . . . . . . . . . . . . . 14 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑅 ∈ Ring)
70 simplr 528 . . . . . . . . . . . . . 14 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑦 ∈ (Base‘𝑅))
7131, 8ringass 13152 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ (𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑚 ∈ (Base‘𝑅))) → ((𝑦(.r𝑅)𝑥)(.r𝑅)𝑚) = (𝑦(.r𝑅)(𝑥(.r𝑅)𝑚)))
7269, 70, 67, 66, 71syl13anc 1240 . . . . . . . . . . . . 13 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → ((𝑦(.r𝑅)𝑥)(.r𝑅)𝑚) = (𝑦(.r𝑅)(𝑥(.r𝑅)𝑚)))
73 simprrl 539 . . . . . . . . . . . . . 14 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (𝑦(.r𝑅)𝑥) = (1r𝑅))
7473oveq1d 5889 . . . . . . . . . . . . 13 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → ((𝑦(.r𝑅)𝑥)(.r𝑅)𝑚) = ((1r𝑅)(.r𝑅)𝑚))
7539ad2antrr 488 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑥𝑈)
76 eqid 2177 . . . . . . . . . . . . . . . . 17 (.r‘(oppr𝑅)) = (.r‘(oppr𝑅))
7731, 8, 51, 76opprmulg 13196 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Ring ∧ 𝑚 ∈ (Base‘𝑅) ∧ 𝑥𝑈) → (𝑚(.r‘(oppr𝑅))𝑥) = (𝑥(.r𝑅)𝑚))
7869, 66, 75, 77syl3anc 1238 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (𝑚(.r‘(oppr𝑅))𝑥) = (𝑥(.r𝑅)𝑚))
79 simprrr 540 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅))
8078, 79eqtr3d 2212 . . . . . . . . . . . . . 14 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (𝑥(.r𝑅)𝑚) = (1r𝑅))
8180oveq2d 5890 . . . . . . . . . . . . 13 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (𝑦(.r𝑅)(𝑥(.r𝑅)𝑚)) = (𝑦(.r𝑅)(1r𝑅)))
8272, 74, 813eqtr3d 2218 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → ((1r𝑅)(.r𝑅)𝑚) = (𝑦(.r𝑅)(1r𝑅)))
8331, 8, 34ringlidm 13159 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑚 ∈ (Base‘𝑅)) → ((1r𝑅)(.r𝑅)𝑚) = 𝑚)
8469, 66, 83syl2anc 411 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → ((1r𝑅)(.r𝑅)𝑚) = 𝑚)
8531, 8, 34ringridm 13160 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑦(.r𝑅)(1r𝑅)) = 𝑦)
8669, 70, 85syl2anc 411 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (𝑦(.r𝑅)(1r𝑅)) = 𝑦)
8782, 84, 863eqtr3d 2218 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑚 = 𝑦)
8868, 87, 803brtr3d 4034 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑦(∥r𝑅)(1r𝑅))
8969, 52syl 14 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (Base‘𝑅) = (Base‘(oppr𝑅)))
90 eqidd 2178 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (∥r‘(oppr𝑅)) = (∥r‘(oppr𝑅)))
9169, 56syl 14 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (oppr𝑅) ∈ SRing)
92 eqidd 2178 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (.r‘(oppr𝑅)) = (.r‘(oppr𝑅)))
9389, 90, 91, 92, 70, 67dvdsrmuld 13218 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑦(∥r‘(oppr𝑅))(𝑥(.r‘(oppr𝑅))𝑦))
9431, 8, 51, 76opprmulg 13196 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑥𝑈𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘(oppr𝑅))𝑦) = (𝑦(.r𝑅)𝑥))
9569, 75, 70, 94syl3anc 1238 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (𝑥(.r‘(oppr𝑅))𝑦) = (𝑦(.r𝑅)𝑥))
9695, 73eqtrd 2210 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (𝑥(.r‘(oppr𝑅))𝑦) = (1r𝑅))
9793, 96breqtrd 4029 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑦(∥r‘(oppr𝑅))(1r𝑅))
981a1i 9 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑈 = (Unit‘𝑅))
99 eqidd 2178 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (1r𝑅) = (1r𝑅))
100 eqidd 2178 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (oppr𝑅) = (oppr𝑅))
10198, 99, 63, 100, 90, 64isunitd 13228 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (𝑦𝑈 ↔ (𝑦(∥r𝑅)(1r𝑅) ∧ 𝑦(∥r‘(oppr𝑅))(1r𝑅))))
10288, 97, 101mpbir2and 944 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑦𝑈)
103102, 73jca 306 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (𝑦𝑈 ∧ (𝑦(.r𝑅)𝑥) = (1r𝑅)))
104103rexlimdvaa 2595 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) → (∃𝑚 ∈ (Base‘𝑅)((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)) → (𝑦𝑈 ∧ (𝑦(.r𝑅)𝑥) = (1r𝑅))))
105104expimpd 363 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → ((𝑦 ∈ (Base‘𝑅) ∧ ∃𝑚 ∈ (Base‘𝑅)((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅))) → (𝑦𝑈 ∧ (𝑦(.r𝑅)𝑥) = (1r𝑅))))
106105reximdv2 2576 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → (∃𝑦 ∈ (Base‘𝑅)∃𝑚 ∈ (Base‘𝑅)((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)) → ∃𝑦𝑈 (𝑦(.r𝑅)𝑥) = (1r𝑅)))
10761, 106biimtrrid 153 . . . 4 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → ((∃𝑦 ∈ (Base‘𝑅)(𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ ∃𝑚 ∈ (Base‘𝑅)(𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)) → ∃𝑦𝑈 (𝑦(.r𝑅)𝑥) = (1r𝑅)))
10860, 107sylbid 150 . . 3 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → ((𝑥(∥r𝑅)(1r𝑅) ∧ 𝑥(∥r‘(oppr𝑅))(1r𝑅)) → ∃𝑦𝑈 (𝑦(.r𝑅)𝑥) = (1r𝑅)))
10948, 108mpd 13 . 2 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → ∃𝑦𝑈 (𝑦(.r𝑅)𝑥) = (1r𝑅))
1106, 19, 20, 33, 35, 42, 109isgrpde 12852 1 (𝑅 ∈ Ring → 𝐺 ∈ Grp)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978   = wceq 1353  wcel 2148  wrex 2456  Vcvv 2737   class class class wbr 4003   Fn wfn 5211  cfv 5216  (class class class)co 5874  Basecbs 12456  s cress 12457  .rcmulr 12531  Grpcgrp 12831  mulGrpcmgp 13083  1rcur 13095  SRingcsrg 13099  Ringcrg 13132  opprcoppr 13192  rcdsr 13208  Unitcui 13209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-pre-ltirr 7922  ax-pre-lttrn 7924  ax-pre-ltadd 7926
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-tpos 6245  df-pnf 7992  df-mnf 7993  df-ltxr 7995  df-inn 8918  df-2 8976  df-3 8977  df-ndx 12459  df-slot 12460  df-base 12462  df-sets 12463  df-iress 12464  df-plusg 12543  df-mulr 12544  df-0g 12697  df-mgm 12729  df-sgrp 12762  df-mnd 12772  df-grp 12834  df-minusg 12835  df-cmn 13043  df-abl 13044  df-mgp 13084  df-ur 13096  df-srg 13100  df-ring 13134  df-oppr 13193  df-dvdsr 13211  df-unit 13212
This theorem is referenced by:  unitabl  13239  unitsubm  13241  invrfvald  13244  unitinvcl  13245  unitinvinv  13246  unitlinv  13248  unitrinv  13249  rdivmuldivd  13266  subrgugrp  13321
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