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Theorem ringi 18876
Description: Properties of a unital ring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
ringi.b 𝐵 = (Base‘𝑅)
ringi.p + = (+g𝑅)
ringi.t · = (.r𝑅)
Assertion
Ref Expression
ringi ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)) ∧ ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍))))

Proof of Theorem ringi
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ringi.b . . . . . . . . . 10 𝐵 = (Base‘𝑅)
2 eqid 2799 . . . . . . . . . 10 (mulGrp‘𝑅) = (mulGrp‘𝑅)
3 ringi.p . . . . . . . . . 10 + = (+g𝑅)
4 ringi.t . . . . . . . . . 10 · = (.r𝑅)
51, 2, 3, 4isring 18867 . . . . . . . . 9 (𝑅 ∈ Ring ↔ (𝑅 ∈ Grp ∧ (mulGrp‘𝑅) ∈ Mnd ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))
65simp3bi 1178 . . . . . . . 8 (𝑅 ∈ Ring → ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))))
76adantr 473 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))))
8 simpr1 1249 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → 𝑥𝐵)
9 rsp 3110 . . . . . . 7 (∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) → (𝑥𝐵 → ∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))
107, 8, 9sylc 65 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))))
11 simpr2 1251 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → 𝑦𝐵)
12 rsp 3110 . . . . . 6 (∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) → (𝑦𝐵 → ∀𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))
1310, 11, 12sylc 65 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ∀𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))))
14 simpr3 1253 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → 𝑧𝐵)
15 rsp 3110 . . . . 5 (∀𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) → (𝑧𝐵 → ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))
1613, 14, 15sylc 65 . . . 4 ((𝑅 ∈ Ring ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))))
1716simpld 489 . . 3 ((𝑅 ∈ Ring ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
1817caovdig 7082 . 2 ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))
1916simprd 490 . . 3 ((𝑅 ∈ Ring ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
2019caovdirg 7085 . 2 ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))
2118, 20jca 508 1 ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)) ∧ ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108   = wceq 1653  wcel 2157  wral 3089  cfv 6101  (class class class)co 6878  Basecbs 16184  +gcplusg 16267  .rcmulr 16268  Mndcmnd 17609  Grpcgrp 17738  mulGrpcmgp 18805  Ringcrg 18863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-nul 4983
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-iota 6064  df-fv 6109  df-ov 6881  df-ring 18865
This theorem is referenced by:  ringdi  18882  ringdir  18883
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