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Theorem srgdilem 12965
Description: Lemma for srgdi 12970 and srgdir 12971. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
srgdilem.b 𝐵 = (Base‘𝑅)
srgdilem.p + = (+g𝑅)
srgdilem.t · = (.r𝑅)
Assertion
Ref Expression
srgdilem ((𝑅 ∈ SRing ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)) ∧ ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍))))

Proof of Theorem srgdilem
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 srgdilem.b . . . . . . . . . . 11 𝐵 = (Base‘𝑅)
2 eqid 2177 . . . . . . . . . . 11 (mulGrp‘𝑅) = (mulGrp‘𝑅)
3 srgdilem.p . . . . . . . . . . 11 + = (+g𝑅)
4 srgdilem.t . . . . . . . . . . 11 · = (.r𝑅)
5 eqid 2177 . . . . . . . . . . 11 (0g𝑅) = (0g𝑅)
61, 2, 3, 4, 5issrg 12961 . . . . . . . . . 10 (𝑅 ∈ SRing ↔ (𝑅 ∈ CMnd ∧ (mulGrp‘𝑅) ∈ Mnd ∧ ∀𝑥𝐵 (∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (((0g𝑅) · 𝑥) = (0g𝑅) ∧ (𝑥 · (0g𝑅)) = (0g𝑅)))))
76simp3bi 1014 . . . . . . . . 9 (𝑅 ∈ SRing → ∀𝑥𝐵 (∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (((0g𝑅) · 𝑥) = (0g𝑅) ∧ (𝑥 · (0g𝑅)) = (0g𝑅))))
87r19.21bi 2565 . . . . . . . 8 ((𝑅 ∈ SRing ∧ 𝑥𝐵) → (∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (((0g𝑅) · 𝑥) = (0g𝑅) ∧ (𝑥 · (0g𝑅)) = (0g𝑅))))
98simpld 112 . . . . . . 7 ((𝑅 ∈ SRing ∧ 𝑥𝐵) → ∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))))
1093ad2antr1 1162 . . . . . 6 ((𝑅 ∈ SRing ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))))
11 simpr2 1004 . . . . . 6 ((𝑅 ∈ SRing ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → 𝑦𝐵)
12 rsp 2524 . . . . . 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 62 . . . . 5 ((𝑅 ∈ SRing ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ∀𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))))
14 simpr3 1005 . . . . 5 ((𝑅 ∈ SRing ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → 𝑧𝐵)
15 rsp 2524 . . . . 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 62 . . . 4 ((𝑅 ∈ SRing ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))))
1716simpld 112 . . 3 ((𝑅 ∈ SRing ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
1817caovdig 6042 . 2 ((𝑅 ∈ SRing ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))
1916simprd 114 . . 3 ((𝑅 ∈ SRing ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
2019caovdirg 6045 . 2 ((𝑅 ∈ SRing ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))
2118, 20jca 306 1 ((𝑅 ∈ SRing ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)) ∧ ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978   = wceq 1353  wcel 2148  wral 2455  cfv 5211  (class class class)co 5868  Basecbs 12432  +gcplusg 12505  .rcmulr 12506  0gc0g 12640  Mndcmnd 12696  CMndccmn 12902  mulGrpcmgp 12944  SRingcsrg 12959
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-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-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-cnex 7880  ax-resscn 7881  ax-1re 7883  ax-addrcl 7886
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  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-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-iota 5173  df-fun 5213  df-fn 5214  df-fv 5219  df-riota 5824  df-ov 5871  df-inn 8896  df-2 8954  df-3 8955  df-ndx 12435  df-slot 12436  df-base 12438  df-plusg 12518  df-mulr 12519  df-0g 12642  df-srg 12960
This theorem is referenced by:  srgdi  12970  srgdir  12971
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