ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  srglmhm GIF version

Theorem srglmhm 12999
Description: Left-multiplication in a semiring by a fixed element of the ring is a monoid homomorphism. (Contributed by AV, 23-Aug-2019.)
Hypotheses
Ref Expression
srglmhm.b 𝐵 = (Base‘𝑅)
srglmhm.t · = (.r𝑅)
Assertion
Ref Expression
srglmhm ((𝑅 ∈ SRing ∧ 𝑋𝐵) → (𝑥𝐵 ↦ (𝑋 · 𝑥)) ∈ (𝑅 MndHom 𝑅))
Distinct variable groups:   𝑥,𝐵   𝑥,𝑅   𝑥,𝑋   𝑥, ·

Proof of Theorem srglmhm
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 srgmnd 12973 . . . 4 (𝑅 ∈ SRing → 𝑅 ∈ Mnd)
21, 1jca 306 . . 3 (𝑅 ∈ SRing → (𝑅 ∈ Mnd ∧ 𝑅 ∈ Mnd))
32adantr 276 . 2 ((𝑅 ∈ SRing ∧ 𝑋𝐵) → (𝑅 ∈ Mnd ∧ 𝑅 ∈ Mnd))
4 srglmhm.b . . . . . 6 𝐵 = (Base‘𝑅)
5 srglmhm.t . . . . . 6 · = (.r𝑅)
64, 5srgcl 12976 . . . . 5 ((𝑅 ∈ SRing ∧ 𝑋𝐵𝑥𝐵) → (𝑋 · 𝑥) ∈ 𝐵)
763expa 1203 . . . 4 (((𝑅 ∈ SRing ∧ 𝑋𝐵) ∧ 𝑥𝐵) → (𝑋 · 𝑥) ∈ 𝐵)
87fmpttd 5667 . . 3 ((𝑅 ∈ SRing ∧ 𝑋𝐵) → (𝑥𝐵 ↦ (𝑋 · 𝑥)):𝐵𝐵)
9 3anass 982 . . . . . . 7 ((𝑋𝐵𝑎𝐵𝑏𝐵) ↔ (𝑋𝐵 ∧ (𝑎𝐵𝑏𝐵)))
10 eqid 2177 . . . . . . . 8 (+g𝑅) = (+g𝑅)
114, 10, 5srgdi 12980 . . . . . . 7 ((𝑅 ∈ SRing ∧ (𝑋𝐵𝑎𝐵𝑏𝐵)) → (𝑋 · (𝑎(+g𝑅)𝑏)) = ((𝑋 · 𝑎)(+g𝑅)(𝑋 · 𝑏)))
129, 11sylan2br 288 . . . . . 6 ((𝑅 ∈ SRing ∧ (𝑋𝐵 ∧ (𝑎𝐵𝑏𝐵))) → (𝑋 · (𝑎(+g𝑅)𝑏)) = ((𝑋 · 𝑎)(+g𝑅)(𝑋 · 𝑏)))
1312anassrs 400 . . . . 5 (((𝑅 ∈ SRing ∧ 𝑋𝐵) ∧ (𝑎𝐵𝑏𝐵)) → (𝑋 · (𝑎(+g𝑅)𝑏)) = ((𝑋 · 𝑎)(+g𝑅)(𝑋 · 𝑏)))
14 eqid 2177 . . . . . 6 (𝑥𝐵 ↦ (𝑋 · 𝑥)) = (𝑥𝐵 ↦ (𝑋 · 𝑥))
15 oveq2 5877 . . . . . 6 (𝑥 = (𝑎(+g𝑅)𝑏) → (𝑋 · 𝑥) = (𝑋 · (𝑎(+g𝑅)𝑏)))
164, 10srgacl 12988 . . . . . . . 8 ((𝑅 ∈ SRing ∧ 𝑎𝐵𝑏𝐵) → (𝑎(+g𝑅)𝑏) ∈ 𝐵)
17163expb 1204 . . . . . . 7 ((𝑅 ∈ SRing ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝑅)𝑏) ∈ 𝐵)
1817adantlr 477 . . . . . 6 (((𝑅 ∈ SRing ∧ 𝑋𝐵) ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝑅)𝑏) ∈ 𝐵)
19 simpll 527 . . . . . . 7 (((𝑅 ∈ SRing ∧ 𝑋𝐵) ∧ (𝑎𝐵𝑏𝐵)) → 𝑅 ∈ SRing)
20 simplr 528 . . . . . . 7 (((𝑅 ∈ SRing ∧ 𝑋𝐵) ∧ (𝑎𝐵𝑏𝐵)) → 𝑋𝐵)
214, 5srgcl 12976 . . . . . . 7 ((𝑅 ∈ SRing ∧ 𝑋𝐵 ∧ (𝑎(+g𝑅)𝑏) ∈ 𝐵) → (𝑋 · (𝑎(+g𝑅)𝑏)) ∈ 𝐵)
2219, 20, 18, 21syl3anc 1238 . . . . . 6 (((𝑅 ∈ SRing ∧ 𝑋𝐵) ∧ (𝑎𝐵𝑏𝐵)) → (𝑋 · (𝑎(+g𝑅)𝑏)) ∈ 𝐵)
2314, 15, 18, 22fvmptd3 5605 . . . . 5 (((𝑅 ∈ SRing ∧ 𝑋𝐵) ∧ (𝑎𝐵𝑏𝐵)) → ((𝑥𝐵 ↦ (𝑋 · 𝑥))‘(𝑎(+g𝑅)𝑏)) = (𝑋 · (𝑎(+g𝑅)𝑏)))
24 oveq2 5877 . . . . . . 7 (𝑥 = 𝑎 → (𝑋 · 𝑥) = (𝑋 · 𝑎))
25 simprl 529 . . . . . . 7 (((𝑅 ∈ SRing ∧ 𝑋𝐵) ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝐵)
264, 5srgcl 12976 . . . . . . . 8 ((𝑅 ∈ SRing ∧ 𝑋𝐵𝑎𝐵) → (𝑋 · 𝑎) ∈ 𝐵)
2719, 20, 25, 26syl3anc 1238 . . . . . . 7 (((𝑅 ∈ SRing ∧ 𝑋𝐵) ∧ (𝑎𝐵𝑏𝐵)) → (𝑋 · 𝑎) ∈ 𝐵)
2814, 24, 25, 27fvmptd3 5605 . . . . . 6 (((𝑅 ∈ SRing ∧ 𝑋𝐵) ∧ (𝑎𝐵𝑏𝐵)) → ((𝑥𝐵 ↦ (𝑋 · 𝑥))‘𝑎) = (𝑋 · 𝑎))
29 oveq2 5877 . . . . . . 7 (𝑥 = 𝑏 → (𝑋 · 𝑥) = (𝑋 · 𝑏))
30 simprr 531 . . . . . . 7 (((𝑅 ∈ SRing ∧ 𝑋𝐵) ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝐵)
314, 5srgcl 12976 . . . . . . . 8 ((𝑅 ∈ SRing ∧ 𝑋𝐵𝑏𝐵) → (𝑋 · 𝑏) ∈ 𝐵)
3219, 20, 30, 31syl3anc 1238 . . . . . . 7 (((𝑅 ∈ SRing ∧ 𝑋𝐵) ∧ (𝑎𝐵𝑏𝐵)) → (𝑋 · 𝑏) ∈ 𝐵)
3314, 29, 30, 32fvmptd3 5605 . . . . . 6 (((𝑅 ∈ SRing ∧ 𝑋𝐵) ∧ (𝑎𝐵𝑏𝐵)) → ((𝑥𝐵 ↦ (𝑋 · 𝑥))‘𝑏) = (𝑋 · 𝑏))
3428, 33oveq12d 5887 . . . . 5 (((𝑅 ∈ SRing ∧ 𝑋𝐵) ∧ (𝑎𝐵𝑏𝐵)) → (((𝑥𝐵 ↦ (𝑋 · 𝑥))‘𝑎)(+g𝑅)((𝑥𝐵 ↦ (𝑋 · 𝑥))‘𝑏)) = ((𝑋 · 𝑎)(+g𝑅)(𝑋 · 𝑏)))
3513, 23, 343eqtr4d 2220 . . . 4 (((𝑅 ∈ SRing ∧ 𝑋𝐵) ∧ (𝑎𝐵𝑏𝐵)) → ((𝑥𝐵 ↦ (𝑋 · 𝑥))‘(𝑎(+g𝑅)𝑏)) = (((𝑥𝐵 ↦ (𝑋 · 𝑥))‘𝑎)(+g𝑅)((𝑥𝐵 ↦ (𝑋 · 𝑥))‘𝑏)))
3635ralrimivva 2559 . . 3 ((𝑅 ∈ SRing ∧ 𝑋𝐵) → ∀𝑎𝐵𝑏𝐵 ((𝑥𝐵 ↦ (𝑋 · 𝑥))‘(𝑎(+g𝑅)𝑏)) = (((𝑥𝐵 ↦ (𝑋 · 𝑥))‘𝑎)(+g𝑅)((𝑥𝐵 ↦ (𝑋 · 𝑥))‘𝑏)))
37 oveq2 5877 . . . . 5 (𝑥 = (0g𝑅) → (𝑋 · 𝑥) = (𝑋 · (0g𝑅)))
38 eqid 2177 . . . . . . 7 (0g𝑅) = (0g𝑅)
394, 38srg0cl 12983 . . . . . 6 (𝑅 ∈ SRing → (0g𝑅) ∈ 𝐵)
4039adantr 276 . . . . 5 ((𝑅 ∈ SRing ∧ 𝑋𝐵) → (0g𝑅) ∈ 𝐵)
414, 5srgcl 12976 . . . . . 6 ((𝑅 ∈ SRing ∧ 𝑋𝐵 ∧ (0g𝑅) ∈ 𝐵) → (𝑋 · (0g𝑅)) ∈ 𝐵)
4240, 41mpd3an3 1338 . . . . 5 ((𝑅 ∈ SRing ∧ 𝑋𝐵) → (𝑋 · (0g𝑅)) ∈ 𝐵)
4314, 37, 40, 42fvmptd3 5605 . . . 4 ((𝑅 ∈ SRing ∧ 𝑋𝐵) → ((𝑥𝐵 ↦ (𝑋 · 𝑥))‘(0g𝑅)) = (𝑋 · (0g𝑅)))
444, 5, 38srgrz 12990 . . . 4 ((𝑅 ∈ SRing ∧ 𝑋𝐵) → (𝑋 · (0g𝑅)) = (0g𝑅))
4543, 44eqtrd 2210 . . 3 ((𝑅 ∈ SRing ∧ 𝑋𝐵) → ((𝑥𝐵 ↦ (𝑋 · 𝑥))‘(0g𝑅)) = (0g𝑅))
468, 36, 453jca 1177 . 2 ((𝑅 ∈ SRing ∧ 𝑋𝐵) → ((𝑥𝐵 ↦ (𝑋 · 𝑥)):𝐵𝐵 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑥𝐵 ↦ (𝑋 · 𝑥))‘(𝑎(+g𝑅)𝑏)) = (((𝑥𝐵 ↦ (𝑋 · 𝑥))‘𝑎)(+g𝑅)((𝑥𝐵 ↦ (𝑋 · 𝑥))‘𝑏)) ∧ ((𝑥𝐵 ↦ (𝑋 · 𝑥))‘(0g𝑅)) = (0g𝑅)))
474, 4, 10, 10, 38, 38ismhm 12740 . 2 ((𝑥𝐵 ↦ (𝑋 · 𝑥)) ∈ (𝑅 MndHom 𝑅) ↔ ((𝑅 ∈ Mnd ∧ 𝑅 ∈ Mnd) ∧ ((𝑥𝐵 ↦ (𝑋 · 𝑥)):𝐵𝐵 ∧ ∀𝑎𝐵𝑏𝐵 ((𝑥𝐵 ↦ (𝑋 · 𝑥))‘(𝑎(+g𝑅)𝑏)) = (((𝑥𝐵 ↦ (𝑋 · 𝑥))‘𝑎)(+g𝑅)((𝑥𝐵 ↦ (𝑋 · 𝑥))‘𝑏)) ∧ ((𝑥𝐵 ↦ (𝑋 · 𝑥))‘(0g𝑅)) = (0g𝑅))))
483, 46, 47sylanbrc 417 1 ((𝑅 ∈ SRing ∧ 𝑋𝐵) → (𝑥𝐵 ↦ (𝑋 · 𝑥)) ∈ (𝑅 MndHom 𝑅))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978   = wceq 1353  wcel 2148  wral 2455  cmpt 4061  wf 5208  cfv 5212  (class class class)co 5869  Basecbs 12442  +gcplusg 12515  .rcmulr 12516  0gc0g 12650  Mndcmnd 12706   MndHom cmhm 12736  SRingcsrg 12969
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-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7890  ax-resscn 7891  ax-1cn 7892  ax-1re 7893  ax-icn 7894  ax-addcl 7895  ax-addrcl 7896  ax-mulcl 7897  ax-addcom 7899  ax-addass 7901  ax-i2m1 7904  ax-0lt1 7905  ax-0id 7907  ax-rnegex 7908  ax-pre-ltirr 7911  ax-pre-ltadd 7915
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 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-map 6644  df-pnf 7981  df-mnf 7982  df-ltxr 7984  df-inn 8906  df-2 8964  df-3 8965  df-ndx 12445  df-slot 12446  df-base 12448  df-sets 12449  df-plusg 12528  df-mulr 12529  df-0g 12652  df-mgm 12664  df-sgrp 12697  df-mnd 12707  df-mhm 12738  df-cmn 12914  df-mgp 12955  df-srg 12970
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator