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Theorem srgrmhm 13000
Description: Right-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
srgrmhm ((𝑅 ∈ SRing ∧ 𝑋𝐵) → (𝑥𝐵 ↦ (𝑥 · 𝑋)) ∈ (𝑅 MndHom 𝑅))
Distinct variable groups:   𝑥,𝐵   𝑥,𝑅   𝑥,𝑋   𝑥, ·

Proof of Theorem srgrmhm
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 . . . . . . 7 𝐵 = (Base‘𝑅)
5 srglmhm.t . . . . . . 7 · = (.r𝑅)
64, 5srgcl 12976 . . . . . 6 ((𝑅 ∈ SRing ∧ 𝑥𝐵𝑋𝐵) → (𝑥 · 𝑋) ∈ 𝐵)
763com23 1209 . . . . 5 ((𝑅 ∈ SRing ∧ 𝑋𝐵𝑥𝐵) → (𝑥 · 𝑋) ∈ 𝐵)
873expa 1203 . . . 4 (((𝑅 ∈ SRing ∧ 𝑋𝐵) ∧ 𝑥𝐵) → (𝑥 · 𝑋) ∈ 𝐵)
98fmpttd 5667 . . 3 ((𝑅 ∈ SRing ∧ 𝑋𝐵) → (𝑥𝐵 ↦ (𝑥 · 𝑋)):𝐵𝐵)
10 3anrot 983 . . . . . . . 8 ((𝑋𝐵𝑎𝐵𝑏𝐵) ↔ (𝑎𝐵𝑏𝐵𝑋𝐵))
11 3anass 982 . . . . . . . 8 ((𝑋𝐵𝑎𝐵𝑏𝐵) ↔ (𝑋𝐵 ∧ (𝑎𝐵𝑏𝐵)))
1210, 11bitr3i 186 . . . . . . 7 ((𝑎𝐵𝑏𝐵𝑋𝐵) ↔ (𝑋𝐵 ∧ (𝑎𝐵𝑏𝐵)))
13 eqid 2177 . . . . . . . 8 (+g𝑅) = (+g𝑅)
144, 13, 5srgdir 12981 . . . . . . 7 ((𝑅 ∈ SRing ∧ (𝑎𝐵𝑏𝐵𝑋𝐵)) → ((𝑎(+g𝑅)𝑏) · 𝑋) = ((𝑎 · 𝑋)(+g𝑅)(𝑏 · 𝑋)))
1512, 14sylan2br 288 . . . . . 6 ((𝑅 ∈ SRing ∧ (𝑋𝐵 ∧ (𝑎𝐵𝑏𝐵))) → ((𝑎(+g𝑅)𝑏) · 𝑋) = ((𝑎 · 𝑋)(+g𝑅)(𝑏 · 𝑋)))
1615anassrs 400 . . . . 5 (((𝑅 ∈ SRing ∧ 𝑋𝐵) ∧ (𝑎𝐵𝑏𝐵)) → ((𝑎(+g𝑅)𝑏) · 𝑋) = ((𝑎 · 𝑋)(+g𝑅)(𝑏 · 𝑋)))
17 eqid 2177 . . . . . 6 (𝑥𝐵 ↦ (𝑥 · 𝑋)) = (𝑥𝐵 ↦ (𝑥 · 𝑋))
18 oveq1 5876 . . . . . 6 (𝑥 = (𝑎(+g𝑅)𝑏) → (𝑥 · 𝑋) = ((𝑎(+g𝑅)𝑏) · 𝑋))
194, 13srgacl 12988 . . . . . . . 8 ((𝑅 ∈ SRing ∧ 𝑎𝐵𝑏𝐵) → (𝑎(+g𝑅)𝑏) ∈ 𝐵)
20193expb 1204 . . . . . . 7 ((𝑅 ∈ SRing ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝑅)𝑏) ∈ 𝐵)
2120adantlr 477 . . . . . 6 (((𝑅 ∈ SRing ∧ 𝑋𝐵) ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝑅)𝑏) ∈ 𝐵)
22 simpll 527 . . . . . . 7 (((𝑅 ∈ SRing ∧ 𝑋𝐵) ∧ (𝑎𝐵𝑏𝐵)) → 𝑅 ∈ SRing)
23 simplr 528 . . . . . . 7 (((𝑅 ∈ SRing ∧ 𝑋𝐵) ∧ (𝑎𝐵𝑏𝐵)) → 𝑋𝐵)
244, 5srgcl 12976 . . . . . . 7 ((𝑅 ∈ SRing ∧ (𝑎(+g𝑅)𝑏) ∈ 𝐵𝑋𝐵) → ((𝑎(+g𝑅)𝑏) · 𝑋) ∈ 𝐵)
2522, 21, 23, 24syl3anc 1238 . . . . . 6 (((𝑅 ∈ SRing ∧ 𝑋𝐵) ∧ (𝑎𝐵𝑏𝐵)) → ((𝑎(+g𝑅)𝑏) · 𝑋) ∈ 𝐵)
2617, 18, 21, 25fvmptd3 5605 . . . . 5 (((𝑅 ∈ SRing ∧ 𝑋𝐵) ∧ (𝑎𝐵𝑏𝐵)) → ((𝑥𝐵 ↦ (𝑥 · 𝑋))‘(𝑎(+g𝑅)𝑏)) = ((𝑎(+g𝑅)𝑏) · 𝑋))
27 oveq1 5876 . . . . . . 7 (𝑥 = 𝑎 → (𝑥 · 𝑋) = (𝑎 · 𝑋))
28 simprl 529 . . . . . . 7 (((𝑅 ∈ SRing ∧ 𝑋𝐵) ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝐵)
294, 5srgcl 12976 . . . . . . . 8 ((𝑅 ∈ SRing ∧ 𝑎𝐵𝑋𝐵) → (𝑎 · 𝑋) ∈ 𝐵)
3022, 28, 23, 29syl3anc 1238 . . . . . . 7 (((𝑅 ∈ SRing ∧ 𝑋𝐵) ∧ (𝑎𝐵𝑏𝐵)) → (𝑎 · 𝑋) ∈ 𝐵)
3117, 27, 28, 30fvmptd3 5605 . . . . . 6 (((𝑅 ∈ SRing ∧ 𝑋𝐵) ∧ (𝑎𝐵𝑏𝐵)) → ((𝑥𝐵 ↦ (𝑥 · 𝑋))‘𝑎) = (𝑎 · 𝑋))
32 oveq1 5876 . . . . . . 7 (𝑥 = 𝑏 → (𝑥 · 𝑋) = (𝑏 · 𝑋))
33 simprr 531 . . . . . . 7 (((𝑅 ∈ SRing ∧ 𝑋𝐵) ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝐵)
344, 5srgcl 12976 . . . . . . . 8 ((𝑅 ∈ SRing ∧ 𝑏𝐵𝑋𝐵) → (𝑏 · 𝑋) ∈ 𝐵)
3522, 33, 23, 34syl3anc 1238 . . . . . . 7 (((𝑅 ∈ SRing ∧ 𝑋𝐵) ∧ (𝑎𝐵𝑏𝐵)) → (𝑏 · 𝑋) ∈ 𝐵)
3617, 32, 33, 35fvmptd3 5605 . . . . . 6 (((𝑅 ∈ SRing ∧ 𝑋𝐵) ∧ (𝑎𝐵𝑏𝐵)) → ((𝑥𝐵 ↦ (𝑥 · 𝑋))‘𝑏) = (𝑏 · 𝑋))
3731, 36oveq12d 5887 . . . . 5 (((𝑅 ∈ SRing ∧ 𝑋𝐵) ∧ (𝑎𝐵𝑏𝐵)) → (((𝑥𝐵 ↦ (𝑥 · 𝑋))‘𝑎)(+g𝑅)((𝑥𝐵 ↦ (𝑥 · 𝑋))‘𝑏)) = ((𝑎 · 𝑋)(+g𝑅)(𝑏 · 𝑋)))
3816, 26, 373eqtr4d 2220 . . . 4 (((𝑅 ∈ SRing ∧ 𝑋𝐵) ∧ (𝑎𝐵𝑏𝐵)) → ((𝑥𝐵 ↦ (𝑥 · 𝑋))‘(𝑎(+g𝑅)𝑏)) = (((𝑥𝐵 ↦ (𝑥 · 𝑋))‘𝑎)(+g𝑅)((𝑥𝐵 ↦ (𝑥 · 𝑋))‘𝑏)))
3938ralrimivva 2559 . . 3 ((𝑅 ∈ SRing ∧ 𝑋𝐵) → ∀𝑎𝐵𝑏𝐵 ((𝑥𝐵 ↦ (𝑥 · 𝑋))‘(𝑎(+g𝑅)𝑏)) = (((𝑥𝐵 ↦ (𝑥 · 𝑋))‘𝑎)(+g𝑅)((𝑥𝐵 ↦ (𝑥 · 𝑋))‘𝑏)))
40 oveq1 5876 . . . . 5 (𝑥 = (0g𝑅) → (𝑥 · 𝑋) = ((0g𝑅) · 𝑋))
41 eqid 2177 . . . . . . 7 (0g𝑅) = (0g𝑅)
424, 41srg0cl 12983 . . . . . 6 (𝑅 ∈ SRing → (0g𝑅) ∈ 𝐵)
4342adantr 276 . . . . 5 ((𝑅 ∈ SRing ∧ 𝑋𝐵) → (0g𝑅) ∈ 𝐵)
44 simpl 109 . . . . . 6 ((𝑅 ∈ SRing ∧ 𝑋𝐵) → 𝑅 ∈ SRing)
45 simpr 110 . . . . . 6 ((𝑅 ∈ SRing ∧ 𝑋𝐵) → 𝑋𝐵)
464, 5srgcl 12976 . . . . . 6 ((𝑅 ∈ SRing ∧ (0g𝑅) ∈ 𝐵𝑋𝐵) → ((0g𝑅) · 𝑋) ∈ 𝐵)
4744, 43, 45, 46syl3anc 1238 . . . . 5 ((𝑅 ∈ SRing ∧ 𝑋𝐵) → ((0g𝑅) · 𝑋) ∈ 𝐵)
4817, 40, 43, 47fvmptd3 5605 . . . 4 ((𝑅 ∈ SRing ∧ 𝑋𝐵) → ((𝑥𝐵 ↦ (𝑥 · 𝑋))‘(0g𝑅)) = ((0g𝑅) · 𝑋))
494, 5, 41srglz 12991 . . . 4 ((𝑅 ∈ SRing ∧ 𝑋𝐵) → ((0g𝑅) · 𝑋) = (0g𝑅))
5048, 49eqtrd 2210 . . 3 ((𝑅 ∈ SRing ∧ 𝑋𝐵) → ((𝑥𝐵 ↦ (𝑥 · 𝑋))‘(0g𝑅)) = (0g𝑅))
519, 39, 503jca 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)))
524, 4, 13, 13, 41, 41ismhm 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))))
533, 51, 52sylanbrc 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)
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