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Theorem ringressid 13169
Description: A ring restricted to its base set is a ring. It will usually be the original ring exactly, of course, but to show that needs additional conditions such as those in strressid 12522. (Contributed by Jim Kingdon, 28-Feb-2025.)
Hypothesis
Ref Expression
ringressid.b 𝐵 = (Base‘𝐺)
Assertion
Ref Expression
ringressid (𝐺 ∈ Ring → (𝐺s 𝐵) ∈ Ring)

Proof of Theorem ringressid
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2178 . . 3 (𝐺 ∈ Ring → (𝐺s 𝐵) = (𝐺s 𝐵))
2 ringressid.b . . . 4 𝐵 = (Base‘𝐺)
32a1i 9 . . 3 (𝐺 ∈ Ring → 𝐵 = (Base‘𝐺))
4 id 19 . . 3 (𝐺 ∈ Ring → 𝐺 ∈ Ring)
5 ssidd 3176 . . 3 (𝐺 ∈ Ring → 𝐵𝐵)
61, 3, 4, 5ressbas2d 12520 . 2 (𝐺 ∈ Ring → 𝐵 = (Base‘(𝐺s 𝐵)))
7 eqidd 2178 . . 3 (𝐺 ∈ Ring → (+g𝐺) = (+g𝐺))
8 basfn 12512 . . . . 5 Base Fn V
9 elex 2748 . . . . 5 (𝐺 ∈ Ring → 𝐺 ∈ V)
10 funfvex 5531 . . . . . 6 ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
1110funfni 5315 . . . . 5 ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
128, 9, 11sylancr 414 . . . 4 (𝐺 ∈ Ring → (Base‘𝐺) ∈ V)
132, 12eqeltrid 2264 . . 3 (𝐺 ∈ Ring → 𝐵 ∈ V)
141, 7, 13, 4ressplusgd 12579 . 2 (𝐺 ∈ Ring → (+g𝐺) = (+g‘(𝐺s 𝐵)))
15 eqid 2177 . . . 4 (𝐺s 𝐵) = (𝐺s 𝐵)
16 eqid 2177 . . . 4 (.r𝐺) = (.r𝐺)
1715, 16ressmulrg 12595 . . 3 ((𝐵 ∈ V ∧ 𝐺 ∈ Ring) → (.r𝐺) = (.r‘(𝐺s 𝐵)))
1813, 17mpancom 422 . 2 (𝐺 ∈ Ring → (.r𝐺) = (.r‘(𝐺s 𝐵)))
19 ringgrp 13115 . . 3 (𝐺 ∈ Ring → 𝐺 ∈ Grp)
202grpressid 12863 . . 3 (𝐺 ∈ Grp → (𝐺s 𝐵) ∈ Grp)
2119, 20syl 14 . 2 (𝐺 ∈ Ring → (𝐺s 𝐵) ∈ Grp)
222, 16ringcl 13127 . 2 ((𝐺 ∈ Ring ∧ 𝑥𝐵𝑦𝐵) → (𝑥(.r𝐺)𝑦) ∈ 𝐵)
232, 16ringass 13130 . 2 ((𝐺 ∈ Ring ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥(.r𝐺)𝑦)(.r𝐺)𝑧) = (𝑥(.r𝐺)(𝑦(.r𝐺)𝑧)))
24 eqid 2177 . . 3 (+g𝐺) = (+g𝐺)
252, 24, 16ringdi 13132 . 2 ((𝐺 ∈ Ring ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → (𝑥(.r𝐺)(𝑦(+g𝐺)𝑧)) = ((𝑥(.r𝐺)𝑦)(+g𝐺)(𝑥(.r𝐺)𝑧)))
262, 24, 16ringdir 13133 . 2 ((𝐺 ∈ Ring ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥(+g𝐺)𝑦)(.r𝐺)𝑧) = ((𝑥(.r𝐺)𝑧)(+g𝐺)(𝑦(.r𝐺)𝑧)))
27 eqid 2177 . . 3 (1r𝐺) = (1r𝐺)
282, 27ringidcl 13134 . 2 (𝐺 ∈ Ring → (1r𝐺) ∈ 𝐵)
292, 16, 27ringlidm 13137 . 2 ((𝐺 ∈ Ring ∧ 𝑥𝐵) → ((1r𝐺)(.r𝐺)𝑥) = 𝑥)
302, 16, 27ringridm 13138 . 2 ((𝐺 ∈ Ring ∧ 𝑥𝐵) → (𝑥(.r𝐺)(1r𝐺)) = 𝑥)
316, 14, 18, 21, 22, 23, 25, 26, 28, 29, 30isringd 13151 1 (𝐺 ∈ Ring → (𝐺s 𝐵) ∈ Ring)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  Vcvv 2737   Fn wfn 5210  cfv 5215  (class class class)co 5872  Basecbs 12454  s cress 12455  +gcplusg 12528  .rcmulr 12529  Grpcgrp 12809  1rcur 13073  Ringcrg 13110
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 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-i2m1 7913  ax-0lt1 7914  ax-0id 7916  ax-rnegex 7917  ax-pre-ltirr 7920  ax-pre-lttrn 7922  ax-pre-ltadd 7924
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 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-pnf 7990  df-mnf 7991  df-ltxr 7993  df-inn 8916  df-2 8974  df-3 8975  df-ndx 12457  df-slot 12458  df-base 12460  df-sets 12461  df-iress 12462  df-plusg 12541  df-mulr 12542  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-grp 12812  df-minusg 12813  df-mgp 13062  df-ur 13074  df-ring 13112
This theorem is referenced by:  subrgid  13282
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