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Theorem srg1zr 12983
Description: The only semiring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.)
Hypotheses
Ref Expression
srg1zr.b 𝐵 = (Base‘𝑅)
srg1zr.p + = (+g𝑅)
srg1zr.t = (.r𝑅)
Assertion
Ref Expression
srg1zr (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → (𝐵 = {𝑍} ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))

Proof of Theorem srg1zr
StepHypRef Expression
1 pm4.24 395 . 2 (𝐵 = {𝑍} ↔ (𝐵 = {𝑍} ∧ 𝐵 = {𝑍}))
2 srgmnd 12963 . . . . . . 7 (𝑅 ∈ SRing → 𝑅 ∈ Mnd)
323ad2ant1 1018 . . . . . 6 ((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) → 𝑅 ∈ Mnd)
43adantr 276 . . . . 5 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → 𝑅 ∈ Mnd)
5 mndmgm 12702 . . . . 5 (𝑅 ∈ Mnd → 𝑅 ∈ Mgm)
64, 5syl 14 . . . 4 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → 𝑅 ∈ Mgm)
7 simpr 110 . . . 4 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → 𝑍𝐵)
8 simpl2 1001 . . . 4 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → + Fn (𝐵 × 𝐵))
9 srg1zr.b . . . . 5 𝐵 = (Base‘𝑅)
10 srg1zr.p . . . . 5 + = (+g𝑅)
119, 10mgmb1mgm1 12666 . . . 4 ((𝑅 ∈ Mgm ∧ 𝑍𝐵+ Fn (𝐵 × 𝐵)) → (𝐵 = {𝑍} ↔ + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
126, 7, 8, 11syl3anc 1238 . . 3 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → (𝐵 = {𝑍} ↔ + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
13 eqid 2177 . . . . . . . 8 (mulGrp‘𝑅) = (mulGrp‘𝑅)
1413, 9mgpbasg 12950 . . . . . . 7 (𝑅 ∈ SRing → 𝐵 = (Base‘(mulGrp‘𝑅)))
15143ad2ant1 1018 . . . . . 6 ((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) → 𝐵 = (Base‘(mulGrp‘𝑅)))
1615adantr 276 . . . . 5 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → 𝐵 = (Base‘(mulGrp‘𝑅)))
1716eqeq1d 2186 . . . 4 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → (𝐵 = {𝑍} ↔ (Base‘(mulGrp‘𝑅)) = {𝑍}))
18 simpl1 1000 . . . . . 6 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → 𝑅 ∈ SRing)
1913srgmgp 12964 . . . . . 6 (𝑅 ∈ SRing → (mulGrp‘𝑅) ∈ Mnd)
20 mndmgm 12702 . . . . . 6 ((mulGrp‘𝑅) ∈ Mnd → (mulGrp‘𝑅) ∈ Mgm)
2118, 19, 203syl 17 . . . . 5 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → (mulGrp‘𝑅) ∈ Mgm)
227, 16eleqtrd 2256 . . . . 5 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → 𝑍 ∈ (Base‘(mulGrp‘𝑅)))
23 srg1zr.t . . . . . . . . . . 11 = (.r𝑅)
2413, 23mgpplusgg 12948 . . . . . . . . . 10 (𝑅 ∈ SRing → = (+g‘(mulGrp‘𝑅)))
2524fneq1d 5301 . . . . . . . . 9 (𝑅 ∈ SRing → ( Fn (𝐵 × 𝐵) ↔ (+g‘(mulGrp‘𝑅)) Fn (𝐵 × 𝐵)))
2625biimpa 296 . . . . . . . 8 ((𝑅 ∈ SRing ∧ Fn (𝐵 × 𝐵)) → (+g‘(mulGrp‘𝑅)) Fn (𝐵 × 𝐵))
27263adant2 1016 . . . . . . 7 ((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) → (+g‘(mulGrp‘𝑅)) Fn (𝐵 × 𝐵))
2827adantr 276 . . . . . 6 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → (+g‘(mulGrp‘𝑅)) Fn (𝐵 × 𝐵))
2916sqxpeqd 4648 . . . . . . 7 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → (𝐵 × 𝐵) = ((Base‘(mulGrp‘𝑅)) × (Base‘(mulGrp‘𝑅))))
3029fneq2d 5302 . . . . . 6 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → ((+g‘(mulGrp‘𝑅)) Fn (𝐵 × 𝐵) ↔ (+g‘(mulGrp‘𝑅)) Fn ((Base‘(mulGrp‘𝑅)) × (Base‘(mulGrp‘𝑅)))))
3128, 30mpbid 147 . . . . 5 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → (+g‘(mulGrp‘𝑅)) Fn ((Base‘(mulGrp‘𝑅)) × (Base‘(mulGrp‘𝑅))))
32 eqid 2177 . . . . . 6 (Base‘(mulGrp‘𝑅)) = (Base‘(mulGrp‘𝑅))
33 eqid 2177 . . . . . 6 (+g‘(mulGrp‘𝑅)) = (+g‘(mulGrp‘𝑅))
3432, 33mgmb1mgm1 12666 . . . . 5 (((mulGrp‘𝑅) ∈ Mgm ∧ 𝑍 ∈ (Base‘(mulGrp‘𝑅)) ∧ (+g‘(mulGrp‘𝑅)) Fn ((Base‘(mulGrp‘𝑅)) × (Base‘(mulGrp‘𝑅)))) → ((Base‘(mulGrp‘𝑅)) = {𝑍} ↔ (+g‘(mulGrp‘𝑅)) = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
3521, 22, 31, 34syl3anc 1238 . . . 4 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → ((Base‘(mulGrp‘𝑅)) = {𝑍} ↔ (+g‘(mulGrp‘𝑅)) = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
3624eqcomd 2183 . . . . . 6 (𝑅 ∈ SRing → (+g‘(mulGrp‘𝑅)) = )
3718, 36syl 14 . . . . 5 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → (+g‘(mulGrp‘𝑅)) = )
3837eqeq1d 2186 . . . 4 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → ((+g‘(mulGrp‘𝑅)) = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ↔ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
3917, 35, 383bitrd 214 . . 3 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → (𝐵 = {𝑍} ↔ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
4012, 39anbi12d 473 . 2 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → ((𝐵 = {𝑍} ∧ 𝐵 = {𝑍}) ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
411, 40bitrid 192 1 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → (𝐵 = {𝑍} ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 978   = wceq 1353  wcel 2148  {csn 3591  cop 3594   × cxp 4620   Fn wfn 5206  cfv 5211  Basecbs 12432  +gcplusg 12505  .rcmulr 12506  Mgmcmgm 12652  Mndcmnd 12696  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-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 4115  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-addcom 7889  ax-addass 7891  ax-i2m1 7894  ax-0lt1 7895  ax-0id 7897  ax-rnegex 7898  ax-pre-ltirr 7901  ax-pre-ltadd 7905
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-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 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-pnf 7971  df-mnf 7972  df-ltxr 7974  df-inn 8896  df-2 8954  df-3 8955  df-ndx 12435  df-slot 12436  df-base 12438  df-sets 12439  df-plusg 12518  df-mulr 12519  df-0g 12642  df-plusf 12653  df-mgm 12654  df-sgrp 12687  df-mnd 12697  df-cmn 12904  df-mgp 12945  df-srg 12960
This theorem is referenced by:  srgen1zr  12984
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