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Theorem srg1zr 18575
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 pm4.24 676 . 2 (𝐵 = {𝑍} ↔ (𝐵 = {𝑍} ∧ 𝐵 = {𝑍}))
2 srgmnd 18555 . . . . . . 7 (𝑅 ∈ SRing → 𝑅 ∈ Mnd)
323ad2ant1 1102 . . . . . 6 ((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) → 𝑅 ∈ Mnd)
43adantr 480 . . . . 5 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → 𝑅 ∈ Mnd)
5 mndmgm 17347 . . . . 5 (𝑅 ∈ Mnd → 𝑅 ∈ Mgm)
64, 5syl 17 . . . 4 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → 𝑅 ∈ Mgm)
7 simpr 476 . . . 4 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → 𝑍𝐵)
8 simpl2 1085 . . . 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 17301 . . . 4 ((𝑅 ∈ Mgm ∧ 𝑍𝐵+ Fn (𝐵 × 𝐵)) → (𝐵 = {𝑍} ↔ + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
126, 7, 8, 11syl3anc 1366 . . 3 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → (𝐵 = {𝑍} ↔ + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
13 simpl1 1084 . . . . . 6 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → 𝑅 ∈ SRing)
14 eqid 2651 . . . . . . 7 (mulGrp‘𝑅) = (mulGrp‘𝑅)
1514srgmgp 18556 . . . . . 6 (𝑅 ∈ SRing → (mulGrp‘𝑅) ∈ Mnd)
16 mndmgm 17347 . . . . . 6 ((mulGrp‘𝑅) ∈ Mnd → (mulGrp‘𝑅) ∈ Mgm)
1713, 15, 163syl 18 . . . . 5 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → (mulGrp‘𝑅) ∈ Mgm)
18 srg1zr.t . . . . . . . . . 10 = (.r𝑅)
1914, 18mgpplusg 18539 . . . . . . . . 9 = (+g‘(mulGrp‘𝑅))
2019fneq1i 6023 . . . . . . . 8 ( Fn (𝐵 × 𝐵) ↔ (+g‘(mulGrp‘𝑅)) Fn (𝐵 × 𝐵))
2120biimpi 206 . . . . . . 7 ( Fn (𝐵 × 𝐵) → (+g‘(mulGrp‘𝑅)) Fn (𝐵 × 𝐵))
22213ad2ant3 1104 . . . . . 6 ((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) → (+g‘(mulGrp‘𝑅)) Fn (𝐵 × 𝐵))
2322adantr 480 . . . . 5 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → (+g‘(mulGrp‘𝑅)) Fn (𝐵 × 𝐵))
2414, 9mgpbas 18541 . . . . . 6 𝐵 = (Base‘(mulGrp‘𝑅))
25 eqid 2651 . . . . . 6 (+g‘(mulGrp‘𝑅)) = (+g‘(mulGrp‘𝑅))
2624, 25mgmb1mgm1 17301 . . . . 5 (((mulGrp‘𝑅) ∈ Mgm ∧ 𝑍𝐵 ∧ (+g‘(mulGrp‘𝑅)) Fn (𝐵 × 𝐵)) → (𝐵 = {𝑍} ↔ (+g‘(mulGrp‘𝑅)) = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
2717, 7, 23, 26syl3anc 1366 . . . 4 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → (𝐵 = {𝑍} ↔ (+g‘(mulGrp‘𝑅)) = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
2819eqcomi 2660 . . . . . 6 (+g‘(mulGrp‘𝑅)) =
2928a1i 11 . . . . 5 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → (+g‘(mulGrp‘𝑅)) = )
3029eqeq1d 2653 . . . 4 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → ((+g‘(mulGrp‘𝑅)) = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ↔ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
3127, 30bitrd 268 . . 3 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → (𝐵 = {𝑍} ↔ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
3212, 31anbi12d 747 . 2 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → ((𝐵 = {𝑍} ∧ 𝐵 = {𝑍}) ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
331, 32syl5bb 272 1 (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → (𝐵 = {𝑍} ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  {csn 4210  cop 4216   × cxp 5141   Fn wfn 5921  cfv 5926  Basecbs 15904  +gcplusg 15988  .rcmulr 15989  Mgmcmgm 17287  Mndcmnd 17341  mulGrpcmgp 18535  SRingcsrg 18551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-plusg 16001  df-plusf 17288  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-cmn 18241  df-mgp 18536  df-srg 18552
This theorem is referenced by:  srgen1zr  18576  ring1zr  19323
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