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

Theorem ismgmid 12685
Description: The identity element of a magma, if it exists, belongs to the base set. (Contributed by Mario Carneiro, 27-Dec-2014.)
Hypotheses
Ref Expression
ismgmid.b 𝐵 = (Base‘𝐺)
ismgmid.o 0 = (0g𝐺)
ismgmid.p + = (+g𝐺)
mgmidcl.e (𝜑 → ∃𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
Assertion
Ref Expression
ismgmid (𝜑 → ((𝑈𝐵 ∧ ∀𝑥𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) ↔ 0 = 𝑈))
Distinct variable groups:   𝑥,𝑒, +   0 ,𝑒,𝑥   𝐵,𝑒,𝑥   𝑒,𝐺,𝑥   𝑈,𝑒,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑒)

Proof of Theorem ismgmid
StepHypRef Expression
1 id 19 . . . 4 (𝑈𝐵𝑈𝐵)
2 mgmidcl.e . . . . 5 (𝜑 → ∃𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
3 mgmidmo 12680 . . . . 5 ∃*𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)
4 reu5 2689 . . . . 5 (∃!𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ (∃𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ∧ ∃*𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
52, 3, 4sylanblrc 416 . . . 4 (𝜑 → ∃!𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
6 oveq1 5876 . . . . . . 7 (𝑒 = 𝑈 → (𝑒 + 𝑥) = (𝑈 + 𝑥))
76eqeq1d 2186 . . . . . 6 (𝑒 = 𝑈 → ((𝑒 + 𝑥) = 𝑥 ↔ (𝑈 + 𝑥) = 𝑥))
87ovanraleqv 5893 . . . . 5 (𝑒 = 𝑈 → (∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ∀𝑥𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)))
98riota2 5847 . . . 4 ((𝑈𝐵 ∧ ∃!𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) → (∀𝑥𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥) ↔ (𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) = 𝑈))
101, 5, 9syl2anr 290 . . 3 ((𝜑𝑈𝐵) → (∀𝑥𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥) ↔ (𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) = 𝑈))
1110pm5.32da 452 . 2 (𝜑 → ((𝑈𝐵 ∧ ∀𝑥𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) ↔ (𝑈𝐵 ∧ (𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) = 𝑈)))
12 riotacl 5839 . . . . 5 (∃!𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) → (𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) ∈ 𝐵)
135, 12syl 14 . . . 4 (𝜑 → (𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) ∈ 𝐵)
14 eleq1 2240 . . . 4 ((𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) = 𝑈 → ((𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) ∈ 𝐵𝑈𝐵))
1513, 14syl5ibcom 155 . . 3 (𝜑 → ((𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) = 𝑈𝑈𝐵))
1615pm4.71rd 394 . 2 (𝜑 → ((𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) = 𝑈 ↔ (𝑈𝐵 ∧ (𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) = 𝑈)))
17 df-riota 5825 . . . 4 (𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) = (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
18 rexm 3522 . . . . . . 7 (∃𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) → ∃𝑒 𝑒𝐵)
192, 18syl 14 . . . . . 6 (𝜑 → ∃𝑒 𝑒𝐵)
20 ismgmid.b . . . . . . . 8 𝐵 = (Base‘𝐺)
2120basmex 12500 . . . . . . 7 (𝑒𝐵𝐺 ∈ V)
2221exlimiv 1598 . . . . . 6 (∃𝑒 𝑒𝐵𝐺 ∈ V)
2319, 22syl 14 . . . . 5 (𝜑𝐺 ∈ V)
24 ismgmid.p . . . . . 6 + = (+g𝐺)
25 ismgmid.o . . . . . 6 0 = (0g𝐺)
2620, 24, 25grpidvalg 12681 . . . . 5 (𝐺 ∈ V → 0 = (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))))
2723, 26syl 14 . . . 4 (𝜑0 = (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))))
2817, 27eqtr4id 2229 . . 3 (𝜑 → (𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) = 0 )
2928eqeq1d 2186 . 2 (𝜑 → ((𝑒𝐵𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) = 𝑈0 = 𝑈))
3011, 16, 293bitr2d 216 1 (𝜑 → ((𝑈𝐵 ∧ ∀𝑥𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) ↔ 0 = 𝑈))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wex 1492  wcel 2148  wral 2455  wrex 2456  ∃!wreu 2457  ∃*wrmo 2458  Vcvv 2737  cio 5172  cfv 5212  crio 5824  (class class class)co 5869  Basecbs 12442  +gcplusg 12515  0gc0g 12650
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-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-cnex 7890  ax-resscn 7891  ax-1re 7893  ax-addrcl 7896
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  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-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  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-iota 5174  df-fun 5214  df-fn 5215  df-fv 5220  df-riota 5825  df-ov 5872  df-inn 8906  df-ndx 12445  df-slot 12446  df-base 12448  df-0g 12652
This theorem is referenced by:  mgmidcl  12686  mgmlrid  12687  ismgmid2  12688  mgmidsssn0  12692  issrgid  12987  isringid  13031
  Copyright terms: Public domain W3C validator