MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpidval Structured version   Visualization version   GIF version

Theorem grpidval 17192
Description: The value of the identity element of a group. (Contributed by NM, 20-Aug-2011.) (Revised by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
grpidval.b 𝐵 = (Base‘𝐺)
grpidval.p + = (+g𝐺)
grpidval.o 0 = (0g𝐺)
Assertion
Ref Expression
grpidval 0 = (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
Distinct variable groups:   𝑥,𝑒,𝐵   𝑒,𝐺,𝑥
Allowed substitution hints:   + (𝑥,𝑒)   0 (𝑥,𝑒)

Proof of Theorem grpidval
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 grpidval.o . 2 0 = (0g𝐺)
2 fveq2 6153 . . . . . . . 8 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3 grpidval.b . . . . . . . 8 𝐵 = (Base‘𝐺)
42, 3syl6eqr 2673 . . . . . . 7 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
54eleq2d 2684 . . . . . 6 (𝑔 = 𝐺 → (𝑒 ∈ (Base‘𝑔) ↔ 𝑒𝐵))
6 fveq2 6153 . . . . . . . . . . 11 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
7 grpidval.p . . . . . . . . . . 11 + = (+g𝐺)
86, 7syl6eqr 2673 . . . . . . . . . 10 (𝑔 = 𝐺 → (+g𝑔) = + )
98oveqd 6627 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑒(+g𝑔)𝑥) = (𝑒 + 𝑥))
109eqeq1d 2623 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑒(+g𝑔)𝑥) = 𝑥 ↔ (𝑒 + 𝑥) = 𝑥))
118oveqd 6627 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑥(+g𝑔)𝑒) = (𝑥 + 𝑒))
1211eqeq1d 2623 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑥(+g𝑔)𝑒) = 𝑥 ↔ (𝑥 + 𝑒) = 𝑥))
1310, 12anbi12d 746 . . . . . . 7 (𝑔 = 𝐺 → (((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥) ↔ ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
144, 13raleqbidv 3144 . . . . . 6 (𝑔 = 𝐺 → (∀𝑥 ∈ (Base‘𝑔)((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥) ↔ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
155, 14anbi12d 746 . . . . 5 (𝑔 = 𝐺 → ((𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥)) ↔ (𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))))
1615iotabidv 5836 . . . 4 (𝑔 = 𝐺 → (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥))) = (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))))
17 df-0g 16034 . . . 4 0g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥))))
18 iotaex 5832 . . . 4 (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))) ∈ V
1916, 17, 18fvmpt 6244 . . 3 (𝐺 ∈ V → (0g𝐺) = (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))))
20 fvprc 6147 . . . 4 𝐺 ∈ V → (0g𝐺) = ∅)
21 euex 2493 . . . . . . 7 (∃!𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) → ∃𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
22 n0i 3901 . . . . . . . . . 10 (𝑒𝐵 → ¬ 𝐵 = ∅)
23 fvprc 6147 . . . . . . . . . . 11 𝐺 ∈ V → (Base‘𝐺) = ∅)
243, 23syl5eq 2667 . . . . . . . . . 10 𝐺 ∈ V → 𝐵 = ∅)
2522, 24nsyl2 142 . . . . . . . . 9 (𝑒𝐵𝐺 ∈ V)
2625adantr 481 . . . . . . . 8 ((𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) → 𝐺 ∈ V)
2726exlimiv 1855 . . . . . . 7 (∃𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) → 𝐺 ∈ V)
2821, 27syl 17 . . . . . 6 (∃!𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) → 𝐺 ∈ V)
2928con3i 150 . . . . 5 𝐺 ∈ V → ¬ ∃!𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
30 iotanul 5830 . . . . 5 (¬ ∃!𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) → (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))) = ∅)
3129, 30syl 17 . . . 4 𝐺 ∈ V → (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))) = ∅)
3220, 31eqtr4d 2658 . . 3 𝐺 ∈ V → (0g𝐺) = (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))))
3319, 32pm2.61i 176 . 2 (0g𝐺) = (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
341, 33eqtri 2643 1 0 = (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384   = wceq 1480  wex 1701  wcel 1987  ∃!weu 2469  wral 2907  Vcvv 3189  c0 3896  cio 5813  cfv 5852  (class class class)co 6610  Basecbs 15792  +gcplusg 15873  0gc0g 16032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5815  df-fun 5854  df-fv 5860  df-ov 6613  df-0g 16034
This theorem is referenced by:  grpidpropd  17193  0g0  17195  ismgmid  17196  oppgid  17718  dfur2  18436  oppr0  18565  oppr1  18566
  Copyright terms: Public domain W3C validator