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

Theorem grpidval 17865
 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 6645 . . . . . . . 8 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3 grpidval.b . . . . . . . 8 𝐵 = (Base‘𝐺)
42, 3eqtr4di 2851 . . . . . . 7 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
54eleq2d 2875 . . . . . 6 (𝑔 = 𝐺 → (𝑒 ∈ (Base‘𝑔) ↔ 𝑒𝐵))
6 fveq2 6645 . . . . . . . . . . 11 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
7 grpidval.p . . . . . . . . . . 11 + = (+g𝐺)
86, 7eqtr4di 2851 . . . . . . . . . 10 (𝑔 = 𝐺 → (+g𝑔) = + )
98oveqd 7152 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑒(+g𝑔)𝑥) = (𝑒 + 𝑥))
109eqeq1d 2800 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑒(+g𝑔)𝑥) = 𝑥 ↔ (𝑒 + 𝑥) = 𝑥))
118oveqd 7152 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑥(+g𝑔)𝑒) = (𝑥 + 𝑒))
1211eqeq1d 2800 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑥(+g𝑔)𝑒) = 𝑥 ↔ (𝑥 + 𝑒) = 𝑥))
1310, 12anbi12d 633 . . . . . . 7 (𝑔 = 𝐺 → (((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥) ↔ ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
144, 13raleqbidv 3354 . . . . . 6 (𝑔 = 𝐺 → (∀𝑥 ∈ (Base‘𝑔)((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥) ↔ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
155, 14anbi12d 633 . . . . 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 6308 . . . 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 16709 . . . 4 0g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥))))
18 iotaex 6304 . . . 4 (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))) ∈ V
1916, 17, 18fvmpt 6745 . . 3 (𝐺 ∈ V → (0g𝐺) = (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))))
20 fvprc 6638 . . . 4 𝐺 ∈ V → (0g𝐺) = ∅)
21 euex 2637 . . . . . 6 (∃!𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) → ∃𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
22 n0i 4249 . . . . . . . . 9 (𝑒𝐵 → ¬ 𝐵 = ∅)
23 fvprc 6638 . . . . . . . . . 10 𝐺 ∈ V → (Base‘𝐺) = ∅)
243, 23syl5eq 2845 . . . . . . . . 9 𝐺 ∈ V → 𝐵 = ∅)
2522, 24nsyl2 143 . . . . . . . 8 (𝑒𝐵𝐺 ∈ V)
2625adantr 484 . . . . . . 7 ((𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) → 𝐺 ∈ V)
2726exlimiv 1931 . . . . . 6 (∃𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) → 𝐺 ∈ V)
2821, 27syl 17 . . . . 5 (∃!𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) → 𝐺 ∈ V)
29 iotanul 6302 . . . . 5 (¬ ∃!𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) → (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))) = ∅)
3028, 29nsyl5 162 . . . 4 𝐺 ∈ V → (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))) = ∅)
3120, 30eqtr4d 2836 . . 3 𝐺 ∈ V → (0g𝐺) = (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))))
3219, 31pm2.61i 185 . 2 (0g𝐺) = (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
331, 32eqtri 2821 1 0 = (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃!weu 2628  ∀wral 3106  Vcvv 3441  ∅c0 4243  ℩cio 6281  ‘cfv 6324  (class class class)co 7135  Basecbs 16477  +gcplusg 16559  0gc0g 16707 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-0g 16709 This theorem is referenced by:  grpidpropd  17866  0g0  17868  ismgmid  17869  sgrpidmnd  17910  oppgid  18479  dfur2  19250  oppr0  19382  oppr1  19383
 Copyright terms: Public domain W3C validator