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Theorem grpidval 17699
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 6538 . . . . . . . 8 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3 grpidval.b . . . . . . . 8 𝐵 = (Base‘𝐺)
42, 3syl6eqr 2849 . . . . . . 7 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
54eleq2d 2868 . . . . . 6 (𝑔 = 𝐺 → (𝑒 ∈ (Base‘𝑔) ↔ 𝑒𝐵))
6 fveq2 6538 . . . . . . . . . . 11 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
7 grpidval.p . . . . . . . . . . 11 + = (+g𝐺)
86, 7syl6eqr 2849 . . . . . . . . . 10 (𝑔 = 𝐺 → (+g𝑔) = + )
98oveqd 7033 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑒(+g𝑔)𝑥) = (𝑒 + 𝑥))
109eqeq1d 2797 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑒(+g𝑔)𝑥) = 𝑥 ↔ (𝑒 + 𝑥) = 𝑥))
118oveqd 7033 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑥(+g𝑔)𝑒) = (𝑥 + 𝑒))
1211eqeq1d 2797 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑥(+g𝑔)𝑒) = 𝑥 ↔ (𝑥 + 𝑒) = 𝑥))
1310, 12anbi12d 630 . . . . . . 7 (𝑔 = 𝐺 → (((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥) ↔ ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
144, 13raleqbidv 3361 . . . . . 6 (𝑔 = 𝐺 → (∀𝑥 ∈ (Base‘𝑔)((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥) ↔ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
155, 14anbi12d 630 . . . . 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 6210 . . . 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 16544 . . . 4 0g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥))))
18 iotaex 6206 . . . 4 (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))) ∈ V
1916, 17, 18fvmpt 6635 . . 3 (𝐺 ∈ V → (0g𝐺) = (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))))
20 fvprc 6531 . . . 4 𝐺 ∈ V → (0g𝐺) = ∅)
21 euex 2622 . . . . . . 7 (∃!𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) → ∃𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
22 n0i 4219 . . . . . . . . . 10 (𝑒𝐵 → ¬ 𝐵 = ∅)
23 fvprc 6531 . . . . . . . . . . 11 𝐺 ∈ V → (Base‘𝐺) = ∅)
243, 23syl5eq 2843 . . . . . . . . . 10 𝐺 ∈ V → 𝐵 = ∅)
2522, 24nsyl2 143 . . . . . . . . 9 (𝑒𝐵𝐺 ∈ V)
2625adantr 481 . . . . . . . 8 ((𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) → 𝐺 ∈ V)
2726exlimiv 1908 . . . . . . 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 157 . . . . 5 𝐺 ∈ V → ¬ ∃!𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
30 iotanul 6204 . . . . 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 2834 . . 3 𝐺 ∈ V → (0g𝐺) = (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))))
3319, 32pm2.61i 183 . 2 (0g𝐺) = (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
341, 33eqtri 2819 1 0 = (℩𝑒(𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1522  wex 1761  wcel 2081  ∃!weu 2611  wral 3105  Vcvv 3437  c0 4211  cio 6187  cfv 6225  (class class class)co 7016  Basecbs 16312  +gcplusg 16394  0gc0g 16542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-iota 6189  df-fun 6227  df-fv 6233  df-ov 7019  df-0g 16544
This theorem is referenced by:  grpidpropd  17700  0g0  17702  ismgmid  17703  oppgid  18225  dfur2  18944  oppr0  19073  oppr1  19074
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