Theorem ismnddef 12685

Description: The predicate "is a monoid", corresponding 1-to-1 to the definition. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 1-Feb-2020.)

Hypotheses

Ref	Expression
ismnddef.b	⊢ 𝐵 = (Base‘𝐺)
ismnddef.p	⊢ + = (+g‘𝐺)

Assertion

Ref	Expression
ismnddef	⊢ (𝐺 ∈ Mnd ↔ (𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))

Distinct variable groups:   𝐵,𝑎,𝑒   + ,𝑎,𝑒

Allowed substitution hints:   𝐺(𝑒,𝑎)

Proof of Theorem ismnddef

Dummy variables 𝑏 𝑔 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		basfn 12486	. . . 4 ⊢ Base Fn V
2		vex 2738	. . . 4 ⊢ 𝑔 ∈ V
3		funfvex 5524	. . . . 5 ⊢ ((Fun Base ∧ 𝑔 ∈ dom Base) → (Base‘𝑔) ∈ V)
4	3	funfni 5308	. . . 4 ⊢ ((Base Fn V ∧ 𝑔 ∈ V) → (Base‘𝑔) ∈ V)
5	1, 2, 4	mp2an 426	. . 3 ⊢ (Base‘𝑔) ∈ V
6		plusgslid 12526	. . . . 5 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
7	6	slotex 12456	. . . 4 ⊢ (𝑔 ∈ V → (+g‘𝑔) ∈ V)
8	7	elv 2739	. . 3 ⊢ (+g‘𝑔) ∈ V
9		fveq2 5507	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
10		ismnddef.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
11	9, 10	eqtr4di 2226	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
12	11	eqeq2d 2187	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑏 = (Base‘𝑔) ↔ 𝑏 = 𝐵))
13		fveq2 5507	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
14		ismnddef.p	. . . . . . 7 ⊢ + = (+g‘𝐺)
15	13, 14	eqtr4di 2226	. . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
16	15	eqeq2d 2187	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑝 = (+g‘𝑔) ↔ 𝑝 = + ))
17	12, 16	anbi12d 473	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑏 = (Base‘𝑔) ∧ 𝑝 = (+g‘𝑔)) ↔ (𝑏 = 𝐵 ∧ 𝑝 = + )))
18		simpl 109	. . . . 5 ⊢ ((𝑏 = 𝐵 ∧ 𝑝 = + ) → 𝑏 = 𝐵)
19		oveq 5871	. . . . . . . . 9 ⊢ (𝑝 = + → (𝑒𝑝𝑎) = (𝑒 + 𝑎))
20	19	eqeq1d 2184	. . . . . . . 8 ⊢ (𝑝 = + → ((𝑒𝑝𝑎) = 𝑎 ↔ (𝑒 + 𝑎) = 𝑎))
21		oveq 5871	. . . . . . . . 9 ⊢ (𝑝 = + → (𝑎𝑝𝑒) = (𝑎 + 𝑒))
22	21	eqeq1d 2184	. . . . . . . 8 ⊢ (𝑝 = + → ((𝑎𝑝𝑒) = 𝑎 ↔ (𝑎 + 𝑒) = 𝑎))
23	20, 22	anbi12d 473	. . . . . . 7 ⊢ (𝑝 = + → (((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
24	23	adantl 277	. . . . . 6 ⊢ ((𝑏 = 𝐵 ∧ 𝑝 = + ) → (((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
25	18, 24	raleqbidv 2682	. . . . 5 ⊢ ((𝑏 = 𝐵 ∧ 𝑝 = + ) → (∀𝑎 ∈ 𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
26	18, 25	rexeqbidv 2683	. . . 4 ⊢ ((𝑏 = 𝐵 ∧ 𝑝 = + ) → (∃𝑒 ∈ 𝑏 ∀𝑎 ∈ 𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
27	17, 26	syl6bi 163	. . 3 ⊢ (𝑔 = 𝐺 → ((𝑏 = (Base‘𝑔) ∧ 𝑝 = (+g‘𝑔)) → (∃𝑒 ∈ 𝑏 ∀𝑎 ∈ 𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎))))
28	5, 8, 27	sbc2iedv 3033	. 2 ⊢ (𝑔 = 𝐺 → ([(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑝]∃𝑒 ∈ 𝑏 ∀𝑎 ∈ 𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
29		df-mnd 12684	. 2 ⊢ Mnd = {𝑔 ∈ Smgrp ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑝]∃𝑒 ∈ 𝑏 ∀𝑎 ∈ 𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎)}
30	28, 29	elrab2 2894	1 ⊢ (𝐺 ∈ Mnd ↔ (𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2146  ∀wral 2453  ∃wrex 2454  Vcvv 2735  [wsbc 2960   Fn wfn 5203  ‘cfv 5208  (class class class)co 5865  Basecbs 12429  +_gcplusg 12493  Smgrpcsgrp 12673  Mndcmnd 12683

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-cnex 7877  ax-resscn 7878  ax-1re 7880  ax-addrcl 7883

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-sbc 2961  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-iota 5170  df-fun 5210  df-fn 5211  df-fv 5216  df-ov 5868  df-inn 8893  df-2 8951  df-ndx 12432  df-slot 12433  df-base 12435  df-plusg 12506  df-mnd 12684

This theorem is referenced by:  ismnd  12686  sgrpidmndm  12687  mndsgrp  12688  mnd1  12710