Theorem ismnddef 13002

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 12679	. . . 4 ⊢ Base Fn V
2		vex 2763	. . . 4 ⊢ 𝑔 ∈ V
3		funfvex 5572	. . . . 5 ⊢ ((Fun Base ∧ 𝑔 ∈ dom Base) → (Base‘𝑔) ∈ V)
4	3	funfni 5355	. . . 4 ⊢ ((Base Fn V ∧ 𝑔 ∈ V) → (Base‘𝑔) ∈ V)
5	1, 2, 4	mp2an 426	. . 3 ⊢ (Base‘𝑔) ∈ V
6		plusgslid 12733	. . . . 5 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
7	6	slotex 12648	. . . 4 ⊢ (𝑔 ∈ V → (+g‘𝑔) ∈ V)
8	7	elv 2764	. . 3 ⊢ (+g‘𝑔) ∈ V
9		fveq2 5555	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
10		ismnddef.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
11	9, 10	eqtr4di 2244	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
12	11	eqeq2d 2205	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑏 = (Base‘𝑔) ↔ 𝑏 = 𝐵))
13		fveq2 5555	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
14		ismnddef.p	. . . . . . 7 ⊢ + = (+g‘𝐺)
15	13, 14	eqtr4di 2244	. . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
16	15	eqeq2d 2205	. . . . 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 5925	. . . . . . . . 9 ⊢ (𝑝 = + → (𝑒𝑝𝑎) = (𝑒 + 𝑎))
20	19	eqeq1d 2202	. . . . . . . 8 ⊢ (𝑝 = + → ((𝑒𝑝𝑎) = 𝑎 ↔ (𝑒 + 𝑎) = 𝑎))
21		oveq 5925	. . . . . . . . 9 ⊢ (𝑝 = + → (𝑎𝑝𝑒) = (𝑎 + 𝑒))
22	21	eqeq1d 2202	. . . . . . . 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 2706	. . . . 5 ⊢ ((𝑏 = 𝐵 ∧ 𝑝 = + ) → (∀𝑎 ∈ 𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
26	18, 25	rexeqbidv 2707	. . . 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	biimtrdi 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 3059	. 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 13001	. 2 ⊢ Mnd = {𝑔 ∈ Smgrp ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑝]∃𝑒 ∈ 𝑏 ∀𝑎 ∈ 𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎)}
30	28, 29	elrab2 2920	1 ⊢ (𝐺 ∈ Mnd ↔ (𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473  Vcvv 2760  [wsbc 2986   Fn wfn 5250  ‘cfv 5255  (class class class)co 5919  Basecbs 12621  +_gcplusg 12698  Smgrpcsgrp 12987  Mndcmnd 13000

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-ov 5922  df-inn 8985  df-2 9043  df-ndx 12624  df-slot 12625  df-base 12627  df-plusg 12711  df-mnd 13001

This theorem is referenced by:  ismnd  13003  sgrpidmndm  13004  mndsgrp  13005  mnd1  13030