Theorem pwmnd 18963

Description: The power set of a class 𝐴 is a monoid under union. (Contributed by AV, 27-Feb-2024.)

Hypotheses

Ref	Expression
pwmnd.b	⊢ (Base‘𝑀) = 𝒫 𝐴
pwmnd.p	⊢ (+g‘𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥 ∪ 𝑦))

Assertion

Ref	Expression
pwmnd	⊢ 𝑀 ∈ Mnd

Distinct variable group:   𝑥,𝐴,𝑦

Allowed substitution hints:   𝑀(𝑥,𝑦)

Proof of Theorem pwmnd

Dummy variables 𝑎 𝑏 𝑐 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwmnd.b	. . . . . 6 ⊢ (Base‘𝑀) = 𝒫 𝐴
2	1	eleq2i 2831	. . . . 5 ⊢ (𝑎 ∈ (Base‘𝑀) ↔ 𝑎 ∈ 𝒫 𝐴)
3	1	eleq2i 2831	. . . . 5 ⊢ (𝑏 ∈ (Base‘𝑀) ↔ 𝑏 ∈ 𝒫 𝐴)
4		pwuncl 7789	. . . . . . 7 ⊢ ((𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴) → (𝑎 ∪ 𝑏) ∈ 𝒫 𝐴)
5		pwmnd.p	. . . . . . . 8 ⊢ (+g‘𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥 ∪ 𝑦))
6	1, 5	pwmndgplus 18961	. . . . . . 7 ⊢ ((𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴) → (𝑎(+g‘𝑀)𝑏) = (𝑎 ∪ 𝑏))
7	1	a1i 11	. . . . . . 7 ⊢ ((𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴) → (Base‘𝑀) = 𝒫 𝐴)
8	4, 6, 7	3eltr4d 2854	. . . . . 6 ⊢ ((𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴) → (𝑎(+g‘𝑀)𝑏) ∈ (Base‘𝑀))
9	1	eleq2i 2831	. . . . . . . 8 ⊢ (𝑐 ∈ (Base‘𝑀) ↔ 𝑐 ∈ 𝒫 𝐴)
10		unass 4182	. . . . . . . . . 10 ⊢ ((𝑎 ∪ 𝑏) ∪ 𝑐) = (𝑎 ∪ (𝑏 ∪ 𝑐))
11	6	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴) ∧ 𝑐 ∈ 𝒫 𝐴) → (𝑎(+g‘𝑀)𝑏) = (𝑎 ∪ 𝑏))
12	11	oveq1d 7446	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴) ∧ 𝑐 ∈ 𝒫 𝐴) → ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = ((𝑎 ∪ 𝑏)(+g‘𝑀)𝑐))
13	1, 5	pwmndgplus 18961	. . . . . . . . . . . 12 ⊢ (((𝑎 ∪ 𝑏) ∈ 𝒫 𝐴 ∧ 𝑐 ∈ 𝒫 𝐴) → ((𝑎 ∪ 𝑏)(+g‘𝑀)𝑐) = ((𝑎 ∪ 𝑏) ∪ 𝑐))
14	4, 13	sylan 580	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴) ∧ 𝑐 ∈ 𝒫 𝐴) → ((𝑎 ∪ 𝑏)(+g‘𝑀)𝑐) = ((𝑎 ∪ 𝑏) ∪ 𝑐))
15	12, 14	eqtrd 2775	. . . . . . . . . 10 ⊢ (((𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴) ∧ 𝑐 ∈ 𝒫 𝐴) → ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = ((𝑎 ∪ 𝑏) ∪ 𝑐))
16	1, 5	pwmndgplus 18961	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ 𝒫 𝐴 ∧ 𝑐 ∈ 𝒫 𝐴) → (𝑏(+g‘𝑀)𝑐) = (𝑏 ∪ 𝑐))
17	16	adantll 714	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴) ∧ 𝑐 ∈ 𝒫 𝐴) → (𝑏(+g‘𝑀)𝑐) = (𝑏 ∪ 𝑐))
18	17	oveq2d 7447	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴) ∧ 𝑐 ∈ 𝒫 𝐴) → (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) = (𝑎(+g‘𝑀)(𝑏 ∪ 𝑐)))
19		simpll 767	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴) ∧ 𝑐 ∈ 𝒫 𝐴) → 𝑎 ∈ 𝒫 𝐴)
20		pwuncl 7789	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ 𝒫 𝐴 ∧ 𝑐 ∈ 𝒫 𝐴) → (𝑏 ∪ 𝑐) ∈ 𝒫 𝐴)
21	20	adantll 714	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴) ∧ 𝑐 ∈ 𝒫 𝐴) → (𝑏 ∪ 𝑐) ∈ 𝒫 𝐴)
22	19, 21	jca 511	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴) ∧ 𝑐 ∈ 𝒫 𝐴) → (𝑎 ∈ 𝒫 𝐴 ∧ (𝑏 ∪ 𝑐) ∈ 𝒫 𝐴))
23	1, 5	pwmndgplus 18961	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝒫 𝐴 ∧ (𝑏 ∪ 𝑐) ∈ 𝒫 𝐴) → (𝑎(+g‘𝑀)(𝑏 ∪ 𝑐)) = (𝑎 ∪ (𝑏 ∪ 𝑐)))
24	22, 23	syl 17	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴) ∧ 𝑐 ∈ 𝒫 𝐴) → (𝑎(+g‘𝑀)(𝑏 ∪ 𝑐)) = (𝑎 ∪ (𝑏 ∪ 𝑐)))
25	18, 24	eqtrd 2775	. . . . . . . . . 10 ⊢ (((𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴) ∧ 𝑐 ∈ 𝒫 𝐴) → (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) = (𝑎 ∪ (𝑏 ∪ 𝑐)))
26	10, 15, 25	3eqtr4a 2801	. . . . . . . . 9 ⊢ (((𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴) ∧ 𝑐 ∈ 𝒫 𝐴) → ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)))
27	26	ex 412	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴) → (𝑐 ∈ 𝒫 𝐴 → ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐))))
28	9, 27	biimtrid 242	. . . . . . 7 ⊢ ((𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴) → (𝑐 ∈ (Base‘𝑀) → ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐))))
29	28	ralrimiv 3143	. . . . . 6 ⊢ ((𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴) → ∀𝑐 ∈ (Base‘𝑀)((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)))
30	8, 29	jca 511	. . . . 5 ⊢ ((𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴) → ((𝑎(+g‘𝑀)𝑏) ∈ (Base‘𝑀) ∧ ∀𝑐 ∈ (Base‘𝑀)((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐))))
31	2, 3, 30	syl2anb 598	. . . 4 ⊢ ((𝑎 ∈ (Base‘𝑀) ∧ 𝑏 ∈ (Base‘𝑀)) → ((𝑎(+g‘𝑀)𝑏) ∈ (Base‘𝑀) ∧ ∀𝑐 ∈ (Base‘𝑀)((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐))))
32	31	rgen2 3197	. . 3 ⊢ ∀𝑎 ∈ (Base‘𝑀)∀𝑏 ∈ (Base‘𝑀)((𝑎(+g‘𝑀)𝑏) ∈ (Base‘𝑀) ∧ ∀𝑐 ∈ (Base‘𝑀)((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)))
33		0ex 5313	. . . . 5 ⊢ ∅ ∈ V
34		eleq1 2827	. . . . . 6 ⊢ (𝑒 = ∅ → (𝑒 ∈ (Base‘𝑀) ↔ ∅ ∈ (Base‘𝑀)))
35		oveq1 7438	. . . . . . . . 9 ⊢ (𝑒 = ∅ → (𝑒(+g‘𝑀)𝑎) = (∅(+g‘𝑀)𝑎))
36	35	eqeq1d 2737	. . . . . . . 8 ⊢ (𝑒 = ∅ → ((𝑒(+g‘𝑀)𝑎) = 𝑎 ↔ (∅(+g‘𝑀)𝑎) = 𝑎))
37		oveq2 7439	. . . . . . . . 9 ⊢ (𝑒 = ∅ → (𝑎(+g‘𝑀)𝑒) = (𝑎(+g‘𝑀)∅))
38	37	eqeq1d 2737	. . . . . . . 8 ⊢ (𝑒 = ∅ → ((𝑎(+g‘𝑀)𝑒) = 𝑎 ↔ (𝑎(+g‘𝑀)∅) = 𝑎))
39	36, 38	anbi12d 632	. . . . . . 7 ⊢ (𝑒 = ∅ → (((𝑒(+g‘𝑀)𝑎) = 𝑎 ∧ (𝑎(+g‘𝑀)𝑒) = 𝑎) ↔ ((∅(+g‘𝑀)𝑎) = 𝑎 ∧ (𝑎(+g‘𝑀)∅) = 𝑎)))
40	39	ralbidv 3176	. . . . . 6 ⊢ (𝑒 = ∅ → (∀𝑎 ∈ (Base‘𝑀)((𝑒(+g‘𝑀)𝑎) = 𝑎 ∧ (𝑎(+g‘𝑀)𝑒) = 𝑎) ↔ ∀𝑎 ∈ (Base‘𝑀)((∅(+g‘𝑀)𝑎) = 𝑎 ∧ (𝑎(+g‘𝑀)∅) = 𝑎)))
41	34, 40	anbi12d 632	. . . . 5 ⊢ (𝑒 = ∅ → ((𝑒 ∈ (Base‘𝑀) ∧ ∀𝑎 ∈ (Base‘𝑀)((𝑒(+g‘𝑀)𝑎) = 𝑎 ∧ (𝑎(+g‘𝑀)𝑒) = 𝑎)) ↔ (∅ ∈ (Base‘𝑀) ∧ ∀𝑎 ∈ (Base‘𝑀)((∅(+g‘𝑀)𝑎) = 𝑎 ∧ (𝑎(+g‘𝑀)∅) = 𝑎))))
42		0elpw 5362	. . . . . . 7 ⊢ ∅ ∈ 𝒫 𝐴
43	42, 1	eleqtrri 2838	. . . . . 6 ⊢ ∅ ∈ (Base‘𝑀)
44	1, 5	pwmndgplus 18961	. . . . . . . . . . 11 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐴) → (∅(+g‘𝑀)𝑎) = (∅ ∪ 𝑎))
45		0un 4402	. . . . . . . . . . 11 ⊢ (∅ ∪ 𝑎) = 𝑎
46	44, 45	eqtrdi 2791	. . . . . . . . . 10 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐴) → (∅(+g‘𝑀)𝑎) = 𝑎)
47	1, 5	pwmndgplus 18961	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝒫 𝐴 ∧ ∅ ∈ 𝒫 𝐴) → (𝑎(+g‘𝑀)∅) = (𝑎 ∪ ∅))
48	47	ancoms 458	. . . . . . . . . . 11 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐴) → (𝑎(+g‘𝑀)∅) = (𝑎 ∪ ∅))
49		un0 4400	. . . . . . . . . . 11 ⊢ (𝑎 ∪ ∅) = 𝑎
50	48, 49	eqtrdi 2791	. . . . . . . . . 10 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐴) → (𝑎(+g‘𝑀)∅) = 𝑎)
51	46, 50	jca 511	. . . . . . . . 9 ⊢ ((∅ ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐴) → ((∅(+g‘𝑀)𝑎) = 𝑎 ∧ (𝑎(+g‘𝑀)∅) = 𝑎))
52	42, 51	mpan 690	. . . . . . . 8 ⊢ (𝑎 ∈ 𝒫 𝐴 → ((∅(+g‘𝑀)𝑎) = 𝑎 ∧ (𝑎(+g‘𝑀)∅) = 𝑎))
53	2, 52	sylbi 217	. . . . . . 7 ⊢ (𝑎 ∈ (Base‘𝑀) → ((∅(+g‘𝑀)𝑎) = 𝑎 ∧ (𝑎(+g‘𝑀)∅) = 𝑎))
54	53	rgen 3061	. . . . . 6 ⊢ ∀𝑎 ∈ (Base‘𝑀)((∅(+g‘𝑀)𝑎) = 𝑎 ∧ (𝑎(+g‘𝑀)∅) = 𝑎)
55	43, 54	pm3.2i 470	. . . . 5 ⊢ (∅ ∈ (Base‘𝑀) ∧ ∀𝑎 ∈ (Base‘𝑀)((∅(+g‘𝑀)𝑎) = 𝑎 ∧ (𝑎(+g‘𝑀)∅) = 𝑎))
56	33, 41, 55	ceqsexv2d 3533	. . . 4 ⊢ ∃𝑒(𝑒 ∈ (Base‘𝑀) ∧ ∀𝑎 ∈ (Base‘𝑀)((𝑒(+g‘𝑀)𝑎) = 𝑎 ∧ (𝑎(+g‘𝑀)𝑒) = 𝑎))
57		df-rex 3069	. . . 4 ⊢ (∃𝑒 ∈ (Base‘𝑀)∀𝑎 ∈ (Base‘𝑀)((𝑒(+g‘𝑀)𝑎) = 𝑎 ∧ (𝑎(+g‘𝑀)𝑒) = 𝑎) ↔ ∃𝑒(𝑒 ∈ (Base‘𝑀) ∧ ∀𝑎 ∈ (Base‘𝑀)((𝑒(+g‘𝑀)𝑎) = 𝑎 ∧ (𝑎(+g‘𝑀)𝑒) = 𝑎)))
58	56, 57	mpbir 231	. . 3 ⊢ ∃𝑒 ∈ (Base‘𝑀)∀𝑎 ∈ (Base‘𝑀)((𝑒(+g‘𝑀)𝑎) = 𝑎 ∧ (𝑎(+g‘𝑀)𝑒) = 𝑎)
59	32, 58	pm3.2i 470	. 2 ⊢ (∀𝑎 ∈ (Base‘𝑀)∀𝑏 ∈ (Base‘𝑀)((𝑎(+g‘𝑀)𝑏) ∈ (Base‘𝑀) ∧ ∀𝑐 ∈ (Base‘𝑀)((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐))) ∧ ∃𝑒 ∈ (Base‘𝑀)∀𝑎 ∈ (Base‘𝑀)((𝑒(+g‘𝑀)𝑎) = 𝑎 ∧ (𝑎(+g‘𝑀)𝑒) = 𝑎))
60		eqid 2735	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
61		eqid 2735	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
62	60, 61	ismnd 18763	. 2 ⊢ (𝑀 ∈ Mnd ↔ (∀𝑎 ∈ (Base‘𝑀)∀𝑏 ∈ (Base‘𝑀)((𝑎(+g‘𝑀)𝑏) ∈ (Base‘𝑀) ∧ ∀𝑐 ∈ (Base‘𝑀)((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐))) ∧ ∃𝑒 ∈ (Base‘𝑀)∀𝑎 ∈ (Base‘𝑀)((𝑒(+g‘𝑀)𝑎) = 𝑎 ∧ (𝑎(+g‘𝑀)𝑒) = 𝑎)))
63	59, 62	mpbir 231	1 ⊢ 𝑀 ∈ Mnd

Syntax hints:   ∧ wa 395   = wceq 1537  ∃wex 1776   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068   ∪ cun 3961  ∅c0 4339  𝒫 cpw 4605  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433  Basecbs 17245  +_gcplusg 17298  Mndcmnd 18760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-mgm 18666  df-sgrp 18745  df-mnd 18761

This theorem is referenced by: (None)