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Theorem pwmnd 18221
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 (+g𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥𝑦))
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.
StepHypRef Expression
1 pwmnd.b . . . . . 6 (Base‘𝑀) = 𝒫 𝐴
21eleq2i 2825 . . . . 5 (𝑎 ∈ (Base‘𝑀) ↔ 𝑎 ∈ 𝒫 𝐴)
31eleq2i 2825 . . . . 5 (𝑏 ∈ (Base‘𝑀) ↔ 𝑏 ∈ 𝒫 𝐴)
4 pwuncl 7514 . . . . . . 7 ((𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴) → (𝑎𝑏) ∈ 𝒫 𝐴)
5 pwmnd.p . . . . . . . 8 (+g𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥𝑦))
61, 5pwmndgplus 18219 . . . . . . 7 ((𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴) → (𝑎(+g𝑀)𝑏) = (𝑎𝑏))
71a1i 11 . . . . . . 7 ((𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴) → (Base‘𝑀) = 𝒫 𝐴)
84, 6, 73eltr4d 2849 . . . . . 6 ((𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴) → (𝑎(+g𝑀)𝑏) ∈ (Base‘𝑀))
91eleq2i 2825 . . . . . . . 8 (𝑐 ∈ (Base‘𝑀) ↔ 𝑐 ∈ 𝒫 𝐴)
10 unass 4057 . . . . . . . . . 10 ((𝑎𝑏) ∪ 𝑐) = (𝑎 ∪ (𝑏𝑐))
116adantr 484 . . . . . . . . . . . 12 (((𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴) ∧ 𝑐 ∈ 𝒫 𝐴) → (𝑎(+g𝑀)𝑏) = (𝑎𝑏))
1211oveq1d 7188 . . . . . . . . . . 11 (((𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴) ∧ 𝑐 ∈ 𝒫 𝐴) → ((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝑎𝑏)(+g𝑀)𝑐))
131, 5pwmndgplus 18219 . . . . . . . . . . . 12 (((𝑎𝑏) ∈ 𝒫 𝐴𝑐 ∈ 𝒫 𝐴) → ((𝑎𝑏)(+g𝑀)𝑐) = ((𝑎𝑏) ∪ 𝑐))
144, 13sylan 583 . . . . . . . . . . 11 (((𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴) ∧ 𝑐 ∈ 𝒫 𝐴) → ((𝑎𝑏)(+g𝑀)𝑐) = ((𝑎𝑏) ∪ 𝑐))
1512, 14eqtrd 2774 . . . . . . . . . 10 (((𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴) ∧ 𝑐 ∈ 𝒫 𝐴) → ((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝑎𝑏) ∪ 𝑐))
161, 5pwmndgplus 18219 . . . . . . . . . . . . 13 ((𝑏 ∈ 𝒫 𝐴𝑐 ∈ 𝒫 𝐴) → (𝑏(+g𝑀)𝑐) = (𝑏𝑐))
1716adantll 714 . . . . . . . . . . . 12 (((𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴) ∧ 𝑐 ∈ 𝒫 𝐴) → (𝑏(+g𝑀)𝑐) = (𝑏𝑐))
1817oveq2d 7189 . . . . . . . . . . 11 (((𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴) ∧ 𝑐 ∈ 𝒫 𝐴) → (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝑎(+g𝑀)(𝑏𝑐)))
19 simpll 767 . . . . . . . . . . . . 13 (((𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴) ∧ 𝑐 ∈ 𝒫 𝐴) → 𝑎 ∈ 𝒫 𝐴)
20 pwuncl 7514 . . . . . . . . . . . . . 14 ((𝑏 ∈ 𝒫 𝐴𝑐 ∈ 𝒫 𝐴) → (𝑏𝑐) ∈ 𝒫 𝐴)
2120adantll 714 . . . . . . . . . . . . 13 (((𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴) ∧ 𝑐 ∈ 𝒫 𝐴) → (𝑏𝑐) ∈ 𝒫 𝐴)
2219, 21jca 515 . . . . . . . . . . . 12 (((𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴) ∧ 𝑐 ∈ 𝒫 𝐴) → (𝑎 ∈ 𝒫 𝐴 ∧ (𝑏𝑐) ∈ 𝒫 𝐴))
231, 5pwmndgplus 18219 . . . . . . . . . . . 12 ((𝑎 ∈ 𝒫 𝐴 ∧ (𝑏𝑐) ∈ 𝒫 𝐴) → (𝑎(+g𝑀)(𝑏𝑐)) = (𝑎 ∪ (𝑏𝑐)))
2422, 23syl 17 . . . . . . . . . . 11 (((𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴) ∧ 𝑐 ∈ 𝒫 𝐴) → (𝑎(+g𝑀)(𝑏𝑐)) = (𝑎 ∪ (𝑏𝑐)))
2518, 24eqtrd 2774 . . . . . . . . . 10 (((𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴) ∧ 𝑐 ∈ 𝒫 𝐴) → (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝑎 ∪ (𝑏𝑐)))
2610, 15, 253eqtr4a 2800 . . . . . . . . 9 (((𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴) ∧ 𝑐 ∈ 𝒫 𝐴) → ((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)))
2726ex 416 . . . . . . . 8 ((𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴) → (𝑐 ∈ 𝒫 𝐴 → ((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐))))
289, 27syl5bi 245 . . . . . . 7 ((𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴) → (𝑐 ∈ (Base‘𝑀) → ((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐))))
2928ralrimiv 3096 . . . . . 6 ((𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴) → ∀𝑐 ∈ (Base‘𝑀)((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)))
308, 29jca 515 . . . . 5 ((𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴) → ((𝑎(+g𝑀)𝑏) ∈ (Base‘𝑀) ∧ ∀𝑐 ∈ (Base‘𝑀)((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐))))
312, 3, 30syl2anb 601 . . . 4 ((𝑎 ∈ (Base‘𝑀) ∧ 𝑏 ∈ (Base‘𝑀)) → ((𝑎(+g𝑀)𝑏) ∈ (Base‘𝑀) ∧ ∀𝑐 ∈ (Base‘𝑀)((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐))))
3231rgen2 3116 . . 3 𝑎 ∈ (Base‘𝑀)∀𝑏 ∈ (Base‘𝑀)((𝑎(+g𝑀)𝑏) ∈ (Base‘𝑀) ∧ ∀𝑐 ∈ (Base‘𝑀)((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)))
33 0ex 5176 . . . . 5 ∅ ∈ V
34 eleq1 2821 . . . . . 6 (𝑒 = ∅ → (𝑒 ∈ (Base‘𝑀) ↔ ∅ ∈ (Base‘𝑀)))
35 oveq1 7180 . . . . . . . . 9 (𝑒 = ∅ → (𝑒(+g𝑀)𝑎) = (∅(+g𝑀)𝑎))
3635eqeq1d 2741 . . . . . . . 8 (𝑒 = ∅ → ((𝑒(+g𝑀)𝑎) = 𝑎 ↔ (∅(+g𝑀)𝑎) = 𝑎))
37 oveq2 7181 . . . . . . . . 9 (𝑒 = ∅ → (𝑎(+g𝑀)𝑒) = (𝑎(+g𝑀)∅))
3837eqeq1d 2741 . . . . . . . 8 (𝑒 = ∅ → ((𝑎(+g𝑀)𝑒) = 𝑎 ↔ (𝑎(+g𝑀)∅) = 𝑎))
3936, 38anbi12d 634 . . . . . . 7 (𝑒 = ∅ → (((𝑒(+g𝑀)𝑎) = 𝑎 ∧ (𝑎(+g𝑀)𝑒) = 𝑎) ↔ ((∅(+g𝑀)𝑎) = 𝑎 ∧ (𝑎(+g𝑀)∅) = 𝑎)))
4039ralbidv 3110 . . . . . 6 (𝑒 = ∅ → (∀𝑎 ∈ (Base‘𝑀)((𝑒(+g𝑀)𝑎) = 𝑎 ∧ (𝑎(+g𝑀)𝑒) = 𝑎) ↔ ∀𝑎 ∈ (Base‘𝑀)((∅(+g𝑀)𝑎) = 𝑎 ∧ (𝑎(+g𝑀)∅) = 𝑎)))
4134, 40anbi12d 634 . . . . 5 (𝑒 = ∅ → ((𝑒 ∈ (Base‘𝑀) ∧ ∀𝑎 ∈ (Base‘𝑀)((𝑒(+g𝑀)𝑎) = 𝑎 ∧ (𝑎(+g𝑀)𝑒) = 𝑎)) ↔ (∅ ∈ (Base‘𝑀) ∧ ∀𝑎 ∈ (Base‘𝑀)((∅(+g𝑀)𝑎) = 𝑎 ∧ (𝑎(+g𝑀)∅) = 𝑎))))
42 0elpw 5223 . . . . . . 7 ∅ ∈ 𝒫 𝐴
4342, 1eleqtrri 2833 . . . . . 6 ∅ ∈ (Base‘𝑀)
441, 5pwmndgplus 18219 . . . . . . . . . . 11 ((∅ ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐴) → (∅(+g𝑀)𝑎) = (∅ ∪ 𝑎))
45 0un 4282 . . . . . . . . . . 11 (∅ ∪ 𝑎) = 𝑎
4644, 45eqtrdi 2790 . . . . . . . . . 10 ((∅ ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐴) → (∅(+g𝑀)𝑎) = 𝑎)
471, 5pwmndgplus 18219 . . . . . . . . . . . 12 ((𝑎 ∈ 𝒫 𝐴 ∧ ∅ ∈ 𝒫 𝐴) → (𝑎(+g𝑀)∅) = (𝑎 ∪ ∅))
4847ancoms 462 . . . . . . . . . . 11 ((∅ ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐴) → (𝑎(+g𝑀)∅) = (𝑎 ∪ ∅))
49 un0 4280 . . . . . . . . . . 11 (𝑎 ∪ ∅) = 𝑎
5048, 49eqtrdi 2790 . . . . . . . . . 10 ((∅ ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐴) → (𝑎(+g𝑀)∅) = 𝑎)
5146, 50jca 515 . . . . . . . . 9 ((∅ ∈ 𝒫 𝐴𝑎 ∈ 𝒫 𝐴) → ((∅(+g𝑀)𝑎) = 𝑎 ∧ (𝑎(+g𝑀)∅) = 𝑎))
5242, 51mpan 690 . . . . . . . 8 (𝑎 ∈ 𝒫 𝐴 → ((∅(+g𝑀)𝑎) = 𝑎 ∧ (𝑎(+g𝑀)∅) = 𝑎))
532, 52sylbi 220 . . . . . . 7 (𝑎 ∈ (Base‘𝑀) → ((∅(+g𝑀)𝑎) = 𝑎 ∧ (𝑎(+g𝑀)∅) = 𝑎))
5453rgen 3064 . . . . . 6 𝑎 ∈ (Base‘𝑀)((∅(+g𝑀)𝑎) = 𝑎 ∧ (𝑎(+g𝑀)∅) = 𝑎)
5543, 54pm3.2i 474 . . . . 5 (∅ ∈ (Base‘𝑀) ∧ ∀𝑎 ∈ (Base‘𝑀)((∅(+g𝑀)𝑎) = 𝑎 ∧ (𝑎(+g𝑀)∅) = 𝑎))
5633, 41, 55ceqsexv2d 3447 . . . 4 𝑒(𝑒 ∈ (Base‘𝑀) ∧ ∀𝑎 ∈ (Base‘𝑀)((𝑒(+g𝑀)𝑎) = 𝑎 ∧ (𝑎(+g𝑀)𝑒) = 𝑎))
57 df-rex 3060 . . . 4 (∃𝑒 ∈ (Base‘𝑀)∀𝑎 ∈ (Base‘𝑀)((𝑒(+g𝑀)𝑎) = 𝑎 ∧ (𝑎(+g𝑀)𝑒) = 𝑎) ↔ ∃𝑒(𝑒 ∈ (Base‘𝑀) ∧ ∀𝑎 ∈ (Base‘𝑀)((𝑒(+g𝑀)𝑎) = 𝑎 ∧ (𝑎(+g𝑀)𝑒) = 𝑎)))
5856, 57mpbir 234 . . 3 𝑒 ∈ (Base‘𝑀)∀𝑎 ∈ (Base‘𝑀)((𝑒(+g𝑀)𝑎) = 𝑎 ∧ (𝑎(+g𝑀)𝑒) = 𝑎)
5932, 58pm3.2i 474 . 2 (∀𝑎 ∈ (Base‘𝑀)∀𝑏 ∈ (Base‘𝑀)((𝑎(+g𝑀)𝑏) ∈ (Base‘𝑀) ∧ ∀𝑐 ∈ (Base‘𝑀)((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐))) ∧ ∃𝑒 ∈ (Base‘𝑀)∀𝑎 ∈ (Base‘𝑀)((𝑒(+g𝑀)𝑎) = 𝑎 ∧ (𝑎(+g𝑀)𝑒) = 𝑎))
60 eqid 2739 . . 3 (Base‘𝑀) = (Base‘𝑀)
61 eqid 2739 . . 3 (+g𝑀) = (+g𝑀)
6260, 61ismnd 18033 . 2 (𝑀 ∈ Mnd ↔ (∀𝑎 ∈ (Base‘𝑀)∀𝑏 ∈ (Base‘𝑀)((𝑎(+g𝑀)𝑏) ∈ (Base‘𝑀) ∧ ∀𝑐 ∈ (Base‘𝑀)((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐))) ∧ ∃𝑒 ∈ (Base‘𝑀)∀𝑎 ∈ (Base‘𝑀)((𝑒(+g𝑀)𝑎) = 𝑎 ∧ (𝑎(+g𝑀)𝑒) = 𝑎)))
6359, 62mpbir 234 1 𝑀 ∈ Mnd
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1542  wex 1786  wcel 2114  wral 3054  wrex 3055  cun 3842  c0 4212  𝒫 cpw 4489  cfv 6340  (class class class)co 7173  cmpo 7175  Basecbs 16589  +gcplusg 16671  Mndcmnd 18030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pr 5297  ax-un 7482
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3401  df-sbc 3682  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-br 5032  df-opab 5094  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-iota 6298  df-fun 6342  df-fv 6348  df-ov 7176  df-oprab 7177  df-mpo 7178  df-mgm 17971  df-sgrp 18020  df-mnd 18031
This theorem is referenced by: (None)
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