Theorem cmnpropd 14096

Description: If two structures have the same group components (properties), one is a commutative monoid iff the other one is. (Contributed by Mario Carneiro, 6-Jan-2015.)

Hypotheses

Ref	Expression
ablpropd.1	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
ablpropd.2	⊢ (𝜑 → 𝐵 = (Base‘𝐿))
ablpropd.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))

Assertion

Ref	Expression
cmnpropd	⊢ (𝜑 → (𝐾 ∈ CMnd ↔ 𝐿 ∈ CMnd))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦

Proof of Theorem cmnpropd

Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ablpropd.1	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
2		ablpropd.2	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
3		ablpropd.3	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
4	1, 2, 3	mndpropd 13737	. . 3 ⊢ (𝜑 → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))
5	3	oveqrspc2v 6085	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢(+g‘𝐾)𝑣) = (𝑢(+g‘𝐿)𝑣))
6	3	oveqrspc2v 6085	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵)) → (𝑣(+g‘𝐾)𝑢) = (𝑣(+g‘𝐿)𝑢))
7	6	ancom2s 568	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑣(+g‘𝐾)𝑢) = (𝑣(+g‘𝐿)𝑢))
8	5, 7	eqeq12d 2249	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → ((𝑢(+g‘𝐾)𝑣) = (𝑣(+g‘𝐾)𝑢) ↔ (𝑢(+g‘𝐿)𝑣) = (𝑣(+g‘𝐿)𝑢)))
9	8	2ralbidva 2566	. . . 4 ⊢ (𝜑 → (∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢(+g‘𝐾)𝑣) = (𝑣(+g‘𝐾)𝑢) ↔ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢(+g‘𝐿)𝑣) = (𝑣(+g‘𝐿)𝑢)))
10	1	raleqdv 2749	. . . . 5 ⊢ (𝜑 → (∀𝑣 ∈ 𝐵 (𝑢(+g‘𝐾)𝑣) = (𝑣(+g‘𝐾)𝑢) ↔ ∀𝑣 ∈ (Base‘𝐾)(𝑢(+g‘𝐾)𝑣) = (𝑣(+g‘𝐾)𝑢)))
11	1, 10	raleqbidv 2759	. . . 4 ⊢ (𝜑 → (∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢(+g‘𝐾)𝑣) = (𝑣(+g‘𝐾)𝑢) ↔ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)(𝑢(+g‘𝐾)𝑣) = (𝑣(+g‘𝐾)𝑢)))
12	2	raleqdv 2749	. . . . 5 ⊢ (𝜑 → (∀𝑣 ∈ 𝐵 (𝑢(+g‘𝐿)𝑣) = (𝑣(+g‘𝐿)𝑢) ↔ ∀𝑣 ∈ (Base‘𝐿)(𝑢(+g‘𝐿)𝑣) = (𝑣(+g‘𝐿)𝑢)))
13	2, 12	raleqbidv 2759	. . . 4 ⊢ (𝜑 → (∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢(+g‘𝐿)𝑣) = (𝑣(+g‘𝐿)𝑢) ↔ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)(𝑢(+g‘𝐿)𝑣) = (𝑣(+g‘𝐿)𝑢)))
14	9, 11, 13	3bitr3d 218	. . 3 ⊢ (𝜑 → (∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)(𝑢(+g‘𝐾)𝑣) = (𝑣(+g‘𝐾)𝑢) ↔ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)(𝑢(+g‘𝐿)𝑣) = (𝑣(+g‘𝐿)𝑢)))
15	4, 14	anbi12d 473	. 2 ⊢ (𝜑 → ((𝐾 ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)(𝑢(+g‘𝐾)𝑣) = (𝑣(+g‘𝐾)𝑢)) ↔ (𝐿 ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)(𝑢(+g‘𝐿)𝑣) = (𝑣(+g‘𝐿)𝑢))))
16		eqid 2234	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
17		eqid 2234	. . 3 ⊢ (+g‘𝐾) = (+g‘𝐾)
18	16, 17	iscmn 14094	. 2 ⊢ (𝐾 ∈ CMnd ↔ (𝐾 ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)(𝑢(+g‘𝐾)𝑣) = (𝑣(+g‘𝐾)𝑢)))
19		eqid 2234	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
20		eqid 2234	. . 3 ⊢ (+g‘𝐿) = (+g‘𝐿)
21	19, 20	iscmn 14094	. 2 ⊢ (𝐿 ∈ CMnd ↔ (𝐿 ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)(𝑢(+g‘𝐿)𝑣) = (𝑣(+g‘𝐿)𝑢)))
22	15, 18, 21	3bitr4g 223	1 ⊢ (𝜑 → (𝐾 ∈ CMnd ↔ 𝐿 ∈ CMnd))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205  ∀wral 2522  ‘cfv 5357  (class class class)co 6058  Basecbs 13296  +_gcplusg 13374  Mndcmnd 13713  CMndccmn 14085

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-ov 6061  df-inn 9255  df-2 9313  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-mgm 13653  df-sgrp 13699  df-mnd 13714  df-cmn 14087

This theorem is referenced by:  ablpropd  14097  crngpropd  14267