Theorem cmnpropd 13875

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 13516	. . 3 ⊢ (𝜑 → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))
5	3	oveqrspc2v 6040	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢(+g‘𝐾)𝑣) = (𝑢(+g‘𝐿)𝑣))
6	3	oveqrspc2v 6040	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵)) → (𝑣(+g‘𝐾)𝑢) = (𝑣(+g‘𝐿)𝑢))
7	6	ancom2s 566	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑣(+g‘𝐾)𝑢) = (𝑣(+g‘𝐿)𝑢))
8	5, 7	eqeq12d 2244	. . . . 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 2552	. . . 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 2734	. . . . 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 2744	. . . 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 2734	. . . . 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 2744	. . . 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 2229	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
17		eqid 2229	. . 3 ⊢ (+g‘𝐾) = (+g‘𝐾)
18	16, 17	iscmn 13873	. 2 ⊢ (𝐾 ∈ CMnd ↔ (𝐾 ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)(𝑢(+g‘𝐾)𝑣) = (𝑣(+g‘𝐾)𝑢)))
19		eqid 2229	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
20		eqid 2229	. . 3 ⊢ (+g‘𝐿) = (+g‘𝐿)
21	19, 20	iscmn 13873	. 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 1395   ∈ wcel 2200  ∀wral 2508  ‘cfv 5324  (class class class)co 6013  Basecbs 13075  +_gcplusg 13153  Mndcmnd 13492  CMndccmn 13864

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-cnex 8116  ax-resscn 8117  ax-1re 8119  ax-addrcl 8122

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-iota 5284  df-fun 5326  df-fn 5327  df-fv 5332  df-ov 6016  df-inn 9137  df-2 9195  df-ndx 13078  df-slot 13079  df-base 13081  df-plusg 13166  df-mgm 13432  df-sgrp 13478  df-mnd 13493  df-cmn 13866

This theorem is referenced by:  ablpropd  13876  crngpropd  14045