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Theorem cmnpropd 12925
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 (𝜑𝐵 = (Base‘𝐾))
ablpropd.2 (𝜑𝐵 = (Base‘𝐿))
ablpropd.3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)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.
StepHypRef Expression
1 ablpropd.1 . . . 4 (𝜑𝐵 = (Base‘𝐾))
2 ablpropd.2 . . . 4 (𝜑𝐵 = (Base‘𝐿))
3 ablpropd.3 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
41, 2, 3mndpropd 12733 . . 3 (𝜑 → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))
53oveqrspc2v 5896 . . . . . 6 ((𝜑 ∧ (𝑢𝐵𝑣𝐵)) → (𝑢(+g𝐾)𝑣) = (𝑢(+g𝐿)𝑣))
63oveqrspc2v 5896 . . . . . . 7 ((𝜑 ∧ (𝑣𝐵𝑢𝐵)) → (𝑣(+g𝐾)𝑢) = (𝑣(+g𝐿)𝑢))
76ancom2s 566 . . . . . 6 ((𝜑 ∧ (𝑢𝐵𝑣𝐵)) → (𝑣(+g𝐾)𝑢) = (𝑣(+g𝐿)𝑢))
85, 7eqeq12d 2192 . . . . 5 ((𝜑 ∧ (𝑢𝐵𝑣𝐵)) → ((𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢) ↔ (𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢)))
982ralbidva 2499 . . . 4 (𝜑 → (∀𝑢𝐵𝑣𝐵 (𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢) ↔ ∀𝑢𝐵𝑣𝐵 (𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢)))
101raleqdv 2678 . . . . 5 (𝜑 → (∀𝑣𝐵 (𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢) ↔ ∀𝑣 ∈ (Base‘𝐾)(𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢)))
111, 10raleqbidv 2684 . . . 4 (𝜑 → (∀𝑢𝐵𝑣𝐵 (𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢) ↔ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)(𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢)))
122raleqdv 2678 . . . . 5 (𝜑 → (∀𝑣𝐵 (𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢) ↔ ∀𝑣 ∈ (Base‘𝐿)(𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢)))
132, 12raleqbidv 2684 . . . 4 (𝜑 → (∀𝑢𝐵𝑣𝐵 (𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢) ↔ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)(𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢)))
149, 11, 133bitr3d 218 . . 3 (𝜑 → (∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)(𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢) ↔ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)(𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢)))
154, 14anbi12d 473 . 2 (𝜑 → ((𝐾 ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)(𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢)) ↔ (𝐿 ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)(𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢))))
16 eqid 2177 . . 3 (Base‘𝐾) = (Base‘𝐾)
17 eqid 2177 . . 3 (+g𝐾) = (+g𝐾)
1816, 17iscmn 12923 . 2 (𝐾 ∈ CMnd ↔ (𝐾 ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)(𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢)))
19 eqid 2177 . . 3 (Base‘𝐿) = (Base‘𝐿)
20 eqid 2177 . . 3 (+g𝐿) = (+g𝐿)
2119, 20iscmn 12923 . 2 (𝐿 ∈ CMnd ↔ (𝐿 ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)(𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢)))
2215, 18, 213bitr4g 223 1 (𝜑 → (𝐾 ∈ CMnd ↔ 𝐿 ∈ CMnd))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wral 2455  cfv 5212  (class class class)co 5869  Basecbs 12445  +gcplusg 12518  Mndcmnd 12709  CMndccmn 12915
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-cnex 7893  ax-resscn 7894  ax-1re 7896  ax-addrcl 7899
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-iota 5174  df-fun 5214  df-fn 5215  df-fv 5220  df-ov 5872  df-inn 8909  df-2 8967  df-ndx 12448  df-slot 12449  df-base 12451  df-plusg 12531  df-mgm 12667  df-sgrp 12700  df-mnd 12710  df-cmn 12917
This theorem is referenced by:  ablpropd  12926  crngpropd  13044
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