![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > cmnpropd | GIF version |
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.) |
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)) |
Ref | Expression |
---|---|
cmnpropd | ⊢ (𝜑 → (𝐾 ∈ CMnd ↔ 𝐿 ∈ CMnd)) |
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 12733 | . . 3 ⊢ (𝜑 → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd)) |
5 | 3 | oveqrspc2v 5896 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢(+g‘𝐾)𝑣) = (𝑢(+g‘𝐿)𝑣)) |
6 | 3 | oveqrspc2v 5896 | . . . . . . 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 2192 | . . . . 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 2499 | . . . 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 2678 | . . . . 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 2684 | . . . 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 2678 | . . . . 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 2684 | . . . 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 2177 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
17 | eqid 2177 | . . 3 ⊢ (+g‘𝐾) = (+g‘𝐾) | |
18 | 16, 17 | iscmn 12923 | . 2 ⊢ (𝐾 ∈ CMnd ↔ (𝐾 ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)(𝑢(+g‘𝐾)𝑣) = (𝑣(+g‘𝐾)𝑢))) |
19 | eqid 2177 | . . 3 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
20 | eqid 2177 | . . 3 ⊢ (+g‘𝐿) = (+g‘𝐿) | |
21 | 19, 20 | iscmn 12923 | . 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)) |
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 |
Copyright terms: Public domain | W3C validator |