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Mirrors > Home > ILE Home > Th. List > mndprop | GIF version |
Description: If two structures have the same group components (properties), one is a monoid iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.) |
Ref | Expression |
---|---|
mndprop.b | ⊢ (Base‘𝐾) = (Base‘𝐿) |
mndprop.p | ⊢ (+g‘𝐾) = (+g‘𝐿) |
Ref | Expression |
---|---|
mndprop | ⊢ (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2190 | . . 3 ⊢ (⊤ → (Base‘𝐾) = (Base‘𝐾)) | |
2 | mndprop.b | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐿) | |
3 | 2 | a1i 9 | . . 3 ⊢ (⊤ → (Base‘𝐾) = (Base‘𝐿)) |
4 | mndprop.p | . . . . 5 ⊢ (+g‘𝐾) = (+g‘𝐿) | |
5 | 4 | oveqi 5904 | . . . 4 ⊢ (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦) |
6 | 5 | a1i 9 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) |
7 | 1, 3, 6 | mndpropd 12867 | . 2 ⊢ (⊤ → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd)) |
8 | 7 | mptru 1373 | 1 ⊢ (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 = wceq 1364 ⊤wtru 1365 ∈ wcel 2160 ‘cfv 5231 (class class class)co 5891 Basecbs 12480 +gcplusg 12555 Mndcmnd 12843 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-cnex 7920 ax-resscn 7921 ax-1re 7923 ax-addrcl 7926 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-iota 5193 df-fun 5233 df-fn 5234 df-fv 5239 df-ov 5894 df-inn 8938 df-2 8996 df-ndx 12483 df-slot 12484 df-base 12486 df-plusg 12568 df-mgm 12798 df-sgrp 12831 df-mnd 12844 |
This theorem is referenced by: (None) |
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