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Mirrors > Home > MPE Home > Th. List > Mathboxes > issmgrpOLD | Structured version Visualization version GIF version |
Description: Obsolete version of issgrp 18473 as of 3-Feb-2020. The predicate "is a semigroup". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
issmgrpOLD.1 | ⊢ 𝑋 = dom dom 𝐺 |
Ref | Expression |
---|---|
issmgrpOLD | ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ SemiGrp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sgrOLD 36124 | . . 3 ⊢ SemiGrp = (Magma ∩ Ass) | |
2 | 1 | eleq2i 2828 | . 2 ⊢ (𝐺 ∈ SemiGrp ↔ 𝐺 ∈ (Magma ∩ Ass)) |
3 | elin 3914 | . . 3 ⊢ (𝐺 ∈ (Magma ∩ Ass) ↔ (𝐺 ∈ Magma ∧ 𝐺 ∈ Ass)) | |
4 | issmgrpOLD.1 | . . . . 5 ⊢ 𝑋 = dom dom 𝐺 | |
5 | 4 | ismgmOLD 36113 | . . . 4 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋)) |
6 | 4 | isass 36109 | . . . 4 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ Ass ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))) |
7 | 5, 6 | anbi12d 631 | . . 3 ⊢ (𝐺 ∈ 𝐴 → ((𝐺 ∈ Magma ∧ 𝐺 ∈ Ass) ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))) |
8 | 3, 7 | bitrid 282 | . 2 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ (Magma ∩ Ass) ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))) |
9 | 2, 8 | bitrid 282 | 1 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ SemiGrp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ∀wral 3061 ∩ cin 3897 × cxp 5618 dom cdm 5620 ⟶wf 6475 (class class class)co 7337 Asscass 36105 Magmacmagm 36111 SemiGrpcsem 36123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pr 5372 ax-un 7650 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-br 5093 df-opab 5155 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-fv 6487 df-ov 7340 df-ass 36106 df-mgmOLD 36112 df-sgrOLD 36124 |
This theorem is referenced by: smgrpmgm 36127 smgrpassOLD 36128 ismndo1 36136 |
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