Mathbox for Jeff Madsen |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > clmgmOLD | Structured version Visualization version GIF version |
Description: Obsolete version of mgmcl 17068 as of 3-Feb-2020. Closure of a magma. (Contributed by FL, 14-Sep-2010.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
clmgmOLD.1 | ⊢ 𝑋 = dom dom 𝐺 |
Ref | Expression |
---|---|
clmgmOLD | ⊢ ((𝐺 ∈ Magma ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clmgmOLD.1 | . . . . 5 ⊢ 𝑋 = dom dom 𝐺 | |
2 | 1 | ismgmOLD 32819 | . . . 4 ⊢ (𝐺 ∈ Magma → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋)) |
3 | fovrn 6702 | . . . . 5 ⊢ ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋) | |
4 | 3 | 3exp 1256 | . . . 4 ⊢ (𝐺:(𝑋 × 𝑋)⟶𝑋 → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (𝐴𝐺𝐵) ∈ 𝑋))) |
5 | 2, 4 | syl6bi 242 | . . 3 ⊢ (𝐺 ∈ Magma → (𝐺 ∈ Magma → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (𝐴𝐺𝐵) ∈ 𝑋)))) |
6 | 5 | pm2.43i 50 | . 2 ⊢ (𝐺 ∈ Magma → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (𝐴𝐺𝐵) ∈ 𝑋))) |
7 | 6 | 3imp 1249 | 1 ⊢ ((𝐺 ∈ Magma ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 × cxp 5036 dom cdm 5038 ⟶wf 5800 (class class class)co 6549 Magmacmagm 32817 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-mgmOLD 32818 |
This theorem is referenced by: exidcl 32845 |
Copyright terms: Public domain | W3C validator |