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Mirrors > Home > MPE Home > Th. List > Mathboxes > exidcl | Structured version Visualization version GIF version |
Description: Closure of the binary operation of a magma with identity. (Contributed by Jeff Madsen, 16-Jun-2011.) |
Ref | Expression |
---|---|
exidcl.1 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
exidcl | ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exidcl.1 | . . . . . . . 8 ⊢ 𝑋 = ran 𝐺 | |
2 | rngopidOLD 35291 | . . . . . . . 8 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ran 𝐺 = dom dom 𝐺) | |
3 | 1, 2 | syl5eq 2845 | . . . . . . 7 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝑋 = dom dom 𝐺) |
4 | 3 | eleq2d 2875 | . . . . . 6 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ dom dom 𝐺)) |
5 | 3 | eleq2d 2875 | . . . . . 6 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (𝐵 ∈ 𝑋 ↔ 𝐵 ∈ dom dom 𝐺)) |
6 | 4, 5 | anbi12d 633 | . . . . 5 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ↔ (𝐴 ∈ dom dom 𝐺 ∧ 𝐵 ∈ dom dom 𝐺))) |
7 | 6 | pm5.32i 578 | . . . 4 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ↔ (𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴 ∈ dom dom 𝐺 ∧ 𝐵 ∈ dom dom 𝐺))) |
8 | inss1 4155 | . . . . . . 7 ⊢ (Magma ∩ ExId ) ⊆ Magma | |
9 | 8 | sseli 3911 | . . . . . 6 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺 ∈ Magma) |
10 | eqid 2798 | . . . . . . 7 ⊢ dom dom 𝐺 = dom dom 𝐺 | |
11 | 10 | clmgmOLD 35289 | . . . . . 6 ⊢ ((𝐺 ∈ Magma ∧ 𝐴 ∈ dom dom 𝐺 ∧ 𝐵 ∈ dom dom 𝐺) → (𝐴𝐺𝐵) ∈ dom dom 𝐺) |
12 | 9, 11 | syl3an1 1160 | . . . . 5 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ dom dom 𝐺 ∧ 𝐵 ∈ dom dom 𝐺) → (𝐴𝐺𝐵) ∈ dom dom 𝐺) |
13 | 12 | 3expb 1117 | . . . 4 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴 ∈ dom dom 𝐺 ∧ 𝐵 ∈ dom dom 𝐺)) → (𝐴𝐺𝐵) ∈ dom dom 𝐺) |
14 | 7, 13 | sylbi 220 | . . 3 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐺𝐵) ∈ dom dom 𝐺) |
15 | 14 | 3impb 1112 | . 2 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ dom dom 𝐺) |
16 | 3 | 3ad2ant1 1130 | . 2 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝑋 = dom dom 𝐺) |
17 | 15, 16 | eleqtrrd 2893 | 1 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∩ cin 3880 dom cdm 5519 ran crn 5520 (class class class)co 7135 ExId cexid 35282 Magmacmagm 35286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fo 6330 df-fv 6332 df-ov 7138 df-exid 35283 df-mgmOLD 35287 |
This theorem is referenced by: (None) |
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