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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 37860 | . . . . . . . 8 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ran 𝐺 = dom dom 𝐺) | |
| 3 | 1, 2 | eqtrid 2789 | . . . . . . 7 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝑋 = dom dom 𝐺) |
| 4 | 3 | eleq2d 2827 | . . . . . 6 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ dom dom 𝐺)) |
| 5 | 3 | eleq2d 2827 | . . . . . 6 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (𝐵 ∈ 𝑋 ↔ 𝐵 ∈ dom dom 𝐺)) |
| 6 | 4, 5 | anbi12d 632 | . . . . 5 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ↔ (𝐴 ∈ dom dom 𝐺 ∧ 𝐵 ∈ dom dom 𝐺))) |
| 7 | 6 | pm5.32i 574 | . . . 4 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ↔ (𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴 ∈ dom dom 𝐺 ∧ 𝐵 ∈ dom dom 𝐺))) |
| 8 | inss1 4237 | . . . . . . 7 ⊢ (Magma ∩ ExId ) ⊆ Magma | |
| 9 | 8 | sseli 3979 | . . . . . 6 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺 ∈ Magma) |
| 10 | eqid 2737 | . . . . . . 7 ⊢ dom dom 𝐺 = dom dom 𝐺 | |
| 11 | 10 | clmgmOLD 37858 | . . . . . 6 ⊢ ((𝐺 ∈ Magma ∧ 𝐴 ∈ dom dom 𝐺 ∧ 𝐵 ∈ dom dom 𝐺) → (𝐴𝐺𝐵) ∈ dom dom 𝐺) |
| 12 | 9, 11 | syl3an1 1164 | . . . . 5 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ dom dom 𝐺 ∧ 𝐵 ∈ dom dom 𝐺) → (𝐴𝐺𝐵) ∈ dom dom 𝐺) |
| 13 | 12 | 3expb 1121 | . . . 4 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴 ∈ dom dom 𝐺 ∧ 𝐵 ∈ dom dom 𝐺)) → (𝐴𝐺𝐵) ∈ dom dom 𝐺) |
| 14 | 7, 13 | sylbi 217 | . . 3 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐺𝐵) ∈ dom dom 𝐺) |
| 15 | 14 | 3impb 1115 | . 2 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ dom dom 𝐺) |
| 16 | 3 | 3ad2ant1 1134 | . 2 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝑋 = dom dom 𝐺) |
| 17 | 15, 16 | eleqtrrd 2844 | 1 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 dom cdm 5685 ran crn 5686 (class class class)co 7431 ExId cexid 37851 Magmacmagm 37855 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-ov 7434 df-exid 37852 df-mgmOLD 37856 |
| This theorem is referenced by: (None) |
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