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| Mirrors > Home > MPE Home > Th. List > unitss | Structured version Visualization version GIF version | ||
| Description: The set of units is contained in the base set. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| Ref | Expression |
|---|---|
| unitcl.1 | ⊢ 𝐵 = (Base‘𝑅) |
| unitcl.2 | ⊢ 𝑈 = (Unit‘𝑅) |
| Ref | Expression |
|---|---|
| unitss | ⊢ 𝑈 ⊆ 𝐵 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | unitcl.1 | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
| 2 | unitcl.2 | . . 3 ⊢ 𝑈 = (Unit‘𝑅) | |
| 3 | 1, 2 | unitcl 20375 | . 2 ⊢ (𝑥 ∈ 𝑈 → 𝑥 ∈ 𝐵) |
| 4 | 3 | ssriv 3987 | 1 ⊢ 𝑈 ⊆ 𝐵 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ⊆ wss 3951 ‘cfv 6561 Basecbs 17247 Unitcui 20355 |
| 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-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 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-pw 4602 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-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-dvdsr 20357 df-unit 20358 |
| This theorem is referenced by: unitgrpbas 20382 unitgrpid 20385 unitsubm 20386 dvrdir 20412 rdivmuldivd 20413 invrpropd 20418 elrhmunit 20510 rhmunitinv 20511 fidomndrng 20774 issubdrg 20781 imadrhmcl 20798 znunithash 21583 dvrcn 24192 nmdvr 24691 nrginvrcnlem 24712 nrginvrcn 24713 dchrelbasd 27283 dchrinvcl 27297 dchrghm 27300 dchr1 27301 dchreq 27302 dchrresb 27303 dchrabs 27304 dchrinv 27305 dchrptlem1 27308 dchrptlem2 27309 dchrpt 27311 dchrsum2 27312 dchrsum 27313 sum2dchr 27318 lgsdchr 27399 rpvmasum2 27556 dvrcan5 33240 isdrng4 33298 dvdsruassoi 33412 lidlunitel 33451 assafld 33688 unitscyglem5 42200 aks5lem7 42201 idomodle 43203 |
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