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Mirrors > Home > MPE Home > Th. List > srngcl | Structured version Visualization version GIF version |
Description: The involution function in a star ring is closed in the ring. (Contributed by Mario Carneiro, 6-Oct-2015.) |
Ref | Expression |
---|---|
srngcl.i | ⊢ ∗ = (*𝑟‘𝑅) |
srngcl.b | ⊢ 𝐵 = (Base‘𝑅) |
Ref | Expression |
---|---|
srngcl | ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵) → ( ∗ ‘𝑋) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | srngcl.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
2 | srngcl.i | . . . 4 ⊢ ∗ = (*𝑟‘𝑅) | |
3 | eqid 2740 | . . . 4 ⊢ (*rf‘𝑅) = (*rf‘𝑅) | |
4 | 1, 2, 3 | stafval 20119 | . . 3 ⊢ (𝑋 ∈ 𝐵 → ((*rf‘𝑅)‘𝑋) = ( ∗ ‘𝑋)) |
5 | 4 | adantl 482 | . 2 ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵) → ((*rf‘𝑅)‘𝑋) = ( ∗ ‘𝑋)) |
6 | 3, 1 | srngf1o 20125 | . . . 4 ⊢ (𝑅 ∈ *-Ring → (*rf‘𝑅):𝐵–1-1-onto→𝐵) |
7 | f1of 6714 | . . . 4 ⊢ ((*rf‘𝑅):𝐵–1-1-onto→𝐵 → (*rf‘𝑅):𝐵⟶𝐵) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝑅 ∈ *-Ring → (*rf‘𝑅):𝐵⟶𝐵) |
9 | 8 | ffvelrnda 6958 | . 2 ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵) → ((*rf‘𝑅)‘𝑋) ∈ 𝐵) |
10 | 5, 9 | eqeltrrd 2842 | 1 ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵) → ( ∗ ‘𝑋) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ⟶wf 6428 –1-1-onto→wf1o 6431 ‘cfv 6432 Basecbs 16923 *𝑟cstv 16975 *rfcstf 20114 *-Ringcsr 20115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-om 7708 df-2nd 7826 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-er 8490 df-map 8609 df-en 8726 df-dom 8727 df-sdom 8728 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-nn 11985 df-2 12047 df-sets 16876 df-slot 16894 df-ndx 16906 df-base 16924 df-plusg 16986 df-0g 17163 df-mhm 18441 df-ghm 18843 df-mgp 19732 df-ur 19749 df-ring 19796 df-rnghom 19970 df-staf 20116 df-srng 20117 |
This theorem is referenced by: srngnvl 20127 ipassr2 20863 cphcjcl 24358 |
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