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Mirrors > Home > MPE Home > Th. List > srngnvl | Structured version Visualization version GIF version |
Description: The involution function in a star ring is an involution. (Contributed by Mario Carneiro, 6-Oct-2015.) |
Ref | Expression |
---|---|
srngcl.i | ⊢ ∗ = (*𝑟‘𝑅) |
srngcl.b | ⊢ 𝐵 = (Base‘𝑅) |
Ref | Expression |
---|---|
srngnvl | ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵) → ( ∗ ‘( ∗ ‘𝑋)) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | srngcl.i | . . . 4 ⊢ ∗ = (*𝑟‘𝑅) | |
2 | srngcl.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
3 | 1, 2 | srngcl 20742 | . . 3 ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵) → ( ∗ ‘𝑋) ∈ 𝐵) |
4 | eqid 2728 | . . . 4 ⊢ (*rf‘𝑅) = (*rf‘𝑅) | |
5 | 2, 1, 4 | stafval 20735 | . . 3 ⊢ (( ∗ ‘𝑋) ∈ 𝐵 → ((*rf‘𝑅)‘( ∗ ‘𝑋)) = ( ∗ ‘( ∗ ‘𝑋))) |
6 | 3, 5 | syl 17 | . 2 ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵) → ((*rf‘𝑅)‘( ∗ ‘𝑋)) = ( ∗ ‘( ∗ ‘𝑋))) |
7 | 4 | srngcnv 20740 | . . . . 5 ⊢ (𝑅 ∈ *-Ring → (*rf‘𝑅) = ◡(*rf‘𝑅)) |
8 | 7 | adantr 479 | . . . 4 ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵) → (*rf‘𝑅) = ◡(*rf‘𝑅)) |
9 | 8 | fveq1d 6904 | . . 3 ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵) → ((*rf‘𝑅)‘((*rf‘𝑅)‘𝑋)) = (◡(*rf‘𝑅)‘((*rf‘𝑅)‘𝑋))) |
10 | 2, 1, 4 | stafval 20735 | . . . . 5 ⊢ (𝑋 ∈ 𝐵 → ((*rf‘𝑅)‘𝑋) = ( ∗ ‘𝑋)) |
11 | 10 | adantl 480 | . . . 4 ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵) → ((*rf‘𝑅)‘𝑋) = ( ∗ ‘𝑋)) |
12 | 11 | fveq2d 6906 | . . 3 ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵) → ((*rf‘𝑅)‘((*rf‘𝑅)‘𝑋)) = ((*rf‘𝑅)‘( ∗ ‘𝑋))) |
13 | 4, 2 | srngf1o 20741 | . . . 4 ⊢ (𝑅 ∈ *-Ring → (*rf‘𝑅):𝐵–1-1-onto→𝐵) |
14 | f1ocnvfv1 7291 | . . . 4 ⊢ (((*rf‘𝑅):𝐵–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → (◡(*rf‘𝑅)‘((*rf‘𝑅)‘𝑋)) = 𝑋) | |
15 | 13, 14 | sylan 578 | . . 3 ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵) → (◡(*rf‘𝑅)‘((*rf‘𝑅)‘𝑋)) = 𝑋) |
16 | 9, 12, 15 | 3eqtr3d 2776 | . 2 ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵) → ((*rf‘𝑅)‘( ∗ ‘𝑋)) = 𝑋) |
17 | 6, 16 | eqtr3d 2770 | 1 ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵) → ( ∗ ‘( ∗ ‘𝑋)) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ◡ccnv 5681 –1-1-onto→wf1o 6552 ‘cfv 6553 Basecbs 17187 *𝑟cstv 17242 *rfcstf 20730 *-Ringcsr 20731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-plusg 17253 df-0g 17430 df-mhm 18747 df-ghm 19175 df-mgp 20082 df-ur 20129 df-ring 20182 df-rhm 20418 df-staf 20732 df-srng 20733 |
This theorem is referenced by: ipassr2 21586 |
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