srngf1o - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > srngf1o		Structured version Visualization version GIF version

Theorem srngf1o 20095

Description: The involution function in a star ring is a bijection. (Contributed by Mario Carneiro, 6-Oct-2015.)

**Hypotheses**
Ref	Expression
srngcnv.i	⊢ ∗ = (*_rf‘𝑅)
srngf1o.b	⊢ 𝐵 = (Base‘𝑅)

**Assertion**
Ref	Expression
srngf1o	⊢ (𝑅 ∈ *-Ring → ∗ :𝐵–1-1-onto→𝐵)

**Proof of Theorem srngf1o**
Step	Hyp	Ref	Expression
1		eqid 2739	. . . 4 ⊢ (opp_r‘𝑅) = (opp_r‘𝑅)
2		srngcnv.i	. . . 4 ⊢ ∗ = (*_rf‘𝑅)
3	1, 2	srngrhm 20092	. . 3 ⊢ (𝑅 ∈ *-Ring → ∗ ∈ (𝑅 RingHom (opp_r‘𝑅)))
4		srngf1o.b	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
5		eqid 2739	. . . 4 ⊢ (Base‘(opp_r‘𝑅)) = (Base‘(opp_r‘𝑅))
6	4, 5	rhmf 19951	. . 3 ⊢ ( ∗ ∈ (𝑅 RingHom (opp_r‘𝑅)) → ∗ :𝐵⟶(Base‘(opp_r‘𝑅)))
7		ffn 6596	. . 3 ⊢ ( ∗ :𝐵⟶(Base‘(opp_r‘𝑅)) → ∗ Fn 𝐵)
8	3, 6, 7	3syl 18	. 2 ⊢ (𝑅 ∈ *-Ring → ∗ Fn 𝐵)
9	2	srngcnv 20094	. . . 4 ⊢ (𝑅 ∈ *-Ring → ∗ = ^◡ ∗ )
10	9	fneq1d 6522	. . 3 ⊢ (𝑅 ∈ *-Ring → ( ∗ Fn 𝐵 ↔ ^◡ ∗ Fn 𝐵))
11	8, 10	mpbid 231	. 2 ⊢ (𝑅 ∈ *-Ring → ^◡ ∗ Fn 𝐵)
12		dff1o4 6720	. 2 ⊢ ( ∗ :𝐵–1-1-onto→𝐵 ↔ ( ∗ Fn 𝐵 ∧ ^◡ ∗ Fn 𝐵))
13	8, 11, 12	sylanbrc 582	1 ⊢ (𝑅 ∈ *-Ring → ∗ :𝐵–1-1-onto→𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2109 ^◡ccnv 5587 Fn wfn 6425 ⟶wf 6426 –1-1-onto→wf1o 6429 ‘cfv 6430 (class class class)co 7268 Basecbs 16893 opp_rcoppr 19842 RingHom crh 19937 *_rfcstf 20084 *-Ringcsr 20085

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-er 8472 df-map 8591 df-en 8708 df-dom 8709 df-sdom 8710 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-2 12019 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-plusg 16956 df-0g 17133 df-mhm 18411 df-ghm 18813 df-mgp 19702 df-ur 19719 df-ring 19766 df-rnghom 19940 df-srng 20087

This theorem is referenced by: srngcl 20096 srngnvl 20097 iporthcom 20821

W3C validator