srngf1o - Metamath Proof Explorer

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Theorem srngf1o 20849

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 2737	. . . 4 ⊢ (opp_r‘𝑅) = (opp_r‘𝑅)
2		srngcnv.i	. . . 4 ⊢ ∗ = (*_rf‘𝑅)
3	1, 2	srngrhm 20846	. . 3 ⊢ (𝑅 ∈ *-Ring → ∗ ∈ (𝑅 RingHom (opp_r‘𝑅)))
4		srngf1o.b	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
5		eqid 2737	. . . 4 ⊢ (Base‘(opp_r‘𝑅)) = (Base‘(opp_r‘𝑅))
6	4, 5	rhmf 20485	. . 3 ⊢ ( ∗ ∈ (𝑅 RingHom (opp_r‘𝑅)) → ∗ :𝐵⟶(Base‘(opp_r‘𝑅)))
7		ffn 6736	. . 3 ⊢ ( ∗ :𝐵⟶(Base‘(opp_r‘𝑅)) → ∗ Fn 𝐵)
8	3, 6, 7	3syl 18	. 2 ⊢ (𝑅 ∈ *-Ring → ∗ Fn 𝐵)
9	2	srngcnv 20848	. . . 4 ⊢ (𝑅 ∈ *-Ring → ∗ = ^◡ ∗ )
10	9	fneq1d 6661	. . 3 ⊢ (𝑅 ∈ *-Ring → ( ∗ Fn 𝐵 ↔ ^◡ ∗ Fn 𝐵))
11	8, 10	mpbid 232	. 2 ⊢ (𝑅 ∈ *-Ring → ^◡ ∗ Fn 𝐵)
12		dff1o4 6856	. 2 ⊢ ( ∗ :𝐵–1-1-onto→𝐵 ↔ ( ∗ Fn 𝐵 ∧ ^◡ ∗ Fn 𝐵))
13	8, 11, 12	sylanbrc 583	1 ⊢ (𝑅 ∈ *-Ring → ∗ :𝐵–1-1-onto→𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ^◡ccnv 5684 Fn wfn 6556 ⟶wf 6557 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 opp_rcoppr 20333 RingHom crh 20469 *_rfcstf 20838 *-Ringcsr 20839

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-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 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-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 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-pss 3971 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-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 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-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-0g 17486 df-mhm 18796 df-ghm 19231 df-mgp 20138 df-ur 20179 df-ring 20232 df-rhm 20472 df-srng 20841

This theorem is referenced by: srngcl 20850 srngnvl 20851 iporthcom 21653

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