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Theorem ringinvnzdiv 20242

Description: In a unital ring, a left invertible element is not a zero divisor. (Contributed by FL, 18-Apr-2010.) (Revised by Jeff Madsen, 18-Apr-2010.) (Revised by AV, 24-Aug-2021.)

**Hypotheses**
Ref	Expression
ringinvnzdiv.b	⊢ 𝐵 = (Base‘𝑅)
ringinvnzdiv.t	⊢ · = (._r‘𝑅)
ringinvnzdiv.u	⊢ 1 = (1_r‘𝑅)
ringinvnzdiv.z	⊢ 0 = (0_g‘𝑅)
ringinvnzdiv.r	⊢ (𝜑 → 𝑅 ∈ Ring)
ringinvnzdiv.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
ringinvnzdiv.a	⊢ (𝜑 → ∃𝑎 ∈ 𝐵 (𝑎 · 𝑋) = 1 )
ringinvnzdiv.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

**Assertion**
Ref	Expression
ringinvnzdiv	⊢ (𝜑 → ((𝑋 · 𝑌) = 0 ↔ 𝑌 = 0 ))

Distinct variable groups: 𝑋,𝑎 0 ,𝑎 1 ,𝑎 · ,𝑎 𝜑,𝑎 𝑌,𝑎 Allowed substitution hints: 𝐵(𝑎) 𝑅(𝑎)

**Proof of Theorem ringinvnzdiv**
Step	Hyp	Ref	Expression
1		ringinvnzdiv.a	. . 3 ⊢ (𝜑 → ∃𝑎 ∈ 𝐵 (𝑎 · 𝑋) = 1 )
2		ringinvnzdiv.r	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Ring)
3		ringinvnzdiv.y	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
4		ringinvnzdiv.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑅)
5		ringinvnzdiv.t	. . . . . . . . . 10 ⊢ · = (._r‘𝑅)
6		ringinvnzdiv.u	. . . . . . . . . 10 ⊢ 1 = (1_r‘𝑅)
7	4, 5, 6	ringlidm 20210	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵) → ( 1 · 𝑌) = 𝑌)
8	2, 3, 7	syl2anc 582	. . . . . . . 8 ⊢ (𝜑 → ( 1 · 𝑌) = 𝑌)
9	8	eqcomd 2731	. . . . . . 7 ⊢ (𝜑 → 𝑌 = ( 1 · 𝑌))
10	9	ad3antrrr 728	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑎 · 𝑋) = 1 ) ∧ (𝑋 · 𝑌) = 0 ) → 𝑌 = ( 1 · 𝑌))
11		oveq1 7424	. . . . . . . . . 10 ⊢ ( 1 = (𝑎 · 𝑋) → ( 1 · 𝑌) = ((𝑎 · 𝑋) · 𝑌))
12	11	eqcoms 2733	. . . . . . . . 9 ⊢ ((𝑎 · 𝑋) = 1 → ( 1 · 𝑌) = ((𝑎 · 𝑋) · 𝑌))
13	12	adantl 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑎 · 𝑋) = 1 ) → ( 1 · 𝑌) = ((𝑎 · 𝑋) · 𝑌))
14	2	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑅 ∈ Ring)
15		simpr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑎 ∈ 𝐵)
16		ringinvnzdiv.x	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
17	16	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑋 ∈ 𝐵)
18	3	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑌 ∈ 𝐵)
19	15, 17, 18	3jca 1125	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑎 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
20	14, 19	jca 510	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑅 ∈ Ring ∧ (𝑎 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)))
21	20	adantr 479	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑎 · 𝑋) = 1 ) → (𝑅 ∈ Ring ∧ (𝑎 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)))
22	4, 5	ringass 20198	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ (𝑎 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑎 · 𝑋) · 𝑌) = (𝑎 · (𝑋 · 𝑌)))
23	21, 22	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑎 · 𝑋) = 1 ) → ((𝑎 · 𝑋) · 𝑌) = (𝑎 · (𝑋 · 𝑌)))
24	13, 23	eqtrd 2765	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑎 · 𝑋) = 1 ) → ( 1 · 𝑌) = (𝑎 · (𝑋 · 𝑌)))
25	24	adantr 479	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑎 · 𝑋) = 1 ) ∧ (𝑋 · 𝑌) = 0 ) → ( 1 · 𝑌) = (𝑎 · (𝑋 · 𝑌)))
26		oveq2 7425	. . . . . . 7 ⊢ ((𝑋 · 𝑌) = 0 → (𝑎 · (𝑋 · 𝑌)) = (𝑎 · 0 ))
27		ringinvnzdiv.z	. . . . . . . . . 10 ⊢ 0 = (0_g‘𝑅)
28	4, 5, 27	ringrz 20235	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑎 ∈ 𝐵) → (𝑎 · 0 ) = 0 )
29	2, 28	sylan 578	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑎 · 0 ) = 0 )
30	29	adantr 479	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑎 · 𝑋) = 1 ) → (𝑎 · 0 ) = 0 )
31	26, 30	sylan9eqr 2787	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑎 · 𝑋) = 1 ) ∧ (𝑋 · 𝑌) = 0 ) → (𝑎 · (𝑋 · 𝑌)) = 0 )
32	10, 25, 31	3eqtrd 2769	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑎 · 𝑋) = 1 ) ∧ (𝑋 · 𝑌) = 0 ) → 𝑌 = 0 )
33	32	exp31 418	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑎 · 𝑋) = 1 → ((𝑋 · 𝑌) = 0 → 𝑌 = 0 )))
34	33	rexlimdva 3145	. . 3 ⊢ (𝜑 → (∃𝑎 ∈ 𝐵 (𝑎 · 𝑋) = 1 → ((𝑋 · 𝑌) = 0 → 𝑌 = 0 )))
35	1, 34	mpd 15	. 2 ⊢ (𝜑 → ((𝑋 · 𝑌) = 0 → 𝑌 = 0 ))
36		oveq2 7425	. . . 4 ⊢ (𝑌 = 0 → (𝑋 · 𝑌) = (𝑋 · 0 ))
37	4, 5, 27	ringrz 20235	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 )
38	2, 16, 37	syl2anc 582	. . . 4 ⊢ (𝜑 → (𝑋 · 0 ) = 0 )
39	36, 38	sylan9eqr 2787	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → (𝑋 · 𝑌) = 0 )
40	39	ex 411	. 2 ⊢ (𝜑 → (𝑌 = 0 → (𝑋 · 𝑌) = 0 ))
41	35, 40	impbid 211	1 ⊢ (𝜑 → ((𝑋 · 𝑌) = 0 ↔ 𝑌 = 0 ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∃wrex 3060 ‘cfv 6547 (class class class)co 7417 Basecbs 17180 ._rcmulr 17234 0_gc0g 17421 1_rcur 20126 Ringcrg 20178

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 2696 ax-sep 5299 ax-nul 5306 ax-pow 5364 ax-pr 5428 ax-un 7739 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3965 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6499 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-riota 7373 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7870 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-plusg 17246 df-0g 17423 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18898 df-minusg 18899 df-cmn 19742 df-abl 19743 df-mgp 20080 df-rng 20098 df-ur 20127 df-ring 20180

This theorem is referenced by: ringunitnzdiv 20342

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