Theorem zerdivemp1x 35799

Description: In a unitary ring a left invertible element is not a zero divisor. See also ringinvnzdiv 19583. (Contributed by Jeff Madsen, 18-Apr-2010.)

Hypotheses

Ref	Expression
zerdivempx.1	⊢ 𝐺 = (1st ‘𝑅)
zerdivempx.2	⊢ 𝐻 = (2nd ‘𝑅)
zerdivempx.3	⊢ 𝑍 = (GId‘𝐺)
zerdivempx.4	⊢ 𝑋 = ran 𝐺
zerdivempx.5	⊢ 𝑈 = (GId‘𝐻)

Assertion

Ref	Expression
zerdivemp1x	⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ ∃𝑎 ∈ 𝑋 (𝑎𝐻𝐴) = 𝑈) → (𝐵 ∈ 𝑋 → ((𝐴𝐻𝐵) = 𝑍 → 𝐵 = 𝑍)))

Distinct variable groups:   𝐴,𝑎   𝐵,𝑎   𝐻,𝑎   𝑅,𝑎   𝑋,𝑎   𝑍,𝑎

Allowed substitution hints:   𝑈(𝑎)   𝐺(𝑎)

Proof of Theorem zerdivemp1x

Step	Hyp	Ref	Expression
1		oveq2 7210	. . . . . . 7 ⊢ ((𝐴𝐻𝐵) = 𝑍 → (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍))
2		simpl1 1193	. . . . . . . . . 10 ⊢ (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ (𝑎𝐻𝐴) = 𝑈 ∧ 𝐴 ∈ 𝑋)) → 𝑅 ∈ RingOps)
3		simpr1 1196	. . . . . . . . . 10 ⊢ (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ (𝑎𝐻𝐴) = 𝑈 ∧ 𝐴 ∈ 𝑋)) → 𝑎 ∈ 𝑋)
4		simpr3 1198	. . . . . . . . . 10 ⊢ (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ (𝑎𝐻𝐴) = 𝑈 ∧ 𝐴 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
5		simpl3 1195	. . . . . . . . . 10 ⊢ (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ (𝑎𝐻𝐴) = 𝑈 ∧ 𝐴 ∈ 𝑋)) → 𝐵 ∈ 𝑋)
6		zerdivempx.1	. . . . . . . . . . 11 ⊢ 𝐺 = (1st ‘𝑅)
7		zerdivempx.2	. . . . . . . . . . 11 ⊢ 𝐻 = (2nd ‘𝑅)
8		zerdivempx.4	. . . . . . . . . . 11 ⊢ 𝑋 = ran 𝐺
9	6, 7, 8	rngoass 35758	. . . . . . . . . 10 ⊢ ((𝑅 ∈ RingOps ∧ (𝑎 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)))
10	2, 3, 4, 5, 9	syl13anc 1374	. . . . . . . . 9 ⊢ (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ (𝑎𝐻𝐴) = 𝑈 ∧ 𝐴 ∈ 𝑋)) → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)))
11		eqtr 2757	. . . . . . . . . . . . 13 ⊢ ((((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)) ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍)) → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍))
12	11	ex 416	. . . . . . . . . . . 12 ⊢ (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)) → ((𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍)))
13		eqtr 2757	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑈𝐻𝐵) = ((𝑎𝐻𝐴)𝐻𝐵) ∧ ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍)) → (𝑈𝐻𝐵) = (𝑎𝐻𝑍))
14		zerdivempx.3	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑍 = (GId‘𝐺)
15	14, 8, 6, 7	rngorz 35775	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑅 ∈ RingOps ∧ 𝑎 ∈ 𝑋) → (𝑎𝐻𝑍) = 𝑍)
16	15	3adant3 1134	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑅 ∈ RingOps ∧ 𝑎 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑎𝐻𝑍) = 𝑍)
17	6	rneqi 5795	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ran 𝐺 = ran (1st ‘𝑅)
18	8, 17	eqtri 2762	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑋 = ran (1st ‘𝑅)
19		zerdivempx.5	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑈 = (GId‘𝐻)
20	7, 18, 19	rngolidm 35789	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑅 ∈ RingOps ∧ 𝐵 ∈ 𝑋) → (𝑈𝐻𝐵) = 𝐵)
21	20	3adant2 1133	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑅 ∈ RingOps ∧ 𝑎 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑈𝐻𝐵) = 𝐵)
22		simp1 1138	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → (𝑈𝐻𝐵) = (𝑎𝐻𝑍))
23		simp2 1139	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → (𝑈𝐻𝐵) = 𝐵)
24		simp3 1140	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → (𝑎𝐻𝑍) = 𝑍)
25	22, 23, 24	3eqtr3d 2782	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → 𝐵 = 𝑍)
26	25	a1d 25	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → (𝐴 ∈ 𝑋 → 𝐵 = 𝑍))
27	26	3exp 1121	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑈𝐻𝐵) = (𝑎𝐻𝑍) → ((𝑈𝐻𝐵) = 𝐵 → ((𝑎𝐻𝑍) = 𝑍 → (𝐴 ∈ 𝑋 → 𝐵 = 𝑍))))
28	27	com14 96	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴 ∈ 𝑋 → ((𝑈𝐻𝐵) = 𝐵 → ((𝑎𝐻𝑍) = 𝑍 → ((𝑈𝐻𝐵) = (𝑎𝐻𝑍) → 𝐵 = 𝑍))))
29	28	com13 88	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑎𝐻𝑍) = 𝑍 → ((𝑈𝐻𝐵) = 𝐵 → (𝐴 ∈ 𝑋 → ((𝑈𝐻𝐵) = (𝑎𝐻𝑍) → 𝐵 = 𝑍))))
30	16, 21, 29	sylc 65	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ RingOps ∧ 𝑎 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ∈ 𝑋 → ((𝑈𝐻𝐵) = (𝑎𝐻𝑍) → 𝐵 = 𝑍)))
31	30	3exp 1121	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ∈ RingOps → (𝑎 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (𝐴 ∈ 𝑋 → ((𝑈𝐻𝐵) = (𝑎𝐻𝑍) → 𝐵 = 𝑍)))))
32	31	com15 101	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑈𝐻𝐵) = (𝑎𝐻𝑍) → (𝑎 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (𝐴 ∈ 𝑋 → (𝑅 ∈ RingOps → 𝐵 = 𝑍)))))
33	32	com24 95	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑈𝐻𝐵) = (𝑎𝐻𝑍) → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (𝑎 ∈ 𝑋 → (𝑅 ∈ RingOps → 𝐵 = 𝑍)))))
34	13, 33	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑈𝐻𝐵) = ((𝑎𝐻𝐴)𝐻𝐵) ∧ ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍)) → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (𝑎 ∈ 𝑋 → (𝑅 ∈ RingOps → 𝐵 = 𝑍)))))
35	34	ex 416	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈𝐻𝐵) = ((𝑎𝐻𝐴)𝐻𝐵) → (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍) → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (𝑎 ∈ 𝑋 → (𝑅 ∈ RingOps → 𝐵 = 𝑍))))))
36	35	eqcoms 2742	. . . . . . . . . . . . . . . 16 ⊢ (((𝑎𝐻𝐴)𝐻𝐵) = (𝑈𝐻𝐵) → (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍) → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (𝑎 ∈ 𝑋 → (𝑅 ∈ RingOps → 𝐵 = 𝑍))))))
37	36	com25 99	. . . . . . . . . . . . . . 15 ⊢ (((𝑎𝐻𝐴)𝐻𝐵) = (𝑈𝐻𝐵) → (𝑎 ∈ 𝑋 → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍) → (𝑅 ∈ RingOps → 𝐵 = 𝑍))))))
38		oveq1 7209	. . . . . . . . . . . . . . 15 ⊢ ((𝑎𝐻𝐴) = 𝑈 → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑈𝐻𝐵))
39	37, 38	syl11 33	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ 𝑋 → ((𝑎𝐻𝐴) = 𝑈 → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍) → (𝑅 ∈ RingOps → 𝐵 = 𝑍))))))
40	39	3imp 1113	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ 𝑋 ∧ (𝑎𝐻𝐴) = 𝑈 ∧ 𝐴 ∈ 𝑋) → (𝐵 ∈ 𝑋 → (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍) → (𝑅 ∈ RingOps → 𝐵 = 𝑍))))
41	40	com13 88	. . . . . . . . . . . 12 ⊢ (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍) → (𝐵 ∈ 𝑋 → ((𝑎 ∈ 𝑋 ∧ (𝑎𝐻𝐴) = 𝑈 ∧ 𝐴 ∈ 𝑋) → (𝑅 ∈ RingOps → 𝐵 = 𝑍))))
42	12, 41	syl6 35	. . . . . . . . . . 11 ⊢ (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)) → ((𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) → (𝐵 ∈ 𝑋 → ((𝑎 ∈ 𝑋 ∧ (𝑎𝐻𝐴) = 𝑈 ∧ 𝐴 ∈ 𝑋) → (𝑅 ∈ RingOps → 𝐵 = 𝑍)))))
43	42	com15 101	. . . . . . . . . 10 ⊢ (𝑅 ∈ RingOps → ((𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) → (𝐵 ∈ 𝑋 → ((𝑎 ∈ 𝑋 ∧ (𝑎𝐻𝐴) = 𝑈 ∧ 𝐴 ∈ 𝑋) → (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)) → 𝐵 = 𝑍)))))
44	43	3imp1 1349	. . . . . . . . 9 ⊢ (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ (𝑎𝐻𝐴) = 𝑈 ∧ 𝐴 ∈ 𝑋)) → (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)) → 𝐵 = 𝑍))
45	10, 44	mpd 15	. . . . . . . 8 ⊢ (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ (𝑎𝐻𝐴) = 𝑈 ∧ 𝐴 ∈ 𝑋)) → 𝐵 = 𝑍)
46	45	3exp1 1354	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → ((𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) → (𝐵 ∈ 𝑋 → ((𝑎 ∈ 𝑋 ∧ (𝑎𝐻𝐴) = 𝑈 ∧ 𝐴 ∈ 𝑋) → 𝐵 = 𝑍))))
47	1, 46	syl5com 31	. . . . . 6 ⊢ ((𝐴𝐻𝐵) = 𝑍 → (𝑅 ∈ RingOps → (𝐵 ∈ 𝑋 → ((𝑎 ∈ 𝑋 ∧ (𝑎𝐻𝐴) = 𝑈 ∧ 𝐴 ∈ 𝑋) → 𝐵 = 𝑍))))
48	47	com14 96	. . . . 5 ⊢ ((𝑎 ∈ 𝑋 ∧ (𝑎𝐻𝐴) = 𝑈 ∧ 𝐴 ∈ 𝑋) → (𝑅 ∈ RingOps → (𝐵 ∈ 𝑋 → ((𝐴𝐻𝐵) = 𝑍 → 𝐵 = 𝑍))))
49	48	3exp 1121	. . . 4 ⊢ (𝑎 ∈ 𝑋 → ((𝑎𝐻𝐴) = 𝑈 → (𝐴 ∈ 𝑋 → (𝑅 ∈ RingOps → (𝐵 ∈ 𝑋 → ((𝐴𝐻𝐵) = 𝑍 → 𝐵 = 𝑍))))))
50	49	rexlimiv 3192	. . 3 ⊢ (∃𝑎 ∈ 𝑋 (𝑎𝐻𝐴) = 𝑈 → (𝐴 ∈ 𝑋 → (𝑅 ∈ RingOps → (𝐵 ∈ 𝑋 → ((𝐴𝐻𝐵) = 𝑍 → 𝐵 = 𝑍)))))
51	50	com13 88	. 2 ⊢ (𝑅 ∈ RingOps → (𝐴 ∈ 𝑋 → (∃𝑎 ∈ 𝑋 (𝑎𝐻𝐴) = 𝑈 → (𝐵 ∈ 𝑋 → ((𝐴𝐻𝐵) = 𝑍 → 𝐵 = 𝑍)))))
52	51	3imp 1113	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ ∃𝑎 ∈ 𝑋 (𝑎𝐻𝐴) = 𝑈) → (𝐵 ∈ 𝑋 → ((𝐴𝐻𝐵) = 𝑍 → 𝐵 = 𝑍)))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ∃wrex 3055  ran crn 5541  ‘cfv 6369  (class class class)co 7202  1^st c1st 7748  2^nd c2nd 7749  GIdcgi 28543  RingOpscrngo 35746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-sep 5181  ax-nul 5188  ax-pr 5311  ax-un 7512

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-nul 4228  df-if 4430  df-sn 4532  df-pr 4534  df-op 4538  df-uni 4810  df-iun 4896  df-br 5044  df-opab 5106  df-mpt 5125  df-id 5444  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-fo 6375  df-fv 6377  df-riota 7159  df-ov 7205  df-1st 7750  df-2nd 7751  df-grpo 28546  df-gid 28547  df-ablo 28598  df-ass 35695  df-exid 35697  df-mgmOLD 35701  df-sgrOLD 35713  df-mndo 35719  df-rngo 35747

This theorem is referenced by:  isdrngo2  35810