Theorem zerdivemp1x 36105

Description: In a unitary ring a left invertible element is not a zero divisor. See also ringinvnzdiv 19832. (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 7283	. . . . . . 7 ⊢ ((𝐴𝐻𝐵) = 𝑍 → (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍))
2		simpl1 1190	. . . . . . . . . 10 ⊢ (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ (𝑎𝐻𝐴) = 𝑈 ∧ 𝐴 ∈ 𝑋)) → 𝑅 ∈ RingOps)
3		simpr1 1193	. . . . . . . . . 10 ⊢ (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ (𝑎𝐻𝐴) = 𝑈 ∧ 𝐴 ∈ 𝑋)) → 𝑎 ∈ 𝑋)
4		simpr3 1195	. . . . . . . . . 10 ⊢ (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ (𝑎𝐻𝐴) = 𝑈 ∧ 𝐴 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
5		simpl3 1192	. . . . . . . . . 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 36064	. . . . . . . . . 10 ⊢ ((𝑅 ∈ RingOps ∧ (𝑎 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)))
10	2, 3, 4, 5, 9	syl13anc 1371	. . . . . . . . 9 ⊢ (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ (𝑎𝐻𝐴) = 𝑈 ∧ 𝐴 ∈ 𝑋)) → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)))
11		eqtr 2761	. . . . . . . . . . . . 13 ⊢ ((((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)) ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍)) → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍))
12	11	ex 413	. . . . . . . . . . . 12 ⊢ (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)) → ((𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍)))
13		eqtr 2761	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑈𝐻𝐵) = ((𝑎𝐻𝐴)𝐻𝐵) ∧ ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍)) → (𝑈𝐻𝐵) = (𝑎𝐻𝑍))
14		zerdivempx.3	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑍 = (GId‘𝐺)
15	14, 8, 6, 7	rngorz 36081	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑅 ∈ RingOps ∧ 𝑎 ∈ 𝑋) → (𝑎𝐻𝑍) = 𝑍)
16	15	3adant3 1131	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑅 ∈ RingOps ∧ 𝑎 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑎𝐻𝑍) = 𝑍)
17	6	rneqi 5846	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ran 𝐺 = ran (1st ‘𝑅)
18	8, 17	eqtri 2766	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑋 = ran (1st ‘𝑅)
19		zerdivempx.5	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑈 = (GId‘𝐻)
20	7, 18, 19	rngolidm 36095	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑅 ∈ RingOps ∧ 𝐵 ∈ 𝑋) → (𝑈𝐻𝐵) = 𝐵)
21	20	3adant2 1130	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑅 ∈ RingOps ∧ 𝑎 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑈𝐻𝐵) = 𝐵)
22		simp1 1135	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → (𝑈𝐻𝐵) = (𝑎𝐻𝑍))
23		simp2 1136	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → (𝑈𝐻𝐵) = 𝐵)
24		simp3 1137	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → (𝑎𝐻𝑍) = 𝑍)
25	22, 23, 24	3eqtr3d 2786	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1118	. . . . . . . . . . . . . . . . . . . . . . . . 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 1118	. . . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈𝐻𝐵) = ((𝑎𝐻𝐴)𝐻𝐵) → (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍) → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (𝑎 ∈ 𝑋 → (𝑅 ∈ RingOps → 𝐵 = 𝑍))))))
36	35	eqcoms 2746	. . . . . . . . . . . . . . . 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 7282	. . . . . . . . . . . . . . 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 1110	. . . . . . . . . . . . 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 1346	. . . . . . . . 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 1351	. . . . . . 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 1118	. . . 4 ⊢ (𝑎 ∈ 𝑋 → ((𝑎𝐻𝐴) = 𝑈 → (𝐴 ∈ 𝑋 → (𝑅 ∈ RingOps → (𝐵 ∈ 𝑋 → ((𝐴𝐻𝐵) = 𝑍 → 𝐵 = 𝑍))))))
50	49	rexlimiv 3209	. . 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 1110	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ ∃𝑎 ∈ 𝑋 (𝑎𝐻𝐴) = 𝑈) → (𝐵 ∈ 𝑋 → ((𝐴𝐻𝐵) = 𝑍 → 𝐵 = 𝑍)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∃wrex 3065  ran crn 5590  ‘cfv 6433  (class class class)co 7275  1^st c1st 7829  2^nd c2nd 7830  GIdcgi 28852  RingOpscrngo 36052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fo 6439  df-fv 6441  df-riota 7232  df-ov 7278  df-1st 7831  df-2nd 7832  df-grpo 28855  df-gid 28856  df-ablo 28907  df-ass 36001  df-exid 36003  df-mgmOLD 36007  df-sgrOLD 36019  df-mndo 36025  df-rngo 36053

This theorem is referenced by:  isdrngo2  36116