Theorem zerdivemp1x 36907

Description: In a unital ring a left invertible element is not a zero divisor. See also ringinvnzdiv 20117. (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 7419	. . . . . . 7 ⊢ ((𝐴𝐻𝐵) = 𝑍 → (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍))
2		simpl1 1191	. . . . . . . . . 10 ⊢ (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ (𝑎𝐻𝐴) = 𝑈 ∧ 𝐴 ∈ 𝑋)) → 𝑅 ∈ RingOps)
3		simpr1 1194	. . . . . . . . . 10 ⊢ (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ (𝑎𝐻𝐴) = 𝑈 ∧ 𝐴 ∈ 𝑋)) → 𝑎 ∈ 𝑋)
4		simpr3 1196	. . . . . . . . . 10 ⊢ (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ (𝑎𝐻𝐴) = 𝑈 ∧ 𝐴 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
5		simpl3 1193	. . . . . . . . . 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 36866	. . . . . . . . . 10 ⊢ ((𝑅 ∈ RingOps ∧ (𝑎 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)))
10	2, 3, 4, 5, 9	syl13anc 1372	. . . . . . . . 9 ⊢ (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ (𝑎𝐻𝐴) = 𝑈 ∧ 𝐴 ∈ 𝑋)) → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)))
11		eqtr 2755	. . . . . . . . . . . . 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 2755	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑈𝐻𝐵) = ((𝑎𝐻𝐴)𝐻𝐵) ∧ ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍)) → (𝑈𝐻𝐵) = (𝑎𝐻𝑍))
14		zerdivempx.3	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑍 = (GId‘𝐺)
15	14, 8, 6, 7	rngorz 36883	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑅 ∈ RingOps ∧ 𝑎 ∈ 𝑋) → (𝑎𝐻𝑍) = 𝑍)
16	15	3adant3 1132	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑅 ∈ RingOps ∧ 𝑎 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑎𝐻𝑍) = 𝑍)
17	6	rneqi 5936	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ran 𝐺 = ran (1st ‘𝑅)
18	8, 17	eqtri 2760	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑋 = ran (1st ‘𝑅)
19		zerdivempx.5	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑈 = (GId‘𝐻)
20	7, 18, 19	rngolidm 36897	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑅 ∈ RingOps ∧ 𝐵 ∈ 𝑋) → (𝑈𝐻𝐵) = 𝐵)
21	20	3adant2 1131	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑅 ∈ RingOps ∧ 𝑎 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑈𝐻𝐵) = 𝐵)
22		simp1 1136	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → (𝑈𝐻𝐵) = (𝑎𝐻𝑍))
23		simp2 1137	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → (𝑈𝐻𝐵) = 𝐵)
24		simp3 1138	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → (𝑎𝐻𝑍) = 𝑍)
25	22, 23, 24	3eqtr3d 2780	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1119	. . . . . . . . . . . . . . . . . . . . . . . . 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 1119	. . . . . . . . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . 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 7418	. . . . . . . . . . . . . . 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 1111	. . . . . . . . . . . . 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 1347	. . . . . . . . 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 1352	. . . . . . 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 1119	. . . 4 ⊢ (𝑎 ∈ 𝑋 → ((𝑎𝐻𝐴) = 𝑈 → (𝐴 ∈ 𝑋 → (𝑅 ∈ RingOps → (𝐵 ∈ 𝑋 → ((𝐴𝐻𝐵) = 𝑍 → 𝐵 = 𝑍))))))
50	49	rexlimiv 3148	. . 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 1111	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ ∃𝑎 ∈ 𝑋 (𝑎𝐻𝐴) = 𝑈) → (𝐵 ∈ 𝑋 → ((𝐴𝐻𝐵) = 𝑍 → 𝐵 = 𝑍)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∃wrex 3070  ran crn 5677  ‘cfv 6543  (class class class)co 7411  1^st c1st 7975  2^nd c2nd 7976  GIdcgi 29781  RingOpscrngo 36854

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-riota 7367  df-ov 7414  df-1st 7977  df-2nd 7978  df-grpo 29784  df-gid 29785  df-ablo 29836  df-ass 36803  df-exid 36805  df-mgmOLD 36809  df-sgrOLD 36821  df-mndo 36827  df-rngo 36855

This theorem is referenced by:  isdrngo2  36918