Theorem zerdivemp1x 35219

Description: In a unitary ring a left invertible element is not a zero divisor. See also ringinvnzdiv 19337. (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 7158	. . . . . . 7 ⊢ ((𝐴𝐻𝐵) = 𝑍 → (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍))
2		simpl1 1187	. . . . . . . . . 10 ⊢ (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ (𝑎𝐻𝐴) = 𝑈 ∧ 𝐴 ∈ 𝑋)) → 𝑅 ∈ RingOps)
3		simpr1 1190	. . . . . . . . . 10 ⊢ (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ (𝑎𝐻𝐴) = 𝑈 ∧ 𝐴 ∈ 𝑋)) → 𝑎 ∈ 𝑋)
4		simpr3 1192	. . . . . . . . . 10 ⊢ (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ (𝑎𝐻𝐴) = 𝑈 ∧ 𝐴 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
5		simpl3 1189	. . . . . . . . . 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 35178	. . . . . . . . . 10 ⊢ ((𝑅 ∈ RingOps ∧ (𝑎 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)))
10	2, 3, 4, 5, 9	syl13anc 1368	. . . . . . . . 9 ⊢ (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵 ∈ 𝑋) ∧ (𝑎 ∈ 𝑋 ∧ (𝑎𝐻𝐴) = 𝑈 ∧ 𝐴 ∈ 𝑋)) → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)))
11		eqtr 2841	. . . . . . . . . . . . 13 ⊢ ((((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)) ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍)) → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍))
12	11	ex 415	. . . . . . . . . . . 12 ⊢ (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)) → ((𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍)))
13		eqtr 2841	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑈𝐻𝐵) = ((𝑎𝐻𝐴)𝐻𝐵) ∧ ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍)) → (𝑈𝐻𝐵) = (𝑎𝐻𝑍))
14		zerdivempx.3	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑍 = (GId‘𝐺)
15	14, 8, 6, 7	rngorz 35195	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑅 ∈ RingOps ∧ 𝑎 ∈ 𝑋) → (𝑎𝐻𝑍) = 𝑍)
16	15	3adant3 1128	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑅 ∈ RingOps ∧ 𝑎 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑎𝐻𝑍) = 𝑍)
17	6	rneqi 5801	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ran 𝐺 = ran (1st ‘𝑅)
18	8, 17	eqtri 2844	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑋 = ran (1st ‘𝑅)
19		zerdivempx.5	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑈 = (GId‘𝐻)
20	7, 18, 19	rngolidm 35209	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑅 ∈ RingOps ∧ 𝐵 ∈ 𝑋) → (𝑈𝐻𝐵) = 𝐵)
21	20	3adant2 1127	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑅 ∈ RingOps ∧ 𝑎 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑈𝐻𝐵) = 𝐵)
22		simp1 1132	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → (𝑈𝐻𝐵) = (𝑎𝐻𝑍))
23		simp2 1133	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → (𝑈𝐻𝐵) = 𝐵)
24		simp3 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → (𝑎𝐻𝑍) = 𝑍)
25	22, 23, 24	3eqtr3d 2864	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1115	. . . . . . . . . . . . . . . . . . . . . . . . 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 1115	. . . . . . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈𝐻𝐵) = ((𝑎𝐻𝐴)𝐻𝐵) → (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍) → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (𝑎 ∈ 𝑋 → (𝑅 ∈ RingOps → 𝐵 = 𝑍))))))
36	35	eqcoms 2829	. . . . . . . . . . . . . . . 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 7157	. . . . . . . . . . . . . . 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 1107	. . . . . . . . . . . . 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 1343	. . . . . . . . 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 1348	. . . . . . 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 1115	. . . 4 ⊢ (𝑎 ∈ 𝑋 → ((𝑎𝐻𝐴) = 𝑈 → (𝐴 ∈ 𝑋 → (𝑅 ∈ RingOps → (𝐵 ∈ 𝑋 → ((𝐴𝐻𝐵) = 𝑍 → 𝐵 = 𝑍))))))
50	49	rexlimiv 3280	. . 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 1107	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ ∃𝑎 ∈ 𝑋 (𝑎𝐻𝐴) = 𝑈) → (𝐵 ∈ 𝑋 → ((𝐴𝐻𝐵) = 𝑍 → 𝐵 = 𝑍)))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∃wrex 3139  ran crn 5550  ‘cfv 6349  (class class class)co 7150  1^st c1st 7681  2^nd c2nd 7682  GIdcgi 28261  RingOpscrngo 35166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-fo 6355  df-fv 6357  df-riota 7108  df-ov 7153  df-1st 7683  df-2nd 7684  df-grpo 28264  df-gid 28265  df-ablo 28316  df-ass 35115  df-exid 35117  df-mgmOLD 35121  df-sgrOLD 35133  df-mndo 35139  df-rngo 35167

This theorem is referenced by:  isdrngo2  35230