Theorem rrgeq0i 20632

Description: Property of a left-regular element. (Contributed by Stefan O'Rear, 22-Mar-2015.)

Hypotheses

Ref	Expression
rrgval.e	⊢ 𝐸 = (RLReg‘𝑅)
rrgval.b	⊢ 𝐵 = (Base‘𝑅)
rrgval.t	⊢ · = (.r‘𝑅)
rrgval.z	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
rrgeq0i	⊢ ((𝑋 ∈ 𝐸 ∧ 𝑌 ∈ 𝐵) → ((𝑋 · 𝑌) = 0 → 𝑌 = 0 ))

Proof of Theorem rrgeq0i

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rrgval.e	. . . 4 ⊢ 𝐸 = (RLReg‘𝑅)
2		rrgval.b	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
3		rrgval.t	. . . 4 ⊢ · = (.r‘𝑅)
4		rrgval.z	. . . 4 ⊢ 0 = (0g‘𝑅)
5	1, 2, 3, 4	isrrg 20631	. . 3 ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑋 · 𝑦) = 0 → 𝑦 = 0 )))
6	5	simprbi 497	. 2 ⊢ (𝑋 ∈ 𝐸 → ∀𝑦 ∈ 𝐵 ((𝑋 · 𝑦) = 0 → 𝑦 = 0 ))
7		oveq2 7323	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
8	7	eqeq1d 2739	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑋 · 𝑦) = 0 ↔ (𝑋 · 𝑌) = 0 ))
9		eqeq1 2741	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑦 = 0 ↔ 𝑌 = 0 ))
10	8, 9	imbi12d 344	. . 3 ⊢ (𝑦 = 𝑌 → (((𝑋 · 𝑦) = 0 → 𝑦 = 0 ) ↔ ((𝑋 · 𝑌) = 0 → 𝑌 = 0 )))
11	10	rspcv 3566	. 2 ⊢ (𝑌 ∈ 𝐵 → (∀𝑦 ∈ 𝐵 ((𝑋 · 𝑦) = 0 → 𝑦 = 0 ) → ((𝑋 · 𝑌) = 0 → 𝑌 = 0 )))
12	6, 11	mpan9 507	1 ⊢ ((𝑋 ∈ 𝐸 ∧ 𝑌 ∈ 𝐵) → ((𝑋 · 𝑌) = 0 → 𝑌 = 0 ))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1540   ∈ wcel 2105  ∀wral 3062  ‘cfv 6465  (class class class)co 7315  Basecbs 16982  ._rcmulr 17033  0_gc0g 17220  RLRegcrlreg 20622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5238  ax-nul 5245  ax-pr 5367

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4268  df-if 4472  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-br 5088  df-opab 5150  df-mpt 5171  df-id 5507  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-iota 6417  df-fun 6467  df-fv 6473  df-ov 7318  df-rlreg 20626

This theorem is referenced by:  rrgeq0  20633  znrrg  20845  deg1mul2  25351