Theorem rrgeq0i 14236

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 14235	. . 3 ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑋 · 𝑦) = 0 → 𝑦 = 0 )))
6	5	simprbi 275	. 2 ⊢ (𝑋 ∈ 𝐸 → ∀𝑦 ∈ 𝐵 ((𝑋 · 𝑦) = 0 → 𝑦 = 0 ))
7		oveq2 6015	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
8	7	eqeq1d 2238	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑋 · 𝑦) = 0 ↔ (𝑋 · 𝑌) = 0 ))
9		eqeq1 2236	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑦 = 0 ↔ 𝑌 = 0 ))
10	8, 9	imbi12d 234	. . 3 ⊢ (𝑦 = 𝑌 → (((𝑋 · 𝑦) = 0 → 𝑦 = 0 ) ↔ ((𝑋 · 𝑌) = 0 → 𝑌 = 0 )))
11	10	rspcv 2903	. 2 ⊢ (𝑌 ∈ 𝐵 → (∀𝑦 ∈ 𝐵 ((𝑋 · 𝑦) = 0 → 𝑦 = 0 ) → ((𝑋 · 𝑌) = 0 → 𝑌 = 0 )))
12	6, 11	mpan9 281	1 ⊢ ((𝑋 ∈ 𝐸 ∧ 𝑌 ∈ 𝐵) → ((𝑋 · 𝑌) = 0 → 𝑌 = 0 ))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  ∀wral 2508  ‘cfv 5318  (class class class)co 6007  Basecbs 13040  ._rcmulr 13119  0_gc0g 13297  RLRegcrlreg 14227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-cnex 8098  ax-resscn 8099  ax-1re 8101  ax-addrcl 8104

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-ov 6010  df-inn 9119  df-ndx 13043  df-slot 13044  df-base 13046  df-rlreg 14230

This theorem is referenced by:  rrgeq0  14237  znrrg  14632