nzrnz - Metamath Proof Explorer

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Theorem nzrnz 20447

Description: One and zero are different in a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)

**Hypotheses**
Ref	Expression
isnzr.o	⊢ 1 = (1_r‘𝑅)
isnzr.z	⊢ 0 = (0_g‘𝑅)

**Assertion**
Ref	Expression
nzrnz	⊢ (𝑅 ∈ NzRing → 1 ≠ 0 )

**Proof of Theorem nzrnz**
Step	Hyp	Ref	Expression
1		isnzr.o	. . 3 ⊢ 1 = (1_r‘𝑅)
2		isnzr.z	. . 3 ⊢ 0 = (0_g‘𝑅)
3	1, 2	isnzr 20446	. 2 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 ≠ 0 ))
4	3	simprbi 496	1 ⊢ (𝑅 ∈ NzRing → 1 ≠ 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 ‘cfv 6542 0_gc0g 17414 1_rcur 20114 Ringcrg 20166 NzRingcnzr 20444

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2937 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-iota 6494 df-fv 6550 df-nzr 20445

This theorem is referenced by: nzrunit 20454 nrhmzr 20467 lringnz 20473 subrgnzr 20526 fidomndrng 21254 uvcf1 21719 lindfind2 21745 nm1 24577 deg1pw 26049 ply1nz 26050 ply1nzb 26051 mon1pid 26082 lgsqrlem4 27275 unitnz 32941 rrgnz 32945 fracfld 32988 drngidl 33143 drngidlhash 33144 drnglidl1ne0 33182 drng0mxidl 33183 qsdrngi 33200 ply1moneq 33254 ply1annnr 33368 algextdeglem4 33382 zrhnm 33564 idomnnzpownz 41597 idomnnzgmulnz 41598 deg1gprod 41606 deg1pow 41607 uvcn0 41766 deg1mhm 42622

	
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