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Mirrors > Home > MPE Home > Th. List > nzrnz | Structured version Visualization version GIF version |
Description: One and zero are different in a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
Ref | Expression |
---|---|
isnzr.o | ⊢ 1 = (1r‘𝑅) |
isnzr.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
nzrnz | ⊢ (𝑅 ∈ NzRing → 1 ≠ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isnzr.o | . . 3 ⊢ 1 = (1r‘𝑅) | |
2 | isnzr.z | . . 3 ⊢ 0 = (0g‘𝑅) | |
3 | 1, 2 | isnzr 19259 | . 2 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 ≠ 0 )) |
4 | 3 | simprbi 480 | 1 ⊢ (𝑅 ∈ NzRing → 1 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ‘cfv 5888 0gc0g 16100 1rcur 18501 Ringcrg 18547 NzRingcnzr 19257 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-nzr 19258 |
This theorem is referenced by: nzrunit 19267 subrgnzr 19268 fidomndrng 19307 uvcf1 20131 lindfind2 20157 nm1 22471 deg1pw 23880 ply1nz 23881 ply1nzb 23882 lgsqrlem4 25074 zrhnm 30013 mon1pid 37783 deg1mhm 37785 nrhmzr 41873 |
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