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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngone0 | Structured version Visualization version GIF version |
Description: The base set of a ring is not empty. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.) |
Ref | Expression |
---|---|
rngone0.1 | ⊢ 𝐺 = (1st ‘𝑅) |
rngone0.2 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
rngone0 | ⊢ (𝑅 ∈ RingOps → 𝑋 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngone0.1 | . . 3 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | 1 | rngogrpo 33709 | . 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp) |
3 | rngone0.2 | . . 3 ⊢ 𝑋 = ran 𝐺 | |
4 | 3 | grpon0 27356 | . 2 ⊢ (𝐺 ∈ GrpOp → 𝑋 ≠ ∅) |
5 | 2, 4 | syl 17 | 1 ⊢ (𝑅 ∈ RingOps → 𝑋 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∅c0 3915 ran crn 5115 ‘cfv 5888 1st c1st 7166 GrpOpcgr 27343 RingOpscrngo 33693 |
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-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
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-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 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-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fo 5894 df-fv 5896 df-ov 6653 df-1st 7168 df-2nd 7169 df-grpo 27347 df-ablo 27399 df-rngo 33694 |
This theorem is referenced by: rngoueqz 33739 |
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