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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngosn6 | Structured version Visualization version GIF version |
Description: Obsolete as of 25-Jan-2020. Use ringen1zr 20306 or srgen1zr 19997 instead. The only unital ring with one element is the zero ring. (Contributed by FL, 15-Feb-2010.) (New usage is discouraged.) |
Ref | Expression |
---|---|
on1el3.1 | ⊢ 𝐺 = (1st ‘𝑅) |
on1el3.2 | ⊢ 𝑋 = ran 𝐺 |
on1el3.3 | ⊢ 𝑍 = (GId‘𝐺) |
Ref | Expression |
---|---|
rngosn6 | ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o ↔ 𝑅 = 〈{〈〈𝑍, 𝑍〉, 𝑍〉}, {〈〈𝑍, 𝑍〉, 𝑍〉}〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | on1el3.1 | . . 3 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | on1el3.2 | . . 3 ⊢ 𝑋 = ran 𝐺 | |
3 | on1el3.3 | . . 3 ⊢ 𝑍 = (GId‘𝐺) | |
4 | 1, 2, 3 | rngo0cl 36592 | . 2 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ 𝑋) |
5 | 1, 2 | rngosn4 36598 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑍 ∈ 𝑋) → (𝑋 ≈ 1o ↔ 𝑅 = 〈{〈〈𝑍, 𝑍〉, 𝑍〉}, {〈〈𝑍, 𝑍〉, 𝑍〉}〉)) |
6 | 4, 5 | mpdan 685 | 1 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o ↔ 𝑅 = 〈{〈〈𝑍, 𝑍〉, 𝑍〉}, {〈〈𝑍, 𝑍〉, 𝑍〉}〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∈ wcel 2106 {csn 4622 〈cop 4628 class class class wbr 5141 ran crn 5670 ‘cfv 6532 1st c1st 7955 1oc1o 8441 ≈ cen 8919 GIdcgi 29606 RingOpscrngo 36567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-1st 7957 df-2nd 7958 df-1o 8448 df-en 8923 df-grpo 29609 df-gid 29610 df-ablo 29661 df-rngo 36568 |
This theorem is referenced by: dvrunz 36627 |
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