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Mirrors > Home > ILE Home > Th. List > crnggrpd | GIF version |
Description: A commutative ring is a group. (Contributed by SN, 16-May-2024.) |
Ref | Expression |
---|---|
crngringd.1 | ⊢ (𝜑 → 𝑅 ∈ CRing) |
Ref | Expression |
---|---|
crnggrpd | ⊢ (𝜑 → 𝑅 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crngringd.1 | . . 3 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
2 | 1 | crngringd 13261 | . 2 ⊢ (𝜑 → 𝑅 ∈ Ring) |
3 | 2 | ringgrpd 13257 | 1 ⊢ (𝜑 → 𝑅 ∈ Grp) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2158 Grpcgrp 12899 CRingccrg 13249 |
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 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-cnex 7916 ax-resscn 7917 ax-1re 7919 ax-addrcl 7922 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-sbc 2975 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-iota 5190 df-fun 5230 df-fn 5231 df-fv 5236 df-ov 5891 df-inn 8934 df-2 8992 df-3 8993 df-ndx 12479 df-slot 12480 df-base 12482 df-plusg 12564 df-mulr 12565 df-ring 13250 df-cring 13251 |
This theorem is referenced by: (None) |
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