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Mirrors > Home > MPE Home > Th. List > subrgnzr | Structured version Visualization version GIF version |
Description: A subring of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 15-Jun-2015.) |
Ref | Expression |
---|---|
subrgnzr.1 | ⊢ 𝑆 = (𝑅 ↾s 𝐴) |
Ref | Expression |
---|---|
subrgnzr | ⊢ ((𝑅 ∈ NzRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → 𝑆 ∈ NzRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subrgnzr.1 | . . . 4 ⊢ 𝑆 = (𝑅 ↾s 𝐴) | |
2 | 1 | subrgring 19541 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring) |
3 | 2 | adantl 484 | . 2 ⊢ ((𝑅 ∈ NzRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → 𝑆 ∈ Ring) |
4 | eqid 2824 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
5 | eqid 2824 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
6 | 4, 5 | nzrnz 20036 | . . . 4 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ≠ (0g‘𝑅)) |
7 | 6 | adantr 483 | . . 3 ⊢ ((𝑅 ∈ NzRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → (1r‘𝑅) ≠ (0g‘𝑅)) |
8 | 1, 4 | subrg1 19548 | . . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (1r‘𝑅) = (1r‘𝑆)) |
9 | 8 | adantl 484 | . . 3 ⊢ ((𝑅 ∈ NzRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → (1r‘𝑅) = (1r‘𝑆)) |
10 | 1, 5 | subrg0 19545 | . . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (0g‘𝑅) = (0g‘𝑆)) |
11 | 10 | adantl 484 | . . 3 ⊢ ((𝑅 ∈ NzRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → (0g‘𝑅) = (0g‘𝑆)) |
12 | 7, 9, 11 | 3netr3d 3095 | . 2 ⊢ ((𝑅 ∈ NzRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → (1r‘𝑆) ≠ (0g‘𝑆)) |
13 | eqid 2824 | . . 3 ⊢ (1r‘𝑆) = (1r‘𝑆) | |
14 | eqid 2824 | . . 3 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
15 | 13, 14 | isnzr 20035 | . 2 ⊢ (𝑆 ∈ NzRing ↔ (𝑆 ∈ Ring ∧ (1r‘𝑆) ≠ (0g‘𝑆))) |
16 | 3, 12, 15 | sylanbrc 585 | 1 ⊢ ((𝑅 ∈ NzRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → 𝑆 ∈ NzRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ‘cfv 6358 (class class class)co 7159 ↾s cress 16487 0gc0g 16716 1rcur 19254 Ringcrg 19300 SubRingcsubrg 19534 NzRingcnzr 20033 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-0g 16718 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-grp 18109 df-subg 18279 df-mgp 19243 df-ur 19255 df-ring 19302 df-subrg 19536 df-nzr 20034 |
This theorem is referenced by: (None) |
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