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Mirrors > Home > MPE Home > Th. List > drngmcl | Structured version Visualization version GIF version |
Description: The product of two nonzero elements of a division ring is nonzero. (Contributed by NM, 7-Sep-2011.) |
Ref | Expression |
---|---|
drngmcl.b | ⊢ 𝐵 = (Base‘𝑅) |
drngmcl.t | ⊢ · = (.r‘𝑅) |
drngmcl.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
drngmcl | ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ (𝐵 ∖ { 0 }) ∧ 𝑌 ∈ (𝐵 ∖ { 0 })) → (𝑋 · 𝑌) ∈ (𝐵 ∖ { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | drngmcl.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
2 | drngmcl.z | . . 3 ⊢ 0 = (0g‘𝑅) | |
3 | eqid 2610 | . . 3 ⊢ ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })) = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })) | |
4 | 1, 2, 3 | drngmgp 18582 | . 2 ⊢ (𝑅 ∈ DivRing → ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })) ∈ Grp) |
5 | difss 3699 | . . . 4 ⊢ (𝐵 ∖ { 0 }) ⊆ 𝐵 | |
6 | eqid 2610 | . . . . . 6 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
7 | 6, 1 | mgpbas 18318 | . . . . 5 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅)) |
8 | 3, 7 | ressbas2 15758 | . . . 4 ⊢ ((𝐵 ∖ { 0 }) ⊆ 𝐵 → (𝐵 ∖ { 0 }) = (Base‘((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })))) |
9 | 5, 8 | ax-mp 5 | . . 3 ⊢ (𝐵 ∖ { 0 }) = (Base‘((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))) |
10 | fvex 6113 | . . . . 5 ⊢ (Base‘𝑅) ∈ V | |
11 | 1, 10 | eqeltri 2684 | . . . 4 ⊢ 𝐵 ∈ V |
12 | difexg 4735 | . . . 4 ⊢ (𝐵 ∈ V → (𝐵 ∖ { 0 }) ∈ V) | |
13 | drngmcl.t | . . . . . 6 ⊢ · = (.r‘𝑅) | |
14 | 6, 13 | mgpplusg 18316 | . . . . 5 ⊢ · = (+g‘(mulGrp‘𝑅)) |
15 | 3, 14 | ressplusg 15818 | . . . 4 ⊢ ((𝐵 ∖ { 0 }) ∈ V → · = (+g‘((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })))) |
16 | 11, 12, 15 | mp2b 10 | . . 3 ⊢ · = (+g‘((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))) |
17 | 9, 16 | grpcl 17253 | . 2 ⊢ ((((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })) ∈ Grp ∧ 𝑋 ∈ (𝐵 ∖ { 0 }) ∧ 𝑌 ∈ (𝐵 ∖ { 0 })) → (𝑋 · 𝑌) ∈ (𝐵 ∖ { 0 })) |
18 | 4, 17 | syl3an1 1351 | 1 ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ (𝐵 ∖ { 0 }) ∧ 𝑌 ∈ (𝐵 ∖ { 0 })) → (𝑋 · 𝑌) ∈ (𝐵 ∖ { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 ⊆ wss 3540 {csn 4125 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 ↾s cress 15696 +gcplusg 15768 .rcmulr 15769 0gc0g 15923 Grpcgrp 17245 mulGrpcmgp 18312 DivRingcdr 18570 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-drng 18572 |
This theorem is referenced by: abvtriv 18664 |
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