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Mirrors > Home > MPE Home > Th. List > Mathboxes > drnginvrcld | Structured version Visualization version GIF version |
Description: Closure of the multiplicative inverse in a division ring. (reccld 11460 analog). (Contributed by SN, 14-Aug-2024.) |
Ref | Expression |
---|---|
drnginvrcld.b | ⊢ 𝐵 = (Base‘𝑅) |
drnginvrcld.0 | ⊢ 0 = (0g‘𝑅) |
drnginvrcld.i | ⊢ 𝐼 = (invr‘𝑅) |
drnginvrcld.r | ⊢ (𝜑 → 𝑅 ∈ DivRing) |
drnginvrcld.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
drnginvrcld.1 | ⊢ (𝜑 → 𝑋 ≠ 0 ) |
Ref | Expression |
---|---|
drnginvrcld | ⊢ (𝜑 → (𝐼‘𝑋) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | drnginvrcld.r | . 2 ⊢ (𝜑 → 𝑅 ∈ DivRing) | |
2 | drnginvrcld.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
3 | drnginvrcld.1 | . 2 ⊢ (𝜑 → 𝑋 ≠ 0 ) | |
4 | drnginvrcld.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
5 | drnginvrcld.0 | . . 3 ⊢ 0 = (0g‘𝑅) | |
6 | drnginvrcld.i | . . 3 ⊢ 𝐼 = (invr‘𝑅) | |
7 | 4, 5, 6 | drnginvrcl 19600 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐼‘𝑋) ∈ 𝐵) |
8 | 1, 2, 3, 7 | syl3anc 1368 | 1 ⊢ (𝜑 → (𝐼‘𝑋) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ≠ wne 2951 ‘cfv 6340 Basecbs 16554 0gc0g 16784 invrcinvr 19505 DivRingcdr 19583 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5160 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 ax-cnex 10644 ax-resscn 10645 ax-1cn 10646 ax-icn 10647 ax-addcl 10648 ax-addrcl 10649 ax-mulcl 10650 ax-mulrcl 10651 ax-mulcom 10652 ax-addass 10653 ax-mulass 10654 ax-distr 10655 ax-i2m1 10656 ax-1ne0 10657 ax-1rid 10658 ax-rnegex 10659 ax-rrecex 10660 ax-cnre 10661 ax-pre-lttri 10662 ax-pre-lttrn 10663 ax-pre-ltadd 10664 ax-pre-mulgt0 10665 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-iun 4888 df-br 5037 df-opab 5099 df-mpt 5117 df-tr 5143 df-id 5434 df-eprel 5439 df-po 5447 df-so 5448 df-fr 5487 df-we 5489 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-pred 6131 df-ord 6177 df-on 6178 df-lim 6179 df-suc 6180 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7586 df-tpos 7908 df-wrecs 7963 df-recs 8024 df-rdg 8062 df-er 8305 df-en 8541 df-dom 8542 df-sdom 8543 df-pnf 10728 df-mnf 10729 df-xr 10730 df-ltxr 10731 df-le 10732 df-sub 10923 df-neg 10924 df-nn 11688 df-2 11750 df-3 11751 df-ndx 16557 df-slot 16558 df-base 16560 df-sets 16561 df-ress 16562 df-plusg 16649 df-mulr 16650 df-0g 16786 df-mgm 17931 df-sgrp 17980 df-mnd 17991 df-grp 18185 df-minusg 18186 df-mgp 19321 df-ur 19333 df-ring 19380 df-oppr 19457 df-dvdsr 19475 df-unit 19476 df-invr 19506 df-drng 19585 |
This theorem is referenced by: drngmulcanad 39795 drngmulcan2ad 39796 drnginvmuld 39797 prjspner1 39995 |
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