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Mirrors > Home > MPE Home > Th. List > rnglidlmmgm | Structured version Visualization version GIF version |
Description: The multiplicative group of a (left) ideal of a non-unital ring is a magma. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption 0 ∈ 𝑈 is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.) |
Ref | Expression |
---|---|
rnglidlabl.l | ⊢ 𝐿 = (LIdeal‘𝑅) |
rnglidlabl.i | ⊢ 𝐼 = (𝑅 ↾s 𝑈) |
rnglidlabl.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
rnglidlmmgm | ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (mulGrp‘𝐼) ∈ Mgm) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1135 | . . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → 𝑅 ∈ Rng) | |
2 | rnglidlabl.l | . . . . . . . . 9 ⊢ 𝐿 = (LIdeal‘𝑅) | |
3 | rnglidlabl.i | . . . . . . . . 9 ⊢ 𝐼 = (𝑅 ↾s 𝑈) | |
4 | 2, 3 | lidlbas 21242 | . . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) = 𝑈) |
5 | eleq1a 2834 | . . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → ((Base‘𝐼) = 𝑈 → (Base‘𝐼) ∈ 𝐿)) | |
6 | 4, 5 | mpd 15 | . . . . . . 7 ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) ∈ 𝐿) |
7 | 6 | 3ad2ant2 1133 | . . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (Base‘𝐼) ∈ 𝐿) |
8 | 4 | eqcomd 2741 | . . . . . . . . 9 ⊢ (𝑈 ∈ 𝐿 → 𝑈 = (Base‘𝐼)) |
9 | 8 | eleq2d 2825 | . . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → ( 0 ∈ 𝑈 ↔ 0 ∈ (Base‘𝐼))) |
10 | 9 | biimpa 476 | . . . . . . 7 ⊢ ((𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → 0 ∈ (Base‘𝐼)) |
11 | 10 | 3adant1 1129 | . . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → 0 ∈ (Base‘𝐼)) |
12 | 1, 7, 11 | 3jca 1127 | . . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (𝑅 ∈ Rng ∧ (Base‘𝐼) ∈ 𝐿 ∧ 0 ∈ (Base‘𝐼))) |
13 | 2, 3 | lidlssbas 21241 | . . . . . . . . 9 ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) ⊆ (Base‘𝑅)) |
14 | 13 | sseld 3994 | . . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → (𝑎 ∈ (Base‘𝐼) → 𝑎 ∈ (Base‘𝑅))) |
15 | 14 | 3ad2ant2 1133 | . . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (𝑎 ∈ (Base‘𝐼) → 𝑎 ∈ (Base‘𝑅))) |
16 | 15 | anim1d 611 | . . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → ((𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼)) → (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝐼)))) |
17 | 16 | imp 406 | . . . . 5 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼))) → (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝐼))) |
18 | rnglidlabl.z | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
19 | eqid 2735 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
20 | eqid 2735 | . . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
21 | 18, 19, 20, 2 | rnglidlmcl 21244 | . . . . 5 ⊢ (((𝑅 ∈ Rng ∧ (Base‘𝐼) ∈ 𝐿 ∧ 0 ∈ (Base‘𝐼)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝐼))) → (𝑎(.r‘𝑅)𝑏) ∈ (Base‘𝐼)) |
22 | 12, 17, 21 | syl2an2r 685 | . . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼))) → (𝑎(.r‘𝑅)𝑏) ∈ (Base‘𝐼)) |
23 | 3, 20 | ressmulr 17353 | . . . . . . . . 9 ⊢ (𝑈 ∈ 𝐿 → (.r‘𝑅) = (.r‘𝐼)) |
24 | 23 | eqcomd 2741 | . . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → (.r‘𝐼) = (.r‘𝑅)) |
25 | 24 | oveqd 7448 | . . . . . . 7 ⊢ (𝑈 ∈ 𝐿 → (𝑎(.r‘𝐼)𝑏) = (𝑎(.r‘𝑅)𝑏)) |
26 | 25 | eleq1d 2824 | . . . . . 6 ⊢ (𝑈 ∈ 𝐿 → ((𝑎(.r‘𝐼)𝑏) ∈ (Base‘𝐼) ↔ (𝑎(.r‘𝑅)𝑏) ∈ (Base‘𝐼))) |
27 | 26 | 3ad2ant2 1133 | . . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → ((𝑎(.r‘𝐼)𝑏) ∈ (Base‘𝐼) ↔ (𝑎(.r‘𝑅)𝑏) ∈ (Base‘𝐼))) |
28 | 27 | adantr 480 | . . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼))) → ((𝑎(.r‘𝐼)𝑏) ∈ (Base‘𝐼) ↔ (𝑎(.r‘𝑅)𝑏) ∈ (Base‘𝐼))) |
29 | 22, 28 | mpbird 257 | . . 3 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼))) → (𝑎(.r‘𝐼)𝑏) ∈ (Base‘𝐼)) |
30 | 29 | ralrimivva 3200 | . 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → ∀𝑎 ∈ (Base‘𝐼)∀𝑏 ∈ (Base‘𝐼)(𝑎(.r‘𝐼)𝑏) ∈ (Base‘𝐼)) |
31 | fvex 6920 | . . 3 ⊢ (mulGrp‘𝐼) ∈ V | |
32 | eqid 2735 | . . . . 5 ⊢ (mulGrp‘𝐼) = (mulGrp‘𝐼) | |
33 | eqid 2735 | . . . . 5 ⊢ (Base‘𝐼) = (Base‘𝐼) | |
34 | 32, 33 | mgpbas 20158 | . . . 4 ⊢ (Base‘𝐼) = (Base‘(mulGrp‘𝐼)) |
35 | eqid 2735 | . . . . 5 ⊢ (.r‘𝐼) = (.r‘𝐼) | |
36 | 32, 35 | mgpplusg 20156 | . . . 4 ⊢ (.r‘𝐼) = (+g‘(mulGrp‘𝐼)) |
37 | 34, 36 | ismgm 18667 | . . 3 ⊢ ((mulGrp‘𝐼) ∈ V → ((mulGrp‘𝐼) ∈ Mgm ↔ ∀𝑎 ∈ (Base‘𝐼)∀𝑏 ∈ (Base‘𝐼)(𝑎(.r‘𝐼)𝑏) ∈ (Base‘𝐼))) |
38 | 31, 37 | mp1i 13 | . 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → ((mulGrp‘𝐼) ∈ Mgm ↔ ∀𝑎 ∈ (Base‘𝐼)∀𝑏 ∈ (Base‘𝐼)(𝑎(.r‘𝐼)𝑏) ∈ (Base‘𝐼))) |
39 | 30, 38 | mpbird 257 | 1 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (mulGrp‘𝐼) ∈ Mgm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 ↾s cress 17274 .rcmulr 17299 0gc0g 17486 Mgmcmgm 18664 mulGrpcmgp 20152 Rngcrng 20170 LIdealclidl 21234 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-abl 19816 df-mgp 20153 df-rng 20171 df-lss 20948 df-sra 21190 df-rgmod 21191 df-lidl 21236 |
This theorem is referenced by: rnglidlmsgrp 21274 |
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