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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isunit3 | Structured version Visualization version GIF version |
Description: Alternate definition of being a unit. (Contributed by Thierry Arnoux, 3-Aug-2025.) |
Ref | Expression |
---|---|
isunit2.b | ⊢ 𝐵 = (Base‘𝑅) |
isunit2.u | ⊢ 𝑈 = (Unit‘𝑅) |
isunit2.m | ⊢ · = (.r‘𝑅) |
isunit2.1 | ⊢ 1 = (1r‘𝑅) |
isunit3.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
isunit3.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
Ref | Expression |
---|---|
isunit3 | ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ ∃𝑦 ∈ 𝐵 ((𝑋 · 𝑦) = 1 ∧ (𝑦 · 𝑋) = 1 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isunit2.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
2 | isunit2.u | . . . 4 ⊢ 𝑈 = (Unit‘𝑅) | |
3 | isunit2.m | . . . 4 ⊢ · = (.r‘𝑅) | |
4 | isunit2.1 | . . . 4 ⊢ 1 = (1r‘𝑅) | |
5 | 1, 2, 3, 4 | isunit2 33212 | . . 3 ⊢ (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ (∃𝑢 ∈ 𝐵 (𝑋 · 𝑢) = 1 ∧ ∃𝑣 ∈ 𝐵 (𝑣 · 𝑋) = 1 ))) |
6 | isunit3.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
7 | 6 | biantrurd 532 | . . 3 ⊢ (𝜑 → ((∃𝑢 ∈ 𝐵 (𝑋 · 𝑢) = 1 ∧ ∃𝑣 ∈ 𝐵 (𝑣 · 𝑋) = 1 ) ↔ (𝑋 ∈ 𝐵 ∧ (∃𝑢 ∈ 𝐵 (𝑋 · 𝑢) = 1 ∧ ∃𝑣 ∈ 𝐵 (𝑣 · 𝑋) = 1 )))) |
8 | 5, 7 | bitr4id 290 | . 2 ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ (∃𝑢 ∈ 𝐵 (𝑋 · 𝑢) = 1 ∧ ∃𝑣 ∈ 𝐵 (𝑣 · 𝑋) = 1 ))) |
9 | eqid 2734 | . . . 4 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
10 | 9, 1 | mgpbas 20162 | . . 3 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅)) |
11 | 9, 4 | ringidval 20205 | . . 3 ⊢ 1 = (0g‘(mulGrp‘𝑅)) |
12 | 9, 3 | mgpplusg 20160 | . . 3 ⊢ · = (+g‘(mulGrp‘𝑅)) |
13 | isunit3.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
14 | 9 | ringmgp 20261 | . . . 4 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd) |
15 | 13, 14 | syl 17 | . . 3 ⊢ (𝜑 → (mulGrp‘𝑅) ∈ Mnd) |
16 | 10, 11, 12, 15, 6 | mndlrinvb 33003 | . 2 ⊢ (𝜑 → ((∃𝑢 ∈ 𝐵 (𝑋 · 𝑢) = 1 ∧ ∃𝑣 ∈ 𝐵 (𝑣 · 𝑋) = 1 ) ↔ ∃𝑦 ∈ 𝐵 ((𝑋 · 𝑦) = 1 ∧ (𝑦 · 𝑋) = 1 ))) |
17 | 8, 16 | bitrd 279 | 1 ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ ∃𝑦 ∈ 𝐵 ((𝑋 · 𝑦) = 1 ∧ (𝑦 · 𝑋) = 1 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2103 ∃wrex 3072 ‘cfv 6572 (class class class)co 7445 Basecbs 17253 .rcmulr 17307 Mndcmnd 18767 mulGrpcmgp 20156 1rcur 20203 Ringcrg 20255 Unitcui 20376 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-2nd 8027 df-tpos 8263 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-2 12352 df-3 12353 df-sets 17206 df-slot 17224 df-ndx 17236 df-base 17254 df-plusg 17319 df-mulr 17320 df-0g 17496 df-mgm 18673 df-sgrp 18752 df-mnd 18768 df-mgp 20157 df-ur 20204 df-ring 20257 df-oppr 20355 df-dvdsr 20378 df-unit 20379 |
This theorem is referenced by: assarrginv 33641 |
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