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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isunit2 | 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) |
Ref | Expression |
---|---|
isunit2 | ⊢ (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ (∃𝑢 ∈ 𝐵 (𝑋 · 𝑢) = 1 ∧ ∃𝑣 ∈ 𝐵 (𝑣 · 𝑋) = 1 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isunit2.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
2 | eqid 2734 | . . . 4 ⊢ (∥r‘𝑅) = (∥r‘𝑅) | |
3 | isunit2.m | . . . 4 ⊢ · = (.r‘𝑅) | |
4 | 1, 2, 3 | dvdsr 20383 | . . 3 ⊢ (𝑋(∥r‘𝑅) 1 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑣 ∈ 𝐵 (𝑣 · 𝑋) = 1 )) |
5 | eqid 2734 | . . . . . 6 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
6 | 5, 1 | opprbas 20362 | . . . . 5 ⊢ 𝐵 = (Base‘(oppr‘𝑅)) |
7 | eqid 2734 | . . . . 5 ⊢ (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅)) | |
8 | eqid 2734 | . . . . 5 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅)) | |
9 | 6, 7, 8 | dvdsr 20383 | . . . 4 ⊢ (𝑋(∥r‘(oppr‘𝑅)) 1 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑢 ∈ 𝐵 (𝑢(.r‘(oppr‘𝑅))𝑋) = 1 )) |
10 | 1, 3, 5, 8 | opprmul 20358 | . . . . . . 7 ⊢ (𝑢(.r‘(oppr‘𝑅))𝑋) = (𝑋 · 𝑢) |
11 | 10 | eqeq1i 2739 | . . . . . 6 ⊢ ((𝑢(.r‘(oppr‘𝑅))𝑋) = 1 ↔ (𝑋 · 𝑢) = 1 ) |
12 | 11 | rexbii 3096 | . . . . 5 ⊢ (∃𝑢 ∈ 𝐵 (𝑢(.r‘(oppr‘𝑅))𝑋) = 1 ↔ ∃𝑢 ∈ 𝐵 (𝑋 · 𝑢) = 1 ) |
13 | 12 | anbi2i 622 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ ∃𝑢 ∈ 𝐵 (𝑢(.r‘(oppr‘𝑅))𝑋) = 1 ) ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑢 ∈ 𝐵 (𝑋 · 𝑢) = 1 )) |
14 | 9, 13 | bitri 275 | . . 3 ⊢ (𝑋(∥r‘(oppr‘𝑅)) 1 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑢 ∈ 𝐵 (𝑋 · 𝑢) = 1 )) |
15 | 4, 14 | anbi12ci 628 | . 2 ⊢ ((𝑋(∥r‘𝑅) 1 ∧ 𝑋(∥r‘(oppr‘𝑅)) 1 ) ↔ ((𝑋 ∈ 𝐵 ∧ ∃𝑢 ∈ 𝐵 (𝑋 · 𝑢) = 1 ) ∧ (𝑋 ∈ 𝐵 ∧ ∃𝑣 ∈ 𝐵 (𝑣 · 𝑋) = 1 ))) |
16 | isunit2.u | . . 3 ⊢ 𝑈 = (Unit‘𝑅) | |
17 | isunit2.1 | . . 3 ⊢ 1 = (1r‘𝑅) | |
18 | 16, 17, 2, 5, 7 | isunit 20394 | . 2 ⊢ (𝑋 ∈ 𝑈 ↔ (𝑋(∥r‘𝑅) 1 ∧ 𝑋(∥r‘(oppr‘𝑅)) 1 )) |
19 | anandi 675 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ (∃𝑢 ∈ 𝐵 (𝑋 · 𝑢) = 1 ∧ ∃𝑣 ∈ 𝐵 (𝑣 · 𝑋) = 1 )) ↔ ((𝑋 ∈ 𝐵 ∧ ∃𝑢 ∈ 𝐵 (𝑋 · 𝑢) = 1 ) ∧ (𝑋 ∈ 𝐵 ∧ ∃𝑣 ∈ 𝐵 (𝑣 · 𝑋) = 1 ))) | |
20 | 15, 18, 19 | 3bitr4i 303 | 1 ⊢ (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ (∃𝑢 ∈ 𝐵 (𝑋 · 𝑢) = 1 ∧ ∃𝑣 ∈ 𝐵 (𝑣 · 𝑋) = 1 ))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2103 ∃wrex 3072 class class class wbr 5169 ‘cfv 6572 (class class class)co 7445 Basecbs 17253 .rcmulr 17307 1rcur 20203 opprcoppr 20354 ∥rcdsr 20375 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-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-mulr 17320 df-oppr 20355 df-dvdsr 20378 df-unit 20379 |
This theorem is referenced by: isunit3 33213 |
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