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Mirrors > Home > MPE Home > Th. List > Mathboxes > algstr | Structured version Visualization version GIF version |
Description: Lemma to shorten proofs of algbase 43075 through algvsca 43079. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.) |
Ref | Expression |
---|---|
algpart.a | ⊢ 𝐴 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉}) |
Ref | Expression |
---|---|
algstr | ⊢ 𝐴 Struct 〈1, 6〉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | algpart.a | . 2 ⊢ 𝐴 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉}) | |
2 | eqid 2734 | . . . 4 ⊢ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} | |
3 | 2 | rngstr 17352 | . . 3 ⊢ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} Struct 〈1, 3〉 |
4 | 5nn 12375 | . . . 4 ⊢ 5 ∈ ℕ | |
5 | scandx 17368 | . . . 4 ⊢ (Scalar‘ndx) = 5 | |
6 | 5lt6 12470 | . . . 4 ⊢ 5 < 6 | |
7 | 6nn 12378 | . . . 4 ⊢ 6 ∈ ℕ | |
8 | vscandx 17373 | . . . 4 ⊢ ( ·𝑠 ‘ndx) = 6 | |
9 | 4, 5, 6, 7, 8 | strle2 17201 | . . 3 ⊢ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉} Struct 〈5, 6〉 |
10 | 3lt5 12467 | . . 3 ⊢ 3 < 5 | |
11 | 3, 9, 10 | strleun 17199 | . 2 ⊢ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉}) Struct 〈1, 6〉 |
12 | 1, 11 | eqbrtri 5190 | 1 ⊢ 𝐴 Struct 〈1, 6〉 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∪ cun 3968 {cpr 4650 {ctp 4652 〈cop 4654 class class class wbr 5169 ‘cfv 6572 1c1 11181 3c3 12345 5c5 12347 6c6 12348 Struct cstr 17188 ndxcnx 17235 Basecbs 17253 +gcplusg 17306 .rcmulr 17307 Scalarcsca 17309 ·𝑠 cvsca 17310 |
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-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-tp 4653 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-1st 8026 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 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-4 12354 df-5 12355 df-6 12356 df-n0 12550 df-z 12636 df-uz 12900 df-fz 13564 df-struct 17189 df-slot 17224 df-ndx 17236 df-base 17254 df-plusg 17319 df-mulr 17320 df-sca 17322 df-vsca 17323 |
This theorem is referenced by: algbase 43075 algaddg 43076 algmulr 43077 algsca 43078 algvsca 43079 |
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