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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlhilslem | Structured version Visualization version GIF version |
Description: Lemma for hlhilsbase2 38080. (Contributed by Mario Carneiro, 28-Jun-2015.) |
Ref | Expression |
---|---|
hlhilslem.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hlhilslem.e | ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊) |
hlhilslem.u | ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) |
hlhilslem.r | ⊢ 𝑅 = (Scalar‘𝑈) |
hlhilslem.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hlhilslem.f | ⊢ 𝐹 = Slot 𝑁 |
hlhilslem.1 | ⊢ 𝑁 ∈ ℕ |
hlhilslem.2 | ⊢ 𝑁 < 4 |
hlhilslem.c | ⊢ 𝐶 = (𝐹‘𝐸) |
Ref | Expression |
---|---|
hlhilslem | ⊢ (𝜑 → 𝐶 = (𝐹‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlhilslem.c | . . 3 ⊢ 𝐶 = (𝐹‘𝐸) | |
2 | hlhilslem.f | . . . . 5 ⊢ 𝐹 = Slot 𝑁 | |
3 | hlhilslem.1 | . . . . 5 ⊢ 𝑁 ∈ ℕ | |
4 | 2, 3 | ndxid 16281 | . . . 4 ⊢ 𝐹 = Slot (𝐹‘ndx) |
5 | 3 | nnrei 11384 | . . . . . 6 ⊢ 𝑁 ∈ ℝ |
6 | hlhilslem.2 | . . . . . 6 ⊢ 𝑁 < 4 | |
7 | 5, 6 | ltneii 10489 | . . . . 5 ⊢ 𝑁 ≠ 4 |
8 | 2, 3 | ndxarg 16280 | . . . . . 6 ⊢ (𝐹‘ndx) = 𝑁 |
9 | starvndx 16396 | . . . . . 6 ⊢ (*𝑟‘ndx) = 4 | |
10 | 8, 9 | neeq12i 3034 | . . . . 5 ⊢ ((𝐹‘ndx) ≠ (*𝑟‘ndx) ↔ 𝑁 ≠ 4) |
11 | 7, 10 | mpbir 223 | . . . 4 ⊢ (𝐹‘ndx) ≠ (*𝑟‘ndx) |
12 | 4, 11 | setsnid 16311 | . . 3 ⊢ (𝐹‘𝐸) = (𝐹‘(𝐸 sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉)) |
13 | 1, 12 | eqtri 2801 | . 2 ⊢ 𝐶 = (𝐹‘(𝐸 sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉)) |
14 | hlhilslem.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
15 | hlhilslem.u | . . . . 5 ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) | |
16 | hlhilslem.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
17 | hlhilslem.e | . . . . 5 ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊) | |
18 | eqid 2777 | . . . . 5 ⊢ ((HGMap‘𝐾)‘𝑊) = ((HGMap‘𝐾)‘𝑊) | |
19 | eqid 2777 | . . . . 5 ⊢ (𝐸 sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉) = (𝐸 sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉) | |
20 | 14, 15, 16, 17, 18, 19 | hlhilsca 38073 | . . . 4 ⊢ (𝜑 → (𝐸 sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉) = (Scalar‘𝑈)) |
21 | hlhilslem.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑈) | |
22 | 20, 21 | syl6eqr 2831 | . . 3 ⊢ (𝜑 → (𝐸 sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉) = 𝑅) |
23 | 22 | fveq2d 6450 | . 2 ⊢ (𝜑 → (𝐹‘(𝐸 sSet 〈(*𝑟‘ndx), ((HGMap‘𝐾)‘𝑊)〉)) = (𝐹‘𝑅)) |
24 | 13, 23 | syl5eq 2825 | 1 ⊢ (𝜑 → 𝐶 = (𝐹‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 〈cop 4403 class class class wbr 4886 ‘cfv 6135 (class class class)co 6922 < clt 10411 ℕcn 11374 4c4 11432 ndxcnx 16252 sSet csts 16253 Slot cslot 16254 *𝑟cstv 16340 Scalarcsca 16341 HLchlt 35488 LHypclh 36122 EDRingcedring 36891 HGMapchg 38021 HLHilchlh 38070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-plusg 16351 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-hlhil 38071 |
This theorem is referenced by: hlhilsbase 38077 hlhilsplus 38078 hlhilsmul 38079 |
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