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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlhilsrnglem | Structured version Visualization version GIF version |
Description: Lemma for hlhilsrng 39094. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
Ref | Expression |
---|---|
hlhillvec.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hlhillvec.u | ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) |
hlhillvec.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hlhildrng.r | ⊢ 𝑅 = (Scalar‘𝑈) |
hlhilsrng.l | ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) |
hlhilsrng.s | ⊢ 𝑆 = (Scalar‘𝐿) |
hlhilsrng.b | ⊢ 𝐵 = (Base‘𝑆) |
hlhilsrng.p | ⊢ + = (+g‘𝑆) |
hlhilsrng.t | ⊢ · = (.r‘𝑆) |
hlhilsrng.g | ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
hlhilsrnglem | ⊢ (𝜑 → 𝑅 ∈ *-Ring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlhillvec.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hlhilsrng.l | . . 3 ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) | |
3 | hlhilsrng.s | . . 3 ⊢ 𝑆 = (Scalar‘𝐿) | |
4 | hlhillvec.u | . . 3 ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) | |
5 | hlhildrng.r | . . 3 ⊢ 𝑅 = (Scalar‘𝑈) | |
6 | hlhillvec.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
7 | hlhilsrng.b | . . 3 ⊢ 𝐵 = (Base‘𝑆) | |
8 | 1, 2, 3, 4, 5, 6, 7 | hlhilsbase2 39082 | . 2 ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) |
9 | hlhilsrng.p | . . 3 ⊢ + = (+g‘𝑆) | |
10 | 1, 2, 3, 4, 5, 6, 9 | hlhilsplus2 39083 | . 2 ⊢ (𝜑 → + = (+g‘𝑅)) |
11 | hlhilsrng.t | . . 3 ⊢ · = (.r‘𝑆) | |
12 | 1, 2, 3, 4, 5, 6, 11 | hlhilsmul2 39084 | . 2 ⊢ (𝜑 → · = (.r‘𝑅)) |
13 | hlhilsrng.g | . . 3 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) | |
14 | 1, 4, 5, 13, 6 | hlhilnvl 39090 | . 2 ⊢ (𝜑 → 𝐺 = (*𝑟‘𝑅)) |
15 | 1, 4, 6, 5 | hlhildrng 39092 | . . 3 ⊢ (𝜑 → 𝑅 ∈ DivRing) |
16 | drngring 19512 | . . 3 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
17 | 15, 16 | syl 17 | . 2 ⊢ (𝜑 → 𝑅 ∈ Ring) |
18 | 6 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
19 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
20 | 1, 2, 3, 7, 13, 18, 19 | hgmapcl 39029 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) ∈ 𝐵) |
21 | 6 | 3ad2ant1 1129 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
22 | simp2 1133 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
23 | simp3 1134 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵) | |
24 | 1, 2, 3, 7, 9, 13, 21, 22, 23 | hgmapadd 39034 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝐺‘(𝑥 + 𝑦)) = ((𝐺‘𝑥) + (𝐺‘𝑦))) |
25 | 1, 2, 3, 7, 11, 13, 21, 22, 23 | hgmapmul 39035 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝐺‘(𝑥 · 𝑦)) = ((𝐺‘𝑦) · (𝐺‘𝑥))) |
26 | 1, 2, 3, 7, 13, 18, 19 | hgmapvv 39066 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘(𝐺‘𝑥)) = 𝑥) |
27 | 8, 10, 12, 14, 17, 20, 24, 25, 26 | issrngd 19635 | 1 ⊢ (𝜑 → 𝑅 ∈ *-Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ‘cfv 6358 Basecbs 16486 +gcplusg 16568 .rcmulr 16569 Scalarcsca 16571 Ringcrg 19300 DivRingcdr 19505 *-Ringcsr 19618 HLchlt 36490 LHypclh 37124 DVecHcdvh 38218 HGMapchg 39023 HLHilchlh 39072 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-riotaBAD 36093 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-ot 4579 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-tpos 7895 df-undef 7942 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-sca 16584 df-vsca 16585 df-ip 16586 df-0g 16718 df-mre 16860 df-mrc 16861 df-acs 16863 df-proset 17541 df-poset 17559 df-plt 17571 df-lub 17587 df-glb 17588 df-join 17589 df-meet 17590 df-p0 17652 df-p1 17653 df-lat 17659 df-clat 17721 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-mhm 17959 df-submnd 17960 df-grp 18109 df-minusg 18110 df-sbg 18111 df-subg 18279 df-ghm 18359 df-cntz 18450 df-oppg 18477 df-lsm 18764 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-ring 19302 df-oppr 19376 df-dvdsr 19394 df-unit 19395 df-invr 19425 df-dvr 19436 df-rnghom 19470 df-drng 19507 df-staf 19619 df-srng 19620 df-lmod 19639 df-lss 19707 df-lsp 19747 df-lvec 19878 df-lsatoms 36116 df-lshyp 36117 df-lcv 36159 df-lfl 36198 df-lkr 36226 df-ldual 36264 df-oposet 36316 df-ol 36318 df-oml 36319 df-covers 36406 df-ats 36407 df-atl 36438 df-cvlat 36462 df-hlat 36491 df-llines 36638 df-lplanes 36639 df-lvols 36640 df-lines 36641 df-psubsp 36643 df-pmap 36644 df-padd 36936 df-lhyp 37128 df-laut 37129 df-ldil 37244 df-ltrn 37245 df-trl 37299 df-tgrp 37883 df-tendo 37895 df-edring 37897 df-dveca 38143 df-disoa 38169 df-dvech 38219 df-dib 38279 df-dic 38313 df-dih 38369 df-doch 38488 df-djh 38535 df-lcdual 38727 df-mapd 38765 df-hvmap 38897 df-hdmap1 38933 df-hdmap 38934 df-hgmap 39024 df-hlhil 39073 |
This theorem is referenced by: hlhilsrng 39094 |
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