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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmapf1oN | Structured version Visualization version GIF version |
Description: Part 12 in [Baer] p. 49. The map from vectors to functionals with closed kernels maps one-to-one onto. Combined with hdmapadd 41178, this shows the map is an automorphism from the additive group of vectors to the additive group of functionals with closed kernels. (Contributed by NM, 30-May-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hdmapf1o.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmapf1o.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmapf1o.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmapf1o.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmapf1o.d | ⊢ 𝐷 = (Base‘𝐶) |
hdmapf1o.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmapf1o.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
Ref | Expression |
---|---|
hdmapf1oN | ⊢ (𝜑 → 𝑆:𝑉–1-1-onto→𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmapf1o.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmapf1o.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hdmapf1o.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
4 | hdmapf1o.s | . . 3 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
5 | hdmapf1o.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
6 | 1, 2, 3, 4, 5 | hdmapfnN 41164 | . 2 ⊢ (𝜑 → 𝑆 Fn 𝑉) |
7 | hdmapf1o.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
8 | hdmapf1o.d | . . 3 ⊢ 𝐷 = (Base‘𝐶) | |
9 | 1, 7, 8, 4, 5 | hdmaprnN 41199 | . 2 ⊢ (𝜑 → ran 𝑆 = 𝐷) |
10 | 5 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
11 | simprl 768 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑥 ∈ 𝑉) | |
12 | simprr 770 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑦 ∈ 𝑉) | |
13 | 1, 2, 3, 4, 10, 11, 12 | hdmap11 41183 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ((𝑆‘𝑥) = (𝑆‘𝑦) ↔ 𝑥 = 𝑦)) |
14 | 13 | biimpd 228 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ((𝑆‘𝑥) = (𝑆‘𝑦) → 𝑥 = 𝑦)) |
15 | 14 | ralrimivva 3199 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ((𝑆‘𝑥) = (𝑆‘𝑦) → 𝑥 = 𝑦)) |
16 | dff1o6 7276 | . 2 ⊢ (𝑆:𝑉–1-1-onto→𝐷 ↔ (𝑆 Fn 𝑉 ∧ ran 𝑆 = 𝐷 ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ((𝑆‘𝑥) = (𝑆‘𝑦) → 𝑥 = 𝑦))) | |
17 | 6, 9, 15, 16 | syl3anbrc 1342 | 1 ⊢ (𝜑 → 𝑆:𝑉–1-1-onto→𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∀wral 3060 ran crn 5677 Fn wfn 6538 –1-1-onto→wf1o 6542 ‘cfv 6543 Basecbs 17151 HLchlt 38684 LHypclh 39319 DVecHcdvh 40413 LCDualclcd 40921 HDMapchdma 41127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-riotaBAD 38287 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-ot 4637 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8217 df-undef 8264 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-n0 12480 df-z 12566 df-uz 12830 df-fz 13492 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-0g 17394 df-mre 17537 df-mrc 17538 df-acs 17540 df-proset 18258 df-poset 18276 df-plt 18293 df-lub 18309 df-glb 18310 df-join 18311 df-meet 18312 df-p0 18388 df-p1 18389 df-lat 18395 df-clat 18462 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19046 df-cntz 19229 df-oppg 19258 df-lsm 19552 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-dvr 20299 df-drng 20585 df-lmod 20704 df-lss 20775 df-lsp 20815 df-lvec 20947 df-lsatoms 38310 df-lshyp 38311 df-lcv 38353 df-lfl 38392 df-lkr 38420 df-ldual 38458 df-oposet 38510 df-ol 38512 df-oml 38513 df-covers 38600 df-ats 38601 df-atl 38632 df-cvlat 38656 df-hlat 38685 df-llines 38833 df-lplanes 38834 df-lvols 38835 df-lines 38836 df-psubsp 38838 df-pmap 38839 df-padd 39131 df-lhyp 39323 df-laut 39324 df-ldil 39439 df-ltrn 39440 df-trl 39494 df-tgrp 40078 df-tendo 40090 df-edring 40092 df-dveca 40338 df-disoa 40364 df-dvech 40414 df-dib 40474 df-dic 40508 df-dih 40564 df-doch 40683 df-djh 40730 df-lcdual 40922 df-mapd 40960 df-hvmap 41092 df-hdmap1 41128 df-hdmap 41129 |
This theorem is referenced by: (None) |
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