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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap1eulemOLDN | Structured version Visualization version GIF version |
Description: Lemma for hdmap1euOLDN 40633. TODO: combine with hdmap1euOLDN 40633 or at least share some hypotheses. (Contributed by NM, 15-May-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hdmap1eulem.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmap1eulem.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmap1eulem.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmap1eulem.s | ⊢ − = (-g‘𝑈) |
hdmap1eulem.o | ⊢ 0 = (0g‘𝑈) |
hdmap1eulem.n | ⊢ 𝑁 = (LSpan‘𝑈) |
hdmap1eulem.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmap1eulem.d | ⊢ 𝐷 = (Base‘𝐶) |
hdmap1eulem.r | ⊢ 𝑅 = (-g‘𝐶) |
hdmap1eulem.q | ⊢ 𝑄 = (0g‘𝐶) |
hdmap1eulem.j | ⊢ 𝐽 = (LSpan‘𝐶) |
hdmap1eulem.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
hdmap1eulem.i | ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) |
hdmap1eulem.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmap1eulem.mn | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
hdmap1eulem.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
hdmap1eulem.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) |
hdmap1eulem.y | ⊢ (𝜑 → 𝑇 ∈ 𝑉) |
hdmap1eulem.l | ⊢ 𝐿 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))))) |
Ref | Expression |
---|---|
hdmap1eulemOLDN | ⊢ (𝜑 → ∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmap1eulem.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmap1eulem.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hdmap1eulem.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
4 | hdmap1eulem.s | . . 3 ⊢ − = (-g‘𝑈) | |
5 | hdmap1eulem.o | . . 3 ⊢ 0 = (0g‘𝑈) | |
6 | hdmap1eulem.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑈) | |
7 | hdmap1eulem.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
8 | hdmap1eulem.d | . . 3 ⊢ 𝐷 = (Base‘𝐶) | |
9 | hdmap1eulem.r | . . 3 ⊢ 𝑅 = (-g‘𝐶) | |
10 | hdmap1eulem.q | . . 3 ⊢ 𝑄 = (0g‘𝐶) | |
11 | hdmap1eulem.j | . . 3 ⊢ 𝐽 = (LSpan‘𝐶) | |
12 | hdmap1eulem.m | . . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
13 | hdmap1eulem.l | . . 3 ⊢ 𝐿 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))))) | |
14 | hdmap1eulem.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
15 | hdmap1eulem.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
16 | hdmap1eulem.mn | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) | |
17 | hdmap1eulem.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
18 | hdmap1eulem.y | . . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑉) | |
19 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 | mapdh9aOLDN 40598 | . 2 ⊢ (𝜑 → ∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → 𝑦 = (𝐿‘〈𝑧, (𝐿‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉))) |
20 | hdmap1eulem.i | . . . . . . . . . 10 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) | |
21 | 14 | ad2antrr 725 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
22 | 17 | ad2antrr 725 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
23 | 15 | ad2antrr 725 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝐹 ∈ 𝐷) |
24 | simplr 768 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑧 ∈ 𝑉) | |
25 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 20, 21, 22, 23, 24, 13 | hdmap1valc 40611 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → (𝐼‘〈𝑋, 𝐹, 𝑧〉) = (𝐿‘〈𝑋, 𝐹, 𝑧〉)) |
26 | 25 | oteq2d 4884 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉 = 〈𝑧, (𝐿‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉) |
27 | 26 | fveq2d 6891 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉) = (𝐼‘〈𝑧, (𝐿‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉)) |
28 | eqid 2733 | . . . . . . . . 9 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
29 | 1, 2, 14 | dvhlmod 39918 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ LMod) |
30 | 29 | ad2antrr 725 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑈 ∈ LMod) |
31 | 17 | eldifad 3958 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
32 | 3, 28, 6, 29, 31, 18 | lspprcl 20576 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑇}) ∈ (LSubSp‘𝑈)) |
33 | 32 | ad2antrr 725 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → (𝑁‘{𝑋, 𝑇}) ∈ (LSubSp‘𝑈)) |
34 | simpr 486 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) | |
35 | 5, 28, 30, 33, 24, 34 | lssneln0 20550 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑧 ∈ (𝑉 ∖ { 0 })) |
36 | 16 | ad2antrr 725 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
37 | 1, 2, 14 | dvhlvec 39917 | . . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈 ∈ LVec) |
38 | 37 | ad2antrr 725 | . . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑈 ∈ LVec) |
39 | 31 | ad2antrr 725 | . . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑋 ∈ 𝑉) |
40 | 18 | ad2antrr 725 | . . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑇 ∈ 𝑉) |
41 | 3, 6, 38, 24, 39, 40, 34 | lspindpi 20732 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) |
42 | 41 | simpld 496 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑋})) |
43 | 42 | necomd 2997 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑧})) |
44 | 10, 13, 1, 12, 2, 3, 4, 5, 6, 7, 8, 9, 11, 21, 23, 36, 22, 24, 43 | mapdhcl 40535 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → (𝐿‘〈𝑋, 𝐹, 𝑧〉) ∈ 𝐷) |
45 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 20, 21, 35, 44, 40, 13 | hdmap1valc 40611 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → (𝐼‘〈𝑧, (𝐿‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉) = (𝐿‘〈𝑧, (𝐿‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉)) |
46 | 27, 45 | eqtrd 2773 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉) = (𝐿‘〈𝑧, (𝐿‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉)) |
47 | 46 | eqeq2d 2744 | . . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → (𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉) ↔ 𝑦 = (𝐿‘〈𝑧, (𝐿‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉))) |
48 | 47 | pm5.74da 803 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉) → ((¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉)) ↔ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → 𝑦 = (𝐿‘〈𝑧, (𝐿‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉)))) |
49 | 48 | ralbidva 3176 | . . 3 ⊢ (𝜑 → (∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉)) ↔ ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → 𝑦 = (𝐿‘〈𝑧, (𝐿‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉)))) |
50 | 49 | reubidv 3395 | . 2 ⊢ (𝜑 → (∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉)) ↔ ∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → 𝑦 = (𝐿‘〈𝑧, (𝐿‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉)))) |
51 | 19, 50 | mpbird 257 | 1 ⊢ (𝜑 → ∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃!wreu 3375 Vcvv 3475 ∖ cdif 3943 ifcif 4526 {csn 4626 {cpr 4628 〈cotp 4634 ↦ cmpt 5229 ‘cfv 6539 ℩crio 7358 (class class class)co 7403 1st c1st 7967 2nd c2nd 7968 Basecbs 17139 0gc0g 17380 -gcsg 18816 LModclmod 20458 LSubSpclss 20529 LSpanclspn 20569 LVecclvec 20700 HLchlt 38157 LHypclh 38792 DVecHcdvh 39886 LCDualclcd 40394 mapdcmpd 40432 HDMap1chdma1 40599 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5283 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-riotaBAD 37760 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-ot 4635 df-uni 4907 df-int 4949 df-iun 4997 df-iin 4998 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-tpos 8205 df-undef 8252 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-1o 8460 df-er 8698 df-map 8817 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-nn 12208 df-2 12270 df-3 12271 df-4 12272 df-5 12273 df-6 12274 df-n0 12468 df-z 12554 df-uz 12818 df-fz 13480 df-struct 17075 df-sets 17092 df-slot 17110 df-ndx 17122 df-base 17140 df-ress 17169 df-plusg 17205 df-mulr 17206 df-sca 17208 df-vsca 17209 df-0g 17382 df-mre 17525 df-mrc 17526 df-acs 17528 df-proset 18243 df-poset 18261 df-plt 18278 df-lub 18294 df-glb 18295 df-join 18296 df-meet 18297 df-p0 18373 df-p1 18374 df-lat 18380 df-clat 18447 df-mgm 18556 df-sgrp 18605 df-mnd 18621 df-submnd 18667 df-grp 18817 df-minusg 18818 df-sbg 18819 df-subg 18996 df-cntz 19174 df-oppg 19202 df-lsm 19496 df-cmn 19642 df-abl 19643 df-mgp 19979 df-ur 19996 df-ring 20048 df-oppr 20138 df-dvdsr 20159 df-unit 20160 df-invr 20190 df-dvr 20203 df-drng 20305 df-lmod 20460 df-lss 20530 df-lsp 20570 df-lvec 20701 df-lsatoms 37783 df-lshyp 37784 df-lcv 37826 df-lfl 37865 df-lkr 37893 df-ldual 37931 df-oposet 37983 df-ol 37985 df-oml 37986 df-covers 38073 df-ats 38074 df-atl 38105 df-cvlat 38129 df-hlat 38158 df-llines 38306 df-lplanes 38307 df-lvols 38308 df-lines 38309 df-psubsp 38311 df-pmap 38312 df-padd 38604 df-lhyp 38796 df-laut 38797 df-ldil 38912 df-ltrn 38913 df-trl 38967 df-tgrp 39551 df-tendo 39563 df-edring 39565 df-dveca 39811 df-disoa 39837 df-dvech 39887 df-dib 39947 df-dic 39981 df-dih 40037 df-doch 40156 df-djh 40203 df-lcdual 40395 df-mapd 40433 df-hdmap1 40601 |
This theorem is referenced by: hdmap1euOLDN 40633 |
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