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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem23 | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 38392. Baer p. 45, line 10: "and so y' meets all our requirements." Our ℎ is Baer's y'. (Contributed by NM, 20-Mar-2015.) |
Ref | Expression |
---|---|
mapdpglem.h | ⊢ 𝐻 = (LHyp‘𝐾) |
mapdpglem.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
mapdpglem.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
mapdpglem.v | ⊢ 𝑉 = (Base‘𝑈) |
mapdpglem.s | ⊢ − = (-g‘𝑈) |
mapdpglem.n | ⊢ 𝑁 = (LSpan‘𝑈) |
mapdpglem.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
mapdpglem.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
mapdpglem.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
mapdpglem.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
mapdpglem1.p | ⊢ ⊕ = (LSSum‘𝐶) |
mapdpglem2.j | ⊢ 𝐽 = (LSpan‘𝐶) |
mapdpglem3.f | ⊢ 𝐹 = (Base‘𝐶) |
mapdpglem3.te | ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) |
mapdpglem3.a | ⊢ 𝐴 = (Scalar‘𝑈) |
mapdpglem3.b | ⊢ 𝐵 = (Base‘𝐴) |
mapdpglem3.t | ⊢ · = ( ·𝑠 ‘𝐶) |
mapdpglem3.r | ⊢ 𝑅 = (-g‘𝐶) |
mapdpglem3.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
mapdpglem3.e | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) |
mapdpglem4.q | ⊢ 𝑄 = (0g‘𝑈) |
mapdpglem.ne | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
mapdpglem4.jt | ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) |
mapdpglem4.z | ⊢ 0 = (0g‘𝐴) |
mapdpglem4.g4 | ⊢ (𝜑 → 𝑔 ∈ 𝐵) |
mapdpglem4.z4 | ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) |
mapdpglem4.t4 | ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) |
mapdpglem4.xn | ⊢ (𝜑 → 𝑋 ≠ 𝑄) |
mapdpglem12.yn | ⊢ (𝜑 → 𝑌 ≠ 𝑄) |
mapdpglem17.ep | ⊢ 𝐸 = (((invr‘𝐴)‘𝑔) · 𝑧) |
Ref | Expression |
---|---|
mapdpglem23 | ⊢ (𝜑 → ∃ℎ ∈ 𝐹 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdpglem.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | mapdpglem.m | . . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
3 | mapdpglem.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | eqid 2795 | . . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
5 | mapdpglem.c | . . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
6 | eqid 2795 | . . . 4 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶) | |
7 | mapdpglem.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
8 | 1, 3, 7 | dvhlmod 37796 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
9 | mapdpglem.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
10 | mapdpglem.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑈) | |
11 | mapdpglem.n | . . . . . 6 ⊢ 𝑁 = (LSpan‘𝑈) | |
12 | 10, 4, 11 | lspsncl 19439 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
13 | 8, 9, 12 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
14 | 1, 2, 3, 4, 5, 6, 7, 13 | mapdcl2 38342 | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶)) |
15 | mapdpglem.s | . . . 4 ⊢ − = (-g‘𝑈) | |
16 | mapdpglem.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
17 | mapdpglem1.p | . . . 4 ⊢ ⊕ = (LSSum‘𝐶) | |
18 | mapdpglem2.j | . . . 4 ⊢ 𝐽 = (LSpan‘𝐶) | |
19 | mapdpglem3.f | . . . 4 ⊢ 𝐹 = (Base‘𝐶) | |
20 | mapdpglem3.te | . . . 4 ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) | |
21 | mapdpglem3.a | . . . 4 ⊢ 𝐴 = (Scalar‘𝑈) | |
22 | mapdpglem3.b | . . . 4 ⊢ 𝐵 = (Base‘𝐴) | |
23 | mapdpglem3.t | . . . 4 ⊢ · = ( ·𝑠 ‘𝐶) | |
24 | mapdpglem3.r | . . . 4 ⊢ 𝑅 = (-g‘𝐶) | |
25 | mapdpglem3.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
26 | mapdpglem3.e | . . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) | |
27 | mapdpglem4.q | . . . 4 ⊢ 𝑄 = (0g‘𝑈) | |
28 | mapdpglem.ne | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
29 | mapdpglem4.jt | . . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) | |
30 | mapdpglem4.z | . . . 4 ⊢ 0 = (0g‘𝐴) | |
31 | mapdpglem4.g4 | . . . 4 ⊢ (𝜑 → 𝑔 ∈ 𝐵) | |
32 | mapdpglem4.z4 | . . . 4 ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) | |
33 | mapdpglem4.t4 | . . . 4 ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) | |
34 | mapdpglem4.xn | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 𝑄) | |
35 | mapdpglem12.yn | . . . 4 ⊢ (𝜑 → 𝑌 ≠ 𝑄) | |
36 | mapdpglem17.ep | . . . 4 ⊢ 𝐸 = (((invr‘𝐴)‘𝑔) · 𝑧) | |
37 | 1, 2, 3, 10, 15, 11, 5, 7, 16, 9, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36 | mapdpglem19 38376 | . . 3 ⊢ (𝜑 → 𝐸 ∈ (𝑀‘(𝑁‘{𝑌}))) |
38 | 19, 6 | lssel 19399 | . . 3 ⊢ (((𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶) ∧ 𝐸 ∈ (𝑀‘(𝑁‘{𝑌}))) → 𝐸 ∈ 𝐹) |
39 | 14, 37, 38 | syl2anc 584 | . 2 ⊢ (𝜑 → 𝐸 ∈ 𝐹) |
40 | 1, 2, 3, 10, 15, 11, 5, 7, 16, 9, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36 | mapdpglem20 38377 | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐸})) |
41 | 1, 2, 3, 10, 15, 11, 5, 7, 16, 9, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36 | mapdpglem22 38379 | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝐸)})) |
42 | sneq 4482 | . . . . . 6 ⊢ (ℎ = 𝐸 → {ℎ} = {𝐸}) | |
43 | 42 | fveq2d 6542 | . . . . 5 ⊢ (ℎ = 𝐸 → (𝐽‘{ℎ}) = (𝐽‘{𝐸})) |
44 | 43 | eqeq2d 2805 | . . . 4 ⊢ (ℎ = 𝐸 → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ↔ (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐸}))) |
45 | oveq2 7024 | . . . . . . 7 ⊢ (ℎ = 𝐸 → (𝐺𝑅ℎ) = (𝐺𝑅𝐸)) | |
46 | 45 | sneqd 4484 | . . . . . 6 ⊢ (ℎ = 𝐸 → {(𝐺𝑅ℎ)} = {(𝐺𝑅𝐸)}) |
47 | 46 | fveq2d 6542 | . . . . 5 ⊢ (ℎ = 𝐸 → (𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝐸)})) |
48 | 47 | eqeq2d 2805 | . . . 4 ⊢ (ℎ = 𝐸 → ((𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)}) ↔ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝐸)}))) |
49 | 44, 48 | anbi12d 630 | . . 3 ⊢ (ℎ = 𝐸 → (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐸}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝐸)})))) |
50 | 49 | rspcev 3559 | . 2 ⊢ ((𝐸 ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐸}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝐸)}))) → ∃ℎ ∈ 𝐹 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)}))) |
51 | 39, 40, 41, 50 | syl12anc 833 | 1 ⊢ (𝜑 → ∃ℎ ∈ 𝐹 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 ∃wrex 3106 {csn 4472 ‘cfv 6225 (class class class)co 7016 Basecbs 16312 Scalarcsca 16397 ·𝑠 cvsca 16398 0gc0g 16542 -gcsg 17863 LSSumclsm 18489 invrcinvr 19111 LModclmod 19324 LSubSpclss 19393 LSpanclspn 19433 HLchlt 36036 LHypclh 36670 DVecHcdvh 37764 LCDualclcd 38272 mapdcmpd 38310 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-riotaBAD 35639 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-iin 4828 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-of 7267 df-om 7437 df-1st 7545 df-2nd 7546 df-tpos 7743 df-undef 7790 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-oadd 7957 df-er 8139 df-map 8258 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-n0 11746 df-z 11830 df-uz 12094 df-fz 12743 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 df-plusg 16407 df-mulr 16408 df-sca 16410 df-vsca 16411 df-0g 16544 df-mre 16686 df-mrc 16687 df-acs 16689 df-proset 17367 df-poset 17385 df-plt 17397 df-lub 17413 df-glb 17414 df-join 17415 df-meet 17416 df-p0 17478 df-p1 17479 df-lat 17485 df-clat 17547 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-submnd 17775 df-grp 17864 df-minusg 17865 df-sbg 17866 df-subg 18030 df-cntz 18188 df-oppg 18215 df-lsm 18491 df-cmn 18635 df-abl 18636 df-mgp 18930 df-ur 18942 df-ring 18989 df-oppr 19063 df-dvdsr 19081 df-unit 19082 df-invr 19112 df-dvr 19123 df-drng 19194 df-lmod 19326 df-lss 19394 df-lsp 19434 df-lvec 19565 df-lsatoms 35662 df-lshyp 35663 df-lcv 35705 df-lfl 35744 df-lkr 35772 df-ldual 35810 df-oposet 35862 df-ol 35864 df-oml 35865 df-covers 35952 df-ats 35953 df-atl 35984 df-cvlat 36008 df-hlat 36037 df-llines 36184 df-lplanes 36185 df-lvols 36186 df-lines 36187 df-psubsp 36189 df-pmap 36190 df-padd 36482 df-lhyp 36674 df-laut 36675 df-ldil 36790 df-ltrn 36791 df-trl 36845 df-tgrp 37429 df-tendo 37441 df-edring 37443 df-dveca 37689 df-disoa 37715 df-dvech 37765 df-dib 37825 df-dic 37859 df-dih 37915 df-doch 38034 df-djh 38081 df-lcdual 38273 df-mapd 38311 |
This theorem is referenced by: mapdpglem24 38390 |
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