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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem31 | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 38844. Baer p. 45 line 19: "...and we have consequently that y' = y'', as we claimed." (Contributed by NM, 23-Mar-2015.) |
Ref | Expression |
---|---|
mapdpg.h | ⊢ 𝐻 = (LHyp‘𝐾) |
mapdpg.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
mapdpg.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
mapdpg.v | ⊢ 𝑉 = (Base‘𝑈) |
mapdpg.s | ⊢ − = (-g‘𝑈) |
mapdpg.z | ⊢ 0 = (0g‘𝑈) |
mapdpg.n | ⊢ 𝑁 = (LSpan‘𝑈) |
mapdpg.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
mapdpg.f | ⊢ 𝐹 = (Base‘𝐶) |
mapdpg.r | ⊢ 𝑅 = (-g‘𝐶) |
mapdpg.j | ⊢ 𝐽 = (LSpan‘𝐶) |
mapdpg.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
mapdpg.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
mapdpg.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
mapdpg.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
mapdpg.ne | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
mapdpg.e | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) |
mapdpgem25.h1 | ⊢ (𝜑 → (ℎ ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})))) |
mapdpgem25.i1 | ⊢ (𝜑 → (𝑖 ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) |
mapdpglem26.a | ⊢ 𝐴 = (Scalar‘𝑈) |
mapdpglem26.b | ⊢ 𝐵 = (Base‘𝐴) |
mapdpglem26.t | ⊢ · = ( ·𝑠 ‘𝐶) |
mapdpglem26.o | ⊢ 𝑂 = (0g‘𝐴) |
mapdpglem28.ve | ⊢ (𝜑 → 𝑣 ∈ 𝐵) |
mapdpglem28.u1 | ⊢ (𝜑 → ℎ = (𝑢 · 𝑖)) |
mapdpglem28.u2 | ⊢ (𝜑 → (𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖))) |
mapdpglem28.ue | ⊢ (𝜑 → 𝑢 ∈ 𝐵) |
Ref | Expression |
---|---|
mapdpglem31 | ⊢ (𝜑 → ℎ = 𝑖) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdpglem28.u1 | . 2 ⊢ (𝜑 → ℎ = (𝑢 · 𝑖)) | |
2 | mapdpg.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | mapdpg.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | mapdpglem26.a | . . . . 5 ⊢ 𝐴 = (Scalar‘𝑈) | |
5 | eqid 2823 | . . . . 5 ⊢ (1r‘𝐴) = (1r‘𝐴) | |
6 | mapdpg.c | . . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
7 | eqid 2823 | . . . . 5 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶) | |
8 | eqid 2823 | . . . . 5 ⊢ (1r‘(Scalar‘𝐶)) = (1r‘(Scalar‘𝐶)) | |
9 | mapdpg.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
10 | 2, 3, 4, 5, 6, 7, 8, 9 | lcd1 38747 | . . . 4 ⊢ (𝜑 → (1r‘(Scalar‘𝐶)) = (1r‘𝐴)) |
11 | 10 | oveq1d 7173 | . . 3 ⊢ (𝜑 → ((1r‘(Scalar‘𝐶)) · 𝑖) = ((1r‘𝐴) · 𝑖)) |
12 | 2, 6, 9 | lcdlmod 38730 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod) |
13 | mapdpgem25.i1 | . . . . 5 ⊢ (𝜑 → (𝑖 ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) | |
14 | 13 | simpld 497 | . . . 4 ⊢ (𝜑 → 𝑖 ∈ 𝐹) |
15 | mapdpg.f | . . . . 5 ⊢ 𝐹 = (Base‘𝐶) | |
16 | mapdpglem26.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝐶) | |
17 | 15, 7, 16, 8 | lmodvs1 19664 | . . . 4 ⊢ ((𝐶 ∈ LMod ∧ 𝑖 ∈ 𝐹) → ((1r‘(Scalar‘𝐶)) · 𝑖) = 𝑖) |
18 | 12, 14, 17 | syl2anc 586 | . . 3 ⊢ (𝜑 → ((1r‘(Scalar‘𝐶)) · 𝑖) = 𝑖) |
19 | mapdpg.m | . . . . . 6 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
20 | mapdpg.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑈) | |
21 | mapdpg.s | . . . . . 6 ⊢ − = (-g‘𝑈) | |
22 | mapdpg.z | . . . . . 6 ⊢ 0 = (0g‘𝑈) | |
23 | mapdpg.n | . . . . . 6 ⊢ 𝑁 = (LSpan‘𝑈) | |
24 | mapdpg.r | . . . . . 6 ⊢ 𝑅 = (-g‘𝐶) | |
25 | mapdpg.j | . . . . . 6 ⊢ 𝐽 = (LSpan‘𝐶) | |
26 | mapdpg.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
27 | mapdpg.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
28 | mapdpg.g | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
29 | mapdpg.ne | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
30 | mapdpg.e | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) | |
31 | mapdpgem25.h1 | . . . . . 6 ⊢ (𝜑 → (ℎ ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})))) | |
32 | mapdpglem26.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐴) | |
33 | mapdpglem26.o | . . . . . 6 ⊢ 𝑂 = (0g‘𝐴) | |
34 | mapdpglem28.ve | . . . . . 6 ⊢ (𝜑 → 𝑣 ∈ 𝐵) | |
35 | mapdpglem28.u2 | . . . . . 6 ⊢ (𝜑 → (𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖))) | |
36 | mapdpglem28.ue | . . . . . 6 ⊢ (𝜑 → 𝑢 ∈ 𝐵) | |
37 | 2, 19, 3, 20, 21, 22, 23, 6, 15, 24, 25, 9, 26, 27, 28, 29, 30, 31, 13, 4, 32, 16, 33, 34, 1, 35, 36 | mapdpglem30 38840 | . . . . 5 ⊢ (𝜑 → (𝑣 = (1r‘𝐴) ∧ 𝑣 = 𝑢)) |
38 | eqtr2 2844 | . . . . 5 ⊢ ((𝑣 = (1r‘𝐴) ∧ 𝑣 = 𝑢) → (1r‘𝐴) = 𝑢) | |
39 | 37, 38 | syl 17 | . . . 4 ⊢ (𝜑 → (1r‘𝐴) = 𝑢) |
40 | 39 | oveq1d 7173 | . . 3 ⊢ (𝜑 → ((1r‘𝐴) · 𝑖) = (𝑢 · 𝑖)) |
41 | 11, 18, 40 | 3eqtr3rd 2867 | . 2 ⊢ (𝜑 → (𝑢 · 𝑖) = 𝑖) |
42 | 1, 41 | eqtrd 2858 | 1 ⊢ (𝜑 → ℎ = 𝑖) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∖ cdif 3935 {csn 4569 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 Scalarcsca 16570 ·𝑠 cvsca 16571 0gc0g 16715 -gcsg 18107 1rcur 19253 LModclmod 19636 LSpanclspn 19745 HLchlt 36488 LHypclh 37122 DVecHcdvh 38216 LCDualclcd 38724 mapdcmpd 38762 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-riotaBAD 36091 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-undef 7941 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-0g 16717 df-mre 16859 df-mrc 16860 df-acs 16862 df-proset 17540 df-poset 17558 df-plt 17570 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-p0 17651 df-p1 17652 df-lat 17658 df-clat 17720 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-subg 18278 df-cntz 18449 df-oppg 18476 df-lsm 18763 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-dvr 19435 df-drng 19506 df-lmod 19638 df-lss 19706 df-lsp 19746 df-lvec 19877 df-lsatoms 36114 df-lshyp 36115 df-lcv 36157 df-lfl 36196 df-lkr 36224 df-ldual 36262 df-oposet 36314 df-ol 36316 df-oml 36317 df-covers 36404 df-ats 36405 df-atl 36436 df-cvlat 36460 df-hlat 36489 df-llines 36636 df-lplanes 36637 df-lvols 36638 df-lines 36639 df-psubsp 36641 df-pmap 36642 df-padd 36934 df-lhyp 37126 df-laut 37127 df-ldil 37242 df-ltrn 37243 df-trl 37297 df-tgrp 37881 df-tendo 37893 df-edring 37895 df-dveca 38141 df-disoa 38167 df-dvech 38217 df-dib 38277 df-dic 38311 df-dih 38367 df-doch 38486 df-djh 38533 df-lcdual 38725 df-mapd 38763 |
This theorem is referenced by: mapdpglem32 38843 |
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