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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem22 | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 41689. Baer p. 45, line 9: "(F(x-y))* = ... = G(x'-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 |
---|---|
mapdpglem22 | ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝐸)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdpglem4.jt | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) | |
2 | mapdpglem.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | mapdpglem.c | . . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
4 | mapdpglem.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
5 | 2, 3, 4 | lcdlvec 41574 | . . 3 ⊢ (𝜑 → 𝐶 ∈ LVec) |
6 | mapdpglem.u | . . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
7 | 2, 6, 4 | dvhlvec 41092 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec) |
8 | mapdpglem3.a | . . . . . . 7 ⊢ 𝐴 = (Scalar‘𝑈) | |
9 | 8 | lvecdrng 21122 | . . . . . 6 ⊢ (𝑈 ∈ LVec → 𝐴 ∈ DivRing) |
10 | 7, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ DivRing) |
11 | mapdpglem4.g4 | . . . . 5 ⊢ (𝜑 → 𝑔 ∈ 𝐵) | |
12 | mapdpglem.m | . . . . . 6 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
13 | mapdpglem.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑈) | |
14 | mapdpglem.s | . . . . . 6 ⊢ − = (-g‘𝑈) | |
15 | mapdpglem.n | . . . . . 6 ⊢ 𝑁 = (LSpan‘𝑈) | |
16 | mapdpglem.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
17 | mapdpglem.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
18 | mapdpglem1.p | . . . . . 6 ⊢ ⊕ = (LSSum‘𝐶) | |
19 | mapdpglem2.j | . . . . . 6 ⊢ 𝐽 = (LSpan‘𝐶) | |
20 | mapdpglem3.f | . . . . . 6 ⊢ 𝐹 = (Base‘𝐶) | |
21 | mapdpglem3.te | . . . . . 6 ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) | |
22 | mapdpglem3.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐴) | |
23 | mapdpglem3.t | . . . . . 6 ⊢ · = ( ·𝑠 ‘𝐶) | |
24 | mapdpglem3.r | . . . . . 6 ⊢ 𝑅 = (-g‘𝐶) | |
25 | mapdpglem3.g | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
26 | mapdpglem3.e | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) | |
27 | mapdpglem4.q | . . . . . 6 ⊢ 𝑄 = (0g‘𝑈) | |
28 | mapdpglem.ne | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
29 | mapdpglem4.z | . . . . . 6 ⊢ 0 = (0g‘𝐴) | |
30 | mapdpglem4.z4 | . . . . . 6 ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) | |
31 | mapdpglem4.t4 | . . . . . 6 ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) | |
32 | mapdpglem4.xn | . . . . . 6 ⊢ (𝜑 → 𝑋 ≠ 𝑄) | |
33 | 2, 12, 6, 13, 14, 15, 3, 4, 16, 17, 18, 19, 20, 21, 8, 22, 23, 24, 25, 26, 27, 28, 1, 29, 11, 30, 31, 32 | mapdpglem11 41665 | . . . . 5 ⊢ (𝜑 → 𝑔 ≠ 0 ) |
34 | eqid 2735 | . . . . . 6 ⊢ (invr‘𝐴) = (invr‘𝐴) | |
35 | 22, 29, 34 | drnginvrcl 20770 | . . . . 5 ⊢ ((𝐴 ∈ DivRing ∧ 𝑔 ∈ 𝐵 ∧ 𝑔 ≠ 0 ) → ((invr‘𝐴)‘𝑔) ∈ 𝐵) |
36 | 10, 11, 33, 35 | syl3anc 1370 | . . . 4 ⊢ (𝜑 → ((invr‘𝐴)‘𝑔) ∈ 𝐵) |
37 | eqid 2735 | . . . . 5 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶) | |
38 | eqid 2735 | . . . . 5 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶)) | |
39 | 2, 6, 8, 22, 3, 37, 38, 4 | lcdsbase 41583 | . . . 4 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) = 𝐵) |
40 | 36, 39 | eleqtrrd 2842 | . . 3 ⊢ (𝜑 → ((invr‘𝐴)‘𝑔) ∈ (Base‘(Scalar‘𝐶))) |
41 | 22, 29, 34 | drnginvrn0 20771 | . . . . 5 ⊢ ((𝐴 ∈ DivRing ∧ 𝑔 ∈ 𝐵 ∧ 𝑔 ≠ 0 ) → ((invr‘𝐴)‘𝑔) ≠ 0 ) |
42 | 10, 11, 33, 41 | syl3anc 1370 | . . . 4 ⊢ (𝜑 → ((invr‘𝐴)‘𝑔) ≠ 0 ) |
43 | eqid 2735 | . . . . 5 ⊢ (0g‘(Scalar‘𝐶)) = (0g‘(Scalar‘𝐶)) | |
44 | 2, 6, 8, 29, 3, 37, 43, 4 | lcd0 41591 | . . . 4 ⊢ (𝜑 → (0g‘(Scalar‘𝐶)) = 0 ) |
45 | 42, 44 | neeqtrrd 3013 | . . 3 ⊢ (𝜑 → ((invr‘𝐴)‘𝑔) ≠ (0g‘(Scalar‘𝐶))) |
46 | 2, 12, 6, 13, 14, 15, 3, 4, 16, 17, 18, 19, 20, 21 | mapdpglem2a 41657 | . . 3 ⊢ (𝜑 → 𝑡 ∈ 𝐹) |
47 | 20, 37, 23, 38, 43, 19 | lspsnvs 21134 | . . 3 ⊢ ((𝐶 ∈ LVec ∧ (((invr‘𝐴)‘𝑔) ∈ (Base‘(Scalar‘𝐶)) ∧ ((invr‘𝐴)‘𝑔) ≠ (0g‘(Scalar‘𝐶))) ∧ 𝑡 ∈ 𝐹) → (𝐽‘{(((invr‘𝐴)‘𝑔) · 𝑡)}) = (𝐽‘{𝑡})) |
48 | 5, 40, 45, 46, 47 | syl121anc 1374 | . 2 ⊢ (𝜑 → (𝐽‘{(((invr‘𝐴)‘𝑔) · 𝑡)}) = (𝐽‘{𝑡})) |
49 | mapdpglem12.yn | . . . . 5 ⊢ (𝜑 → 𝑌 ≠ 𝑄) | |
50 | mapdpglem17.ep | . . . . 5 ⊢ 𝐸 = (((invr‘𝐴)‘𝑔) · 𝑧) | |
51 | 2, 12, 6, 13, 14, 15, 3, 4, 16, 17, 18, 19, 20, 21, 8, 22, 23, 24, 25, 26, 27, 28, 1, 29, 11, 30, 31, 32, 49, 50 | mapdpglem21 41675 | . . . 4 ⊢ (𝜑 → (((invr‘𝐴)‘𝑔) · 𝑡) = (𝐺𝑅𝐸)) |
52 | 51 | sneqd 4643 | . . 3 ⊢ (𝜑 → {(((invr‘𝐴)‘𝑔) · 𝑡)} = {(𝐺𝑅𝐸)}) |
53 | 52 | fveq2d 6911 | . 2 ⊢ (𝜑 → (𝐽‘{(((invr‘𝐴)‘𝑔) · 𝑡)}) = (𝐽‘{(𝐺𝑅𝐸)})) |
54 | 1, 48, 53 | 3eqtr2d 2781 | 1 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝐸)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 {csn 4631 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 Scalarcsca 17301 ·𝑠 cvsca 17302 0gc0g 17486 -gcsg 18966 LSSumclsm 19667 invrcinvr 20404 DivRingcdr 20746 LSpanclspn 20987 LVecclvec 21119 HLchlt 39332 LHypclh 39967 DVecHcdvh 41061 LCDualclcd 41569 mapdcmpd 41607 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-riotaBAD 38935 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-undef 8297 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-0g 17488 df-mre 17631 df-mrc 17632 df-acs 17634 df-proset 18352 df-poset 18371 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-p1 18484 df-lat 18490 df-clat 18557 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cntz 19348 df-oppg 19377 df-lsm 19669 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-nzr 20530 df-rlreg 20711 df-domn 20712 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lvec 21120 df-lsatoms 38958 df-lshyp 38959 df-lcv 39001 df-lfl 39040 df-lkr 39068 df-ldual 39106 df-oposet 39158 df-ol 39160 df-oml 39161 df-covers 39248 df-ats 39249 df-atl 39280 df-cvlat 39304 df-hlat 39333 df-llines 39481 df-lplanes 39482 df-lvols 39483 df-lines 39484 df-psubsp 39486 df-pmap 39487 df-padd 39779 df-lhyp 39971 df-laut 39972 df-ldil 40087 df-ltrn 40088 df-trl 40142 df-tgrp 40726 df-tendo 40738 df-edring 40740 df-dveca 40986 df-disoa 41012 df-dvech 41062 df-dib 41122 df-dic 41156 df-dih 41212 df-doch 41331 df-djh 41378 df-lcdual 41570 df-mapd 41608 |
This theorem is referenced by: mapdpglem23 41677 |
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