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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem3 | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 40169. Baer p. 45, line 3: "infer ... the existence of a number g in G and of an element z in (Fy)* such that t = gx'-z." (We scope $d 𝑔𝑤𝑧𝜑 locally to avoid clashes with later substitutions into 𝜑.) (Contributed by NM, 18-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})) |
Ref | Expression |
---|---|
mapdpglem3 | ⊢ (𝜑 → ∃𝑔 ∈ 𝐵 ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdpglem3.te | . . . 4 ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) | |
2 | mapdpglem3.e | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) | |
3 | 2 | oveq1d 7372 | . . . 4 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌}))) = ((𝐽‘{𝐺}) ⊕ (𝑀‘(𝑁‘{𝑌})))) |
4 | 1, 3 | eleqtrd 2840 | . . 3 ⊢ (𝜑 → 𝑡 ∈ ((𝐽‘{𝐺}) ⊕ (𝑀‘(𝑁‘{𝑌})))) |
5 | r19.41v 3185 | . . . . . . 7 ⊢ (∃𝑔 ∈ 𝐵 (𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)) ↔ (∃𝑔 ∈ 𝐵 𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧))) | |
6 | mapdpglem.h | . . . . . . . . . . 11 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | mapdpglem.c | . . . . . . . . . . 11 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
8 | mapdpglem.k | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
9 | 6, 7, 8 | lcdlmod 40055 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ LMod) |
10 | mapdpglem3.g | . . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
11 | eqid 2736 | . . . . . . . . . . 11 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶) | |
12 | eqid 2736 | . . . . . . . . . . 11 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶)) | |
13 | mapdpglem3.f | . . . . . . . . . . 11 ⊢ 𝐹 = (Base‘𝐶) | |
14 | mapdpglem3.t | . . . . . . . . . . 11 ⊢ · = ( ·𝑠 ‘𝐶) | |
15 | mapdpglem2.j | . . . . . . . . . . 11 ⊢ 𝐽 = (LSpan‘𝐶) | |
16 | 11, 12, 13, 14, 15 | lspsnel 20464 | . . . . . . . . . 10 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝑤 ∈ (𝐽‘{𝐺}) ↔ ∃𝑔 ∈ (Base‘(Scalar‘𝐶))𝑤 = (𝑔 · 𝐺))) |
17 | 9, 10, 16 | syl2anc 584 | . . . . . . . . 9 ⊢ (𝜑 → (𝑤 ∈ (𝐽‘{𝐺}) ↔ ∃𝑔 ∈ (Base‘(Scalar‘𝐶))𝑤 = (𝑔 · 𝐺))) |
18 | mapdpglem.u | . . . . . . . . . . 11 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
19 | mapdpglem3.a | . . . . . . . . . . 11 ⊢ 𝐴 = (Scalar‘𝑈) | |
20 | mapdpglem3.b | . . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐴) | |
21 | 6, 18, 19, 20, 7, 11, 12, 8 | lcdsbase 40063 | . . . . . . . . . 10 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) = 𝐵) |
22 | 21 | rexeqdv 3314 | . . . . . . . . 9 ⊢ (𝜑 → (∃𝑔 ∈ (Base‘(Scalar‘𝐶))𝑤 = (𝑔 · 𝐺) ↔ ∃𝑔 ∈ 𝐵 𝑤 = (𝑔 · 𝐺))) |
23 | 17, 22 | bitrd 278 | . . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ (𝐽‘{𝐺}) ↔ ∃𝑔 ∈ 𝐵 𝑤 = (𝑔 · 𝐺))) |
24 | 23 | anbi1d 630 | . . . . . . 7 ⊢ (𝜑 → ((𝑤 ∈ (𝐽‘{𝐺}) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)) ↔ (∃𝑔 ∈ 𝐵 𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)))) |
25 | 5, 24 | bitr4id 289 | . . . . . 6 ⊢ (𝜑 → (∃𝑔 ∈ 𝐵 (𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)) ↔ (𝑤 ∈ (𝐽‘{𝐺}) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)))) |
26 | 25 | exbidv 1924 | . . . . 5 ⊢ (𝜑 → (∃𝑤∃𝑔 ∈ 𝐵 (𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)) ↔ ∃𝑤(𝑤 ∈ (𝐽‘{𝐺}) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)))) |
27 | df-rex 3074 | . . . . 5 ⊢ (∃𝑤 ∈ (𝐽‘{𝐺})∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧) ↔ ∃𝑤(𝑤 ∈ (𝐽‘{𝐺}) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧))) | |
28 | 26, 27 | bitr4di 288 | . . . 4 ⊢ (𝜑 → (∃𝑤∃𝑔 ∈ 𝐵 (𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)) ↔ ∃𝑤 ∈ (𝐽‘{𝐺})∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧))) |
29 | mapdpglem3.r | . . . . 5 ⊢ 𝑅 = (-g‘𝐶) | |
30 | mapdpglem1.p | . . . . 5 ⊢ ⊕ = (LSSum‘𝐶) | |
31 | eqid 2736 | . . . . . . . 8 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶) | |
32 | 31 | lsssssubg 20419 | . . . . . . 7 ⊢ (𝐶 ∈ LMod → (LSubSp‘𝐶) ⊆ (SubGrp‘𝐶)) |
33 | 9, 32 | syl 17 | . . . . . 6 ⊢ (𝜑 → (LSubSp‘𝐶) ⊆ (SubGrp‘𝐶)) |
34 | 13, 31, 15 | lspsncl 20438 | . . . . . . 7 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐽‘{𝐺}) ∈ (LSubSp‘𝐶)) |
35 | 9, 10, 34 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (𝐽‘{𝐺}) ∈ (LSubSp‘𝐶)) |
36 | 33, 35 | sseldd 3945 | . . . . 5 ⊢ (𝜑 → (𝐽‘{𝐺}) ∈ (SubGrp‘𝐶)) |
37 | mapdpglem.m | . . . . . . 7 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
38 | eqid 2736 | . . . . . . 7 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
39 | 6, 18, 8 | dvhlmod 39573 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod) |
40 | mapdpglem.y | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
41 | mapdpglem.v | . . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑈) | |
42 | mapdpglem.n | . . . . . . . . 9 ⊢ 𝑁 = (LSpan‘𝑈) | |
43 | 41, 38, 42 | lspsncl 20438 | . . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
44 | 39, 40, 43 | syl2anc 584 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
45 | 6, 37, 18, 38, 7, 31, 8, 44 | mapdcl2 40119 | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶)) |
46 | 33, 45 | sseldd 3945 | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ∈ (SubGrp‘𝐶)) |
47 | 29, 30, 36, 46 | lsmelvalm 19433 | . . . 4 ⊢ (𝜑 → (𝑡 ∈ ((𝐽‘{𝐺}) ⊕ (𝑀‘(𝑁‘{𝑌}))) ↔ ∃𝑤 ∈ (𝐽‘{𝐺})∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧))) |
48 | 28, 47 | bitr4d 281 | . . 3 ⊢ (𝜑 → (∃𝑤∃𝑔 ∈ 𝐵 (𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)) ↔ 𝑡 ∈ ((𝐽‘{𝐺}) ⊕ (𝑀‘(𝑁‘{𝑌}))))) |
49 | 4, 48 | mpbird 256 | . 2 ⊢ (𝜑 → ∃𝑤∃𝑔 ∈ 𝐵 (𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧))) |
50 | ovex 7390 | . . . . 5 ⊢ (𝑔 · 𝐺) ∈ V | |
51 | oveq1 7364 | . . . . . . 7 ⊢ (𝑤 = (𝑔 · 𝐺) → (𝑤𝑅𝑧) = ((𝑔 · 𝐺)𝑅𝑧)) | |
52 | 51 | eqeq2d 2747 | . . . . . 6 ⊢ (𝑤 = (𝑔 · 𝐺) → (𝑡 = (𝑤𝑅𝑧) ↔ 𝑡 = ((𝑔 · 𝐺)𝑅𝑧))) |
53 | 52 | rexbidv 3175 | . . . . 5 ⊢ (𝑤 = (𝑔 · 𝐺) → (∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧) ↔ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = ((𝑔 · 𝐺)𝑅𝑧))) |
54 | 50, 53 | ceqsexv 3494 | . . . 4 ⊢ (∃𝑤(𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)) ↔ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) |
55 | 54 | rexbii 3097 | . . 3 ⊢ (∃𝑔 ∈ 𝐵 ∃𝑤(𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)) ↔ ∃𝑔 ∈ 𝐵 ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) |
56 | rexcom4 3271 | . . 3 ⊢ (∃𝑔 ∈ 𝐵 ∃𝑤(𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)) ↔ ∃𝑤∃𝑔 ∈ 𝐵 (𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧))) | |
57 | 55, 56 | bitr3i 276 | . 2 ⊢ (∃𝑔 ∈ 𝐵 ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = ((𝑔 · 𝐺)𝑅𝑧) ↔ ∃𝑤∃𝑔 ∈ 𝐵 (𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧))) |
58 | 49, 57 | sylibr 233 | 1 ⊢ (𝜑 → ∃𝑔 ∈ 𝐵 ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ∃wrex 3073 ⊆ wss 3910 {csn 4586 ‘cfv 6496 (class class class)co 7357 Basecbs 17083 Scalarcsca 17136 ·𝑠 cvsca 17137 -gcsg 18750 SubGrpcsubg 18922 LSSumclsm 19416 LModclmod 20322 LSubSpclss 20392 LSpanclspn 20432 HLchlt 37812 LHypclh 38447 DVecHcdvh 39541 LCDualclcd 40049 mapdcmpd 40087 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-riotaBAD 37415 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-tpos 8157 df-undef 8204 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-sca 17149 df-vsca 17150 df-0g 17323 df-mre 17466 df-mrc 17467 df-acs 17469 df-proset 18184 df-poset 18202 df-plt 18219 df-lub 18235 df-glb 18236 df-join 18237 df-meet 18238 df-p0 18314 df-p1 18315 df-lat 18321 df-clat 18388 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-subg 18925 df-cntz 19097 df-oppg 19124 df-lsm 19418 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-dvr 20112 df-drng 20187 df-lmod 20324 df-lss 20393 df-lsp 20433 df-lvec 20564 df-lsatoms 37438 df-lshyp 37439 df-lcv 37481 df-lfl 37520 df-lkr 37548 df-ldual 37586 df-oposet 37638 df-ol 37640 df-oml 37641 df-covers 37728 df-ats 37729 df-atl 37760 df-cvlat 37784 df-hlat 37813 df-llines 37961 df-lplanes 37962 df-lvols 37963 df-lines 37964 df-psubsp 37966 df-pmap 37967 df-padd 38259 df-lhyp 38451 df-laut 38452 df-ldil 38567 df-ltrn 38568 df-trl 38622 df-tgrp 39206 df-tendo 39218 df-edring 39220 df-dveca 39466 df-disoa 39492 df-dvech 39542 df-dib 39602 df-dic 39636 df-dih 39692 df-doch 39811 df-djh 39858 df-lcdual 40050 df-mapd 40088 |
This theorem is referenced by: mapdpglem24 40167 |
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