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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem3 | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 37726. 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 6894 | . . . 4 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌}))) = ((𝐽‘{𝐺}) ⊕ (𝑀‘(𝑁‘{𝑌})))) |
4 | 1, 3 | eleqtrd 2881 | . . 3 ⊢ (𝜑 → 𝑡 ∈ ((𝐽‘{𝐺}) ⊕ (𝑀‘(𝑁‘{𝑌})))) |
5 | mapdpglem.h | . . . . . . . . . . 11 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | mapdpglem.c | . . . . . . . . . . 11 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
7 | mapdpglem.k | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
8 | 5, 6, 7 | lcdlmod 37612 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ LMod) |
9 | mapdpglem3.g | . . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
10 | eqid 2800 | . . . . . . . . . . 11 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶) | |
11 | eqid 2800 | . . . . . . . . . . 11 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶)) | |
12 | mapdpglem3.f | . . . . . . . . . . 11 ⊢ 𝐹 = (Base‘𝐶) | |
13 | mapdpglem3.t | . . . . . . . . . . 11 ⊢ · = ( ·𝑠 ‘𝐶) | |
14 | mapdpglem2.j | . . . . . . . . . . 11 ⊢ 𝐽 = (LSpan‘𝐶) | |
15 | 10, 11, 12, 13, 14 | lspsnel 19323 | . . . . . . . . . 10 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝑤 ∈ (𝐽‘{𝐺}) ↔ ∃𝑔 ∈ (Base‘(Scalar‘𝐶))𝑤 = (𝑔 · 𝐺))) |
16 | 8, 9, 15 | syl2anc 580 | . . . . . . . . 9 ⊢ (𝜑 → (𝑤 ∈ (𝐽‘{𝐺}) ↔ ∃𝑔 ∈ (Base‘(Scalar‘𝐶))𝑤 = (𝑔 · 𝐺))) |
17 | mapdpglem.u | . . . . . . . . . . 11 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
18 | mapdpglem3.a | . . . . . . . . . . 11 ⊢ 𝐴 = (Scalar‘𝑈) | |
19 | mapdpglem3.b | . . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐴) | |
20 | 5, 17, 18, 19, 6, 10, 11, 7 | lcdsbase 37620 | . . . . . . . . . 10 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) = 𝐵) |
21 | 20 | rexeqdv 3329 | . . . . . . . . 9 ⊢ (𝜑 → (∃𝑔 ∈ (Base‘(Scalar‘𝐶))𝑤 = (𝑔 · 𝐺) ↔ ∃𝑔 ∈ 𝐵 𝑤 = (𝑔 · 𝐺))) |
22 | 16, 21 | bitrd 271 | . . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ (𝐽‘{𝐺}) ↔ ∃𝑔 ∈ 𝐵 𝑤 = (𝑔 · 𝐺))) |
23 | 22 | anbi1d 624 | . . . . . . 7 ⊢ (𝜑 → ((𝑤 ∈ (𝐽‘{𝐺}) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)) ↔ (∃𝑔 ∈ 𝐵 𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)))) |
24 | r19.41v 3271 | . . . . . . 7 ⊢ (∃𝑔 ∈ 𝐵 (𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)) ↔ (∃𝑔 ∈ 𝐵 𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧))) | |
25 | 23, 24 | syl6rbbr 282 | . . . . . 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 2017 | . . . . 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 3096 | . . . . 5 ⊢ (∃𝑤 ∈ (𝐽‘{𝐺})∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧) ↔ ∃𝑤(𝑤 ∈ (𝐽‘{𝐺}) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧))) | |
28 | 26, 27 | syl6bbr 281 | . . . 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 2800 | . . . . . . . 8 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶) | |
32 | 31 | lsssssubg 19278 | . . . . . . 7 ⊢ (𝐶 ∈ LMod → (LSubSp‘𝐶) ⊆ (SubGrp‘𝐶)) |
33 | 8, 32 | syl 17 | . . . . . 6 ⊢ (𝜑 → (LSubSp‘𝐶) ⊆ (SubGrp‘𝐶)) |
34 | 12, 31, 14 | lspsncl 19297 | . . . . . . 7 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐽‘{𝐺}) ∈ (LSubSp‘𝐶)) |
35 | 8, 9, 34 | syl2anc 580 | . . . . . 6 ⊢ (𝜑 → (𝐽‘{𝐺}) ∈ (LSubSp‘𝐶)) |
36 | 33, 35 | sseldd 3800 | . . . . 5 ⊢ (𝜑 → (𝐽‘{𝐺}) ∈ (SubGrp‘𝐶)) |
37 | mapdpglem.m | . . . . . . 7 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
38 | eqid 2800 | . . . . . . 7 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
39 | 5, 17, 7 | dvhlmod 37130 | . . . . . . . 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 19297 | . . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
44 | 39, 40, 43 | syl2anc 580 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
45 | 5, 37, 17, 38, 6, 31, 7, 44 | mapdcl2 37676 | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶)) |
46 | 33, 45 | sseldd 3800 | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ∈ (SubGrp‘𝐶)) |
47 | 29, 30, 36, 46 | lsmelvalm 18378 | . . . 4 ⊢ (𝜑 → (𝑡 ∈ ((𝐽‘{𝐺}) ⊕ (𝑀‘(𝑁‘{𝑌}))) ↔ ∃𝑤 ∈ (𝐽‘{𝐺})∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧))) |
48 | 28, 47 | bitr4d 274 | . . 3 ⊢ (𝜑 → (∃𝑤∃𝑔 ∈ 𝐵 (𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)) ↔ 𝑡 ∈ ((𝐽‘{𝐺}) ⊕ (𝑀‘(𝑁‘{𝑌}))))) |
49 | 4, 48 | mpbird 249 | . 2 ⊢ (𝜑 → ∃𝑤∃𝑔 ∈ 𝐵 (𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧))) |
50 | ovex 6911 | . . . . 5 ⊢ (𝑔 · 𝐺) ∈ V | |
51 | oveq1 6886 | . . . . . . 7 ⊢ (𝑤 = (𝑔 · 𝐺) → (𝑤𝑅𝑧) = ((𝑔 · 𝐺)𝑅𝑧)) | |
52 | 51 | eqeq2d 2810 | . . . . . 6 ⊢ (𝑤 = (𝑔 · 𝐺) → (𝑡 = (𝑤𝑅𝑧) ↔ 𝑡 = ((𝑔 · 𝐺)𝑅𝑧))) |
53 | 52 | rexbidv 3234 | . . . . 5 ⊢ (𝑤 = (𝑔 · 𝐺) → (∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧) ↔ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = ((𝑔 · 𝐺)𝑅𝑧))) |
54 | 50, 53 | ceqsexv 3431 | . . . 4 ⊢ (∃𝑤(𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)) ↔ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) |
55 | 54 | rexbii 3223 | . . 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 3414 | . . 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 269 | . 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 226 | 1 ⊢ (𝜑 → ∃𝑔 ∈ 𝐵 ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∃wex 1875 ∈ wcel 2157 ∃wrex 3091 ⊆ wss 3770 {csn 4369 ‘cfv 6102 (class class class)co 6879 Basecbs 16183 Scalarcsca 16269 ·𝑠 cvsca 16270 -gcsg 17739 SubGrpcsubg 17900 LSSumclsm 18361 LModclmod 19180 LSubSpclss 19249 LSpanclspn 19291 HLchlt 35370 LHypclh 36004 DVecHcdvh 37098 LCDualclcd 37606 mapdcmpd 37644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-rep 4965 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 ax-cnex 10281 ax-resscn 10282 ax-1cn 10283 ax-icn 10284 ax-addcl 10285 ax-addrcl 10286 ax-mulcl 10287 ax-mulrcl 10288 ax-mulcom 10289 ax-addass 10290 ax-mulass 10291 ax-distr 10292 ax-i2m1 10293 ax-1ne0 10294 ax-1rid 10295 ax-rnegex 10296 ax-rrecex 10297 ax-cnre 10298 ax-pre-lttri 10299 ax-pre-lttrn 10300 ax-pre-ltadd 10301 ax-pre-mulgt0 10302 ax-riotaBAD 34973 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-pss 3786 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-tp 4374 df-op 4376 df-uni 4630 df-int 4669 df-iun 4713 df-iin 4714 df-br 4845 df-opab 4907 df-mpt 4924 df-tr 4947 df-id 5221 df-eprel 5226 df-po 5234 df-so 5235 df-fr 5272 df-we 5274 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-pred 5899 df-ord 5945 df-on 5946 df-lim 5947 df-suc 5948 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 df-riota 6840 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-of 7132 df-om 7301 df-1st 7402 df-2nd 7403 df-tpos 7591 df-undef 7638 df-wrecs 7646 df-recs 7708 df-rdg 7746 df-1o 7800 df-oadd 7804 df-er 7983 df-map 8098 df-en 8197 df-dom 8198 df-sdom 8199 df-fin 8200 df-pnf 10366 df-mnf 10367 df-xr 10368 df-ltxr 10369 df-le 10370 df-sub 10559 df-neg 10560 df-nn 11314 df-2 11375 df-3 11376 df-4 11377 df-5 11378 df-6 11379 df-n0 11580 df-z 11666 df-uz 11930 df-fz 12580 df-struct 16185 df-ndx 16186 df-slot 16187 df-base 16189 df-sets 16190 df-ress 16191 df-plusg 16279 df-mulr 16280 df-sca 16282 df-vsca 16283 df-0g 16416 df-mre 16560 df-mrc 16561 df-acs 16563 df-proset 17242 df-poset 17260 df-plt 17272 df-lub 17288 df-glb 17289 df-join 17290 df-meet 17291 df-p0 17353 df-p1 17354 df-lat 17360 df-clat 17422 df-mgm 17556 df-sgrp 17598 df-mnd 17609 df-submnd 17650 df-grp 17740 df-minusg 17741 df-sbg 17742 df-subg 17903 df-cntz 18061 df-oppg 18087 df-lsm 18363 df-cmn 18509 df-abl 18510 df-mgp 18805 df-ur 18817 df-ring 18864 df-oppr 18938 df-dvdsr 18956 df-unit 18957 df-invr 18987 df-dvr 18998 df-drng 19066 df-lmod 19182 df-lss 19250 df-lsp 19292 df-lvec 19423 df-lsatoms 34996 df-lshyp 34997 df-lcv 35039 df-lfl 35078 df-lkr 35106 df-ldual 35144 df-oposet 35196 df-ol 35198 df-oml 35199 df-covers 35286 df-ats 35287 df-atl 35318 df-cvlat 35342 df-hlat 35371 df-llines 35518 df-lplanes 35519 df-lvols 35520 df-lines 35521 df-psubsp 35523 df-pmap 35524 df-padd 35816 df-lhyp 36008 df-laut 36009 df-ldil 36124 df-ltrn 36125 df-trl 36179 df-tgrp 36763 df-tendo 36775 df-edring 36777 df-dveca 37023 df-disoa 37049 df-dvech 37099 df-dib 37159 df-dic 37193 df-dih 37249 df-doch 37368 df-djh 37415 df-lcdual 37607 df-mapd 37645 |
This theorem is referenced by: mapdpglem24 37724 |
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