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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap1l6a | Structured version Visualization version GIF version |
Description: Lemma for hdmap1l6 41804. Part (6) in [Baer] p. 47, case 1. (Contributed by NM, 23-Apr-2015.) |
Ref | Expression |
---|---|
hdmap1l6.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmap1l6.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmap1l6.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmap1l6.p | ⊢ + = (+g‘𝑈) |
hdmap1l6.s | ⊢ − = (-g‘𝑈) |
hdmap1l6c.o | ⊢ 0 = (0g‘𝑈) |
hdmap1l6.n | ⊢ 𝑁 = (LSpan‘𝑈) |
hdmap1l6.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmap1l6.d | ⊢ 𝐷 = (Base‘𝐶) |
hdmap1l6.a | ⊢ ✚ = (+g‘𝐶) |
hdmap1l6.r | ⊢ 𝑅 = (-g‘𝐶) |
hdmap1l6.q | ⊢ 𝑄 = (0g‘𝐶) |
hdmap1l6.l | ⊢ 𝐿 = (LSpan‘𝐶) |
hdmap1l6.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
hdmap1l6.i | ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) |
hdmap1l6.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmap1l6.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) |
hdmap1l6cl.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
hdmap1l6.mn | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹})) |
hdmap1l6e.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
hdmap1l6e.z | ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) |
hdmap1l6e.xn | ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) |
hdmap1l6.yz | ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) |
hdmap1l6.fg | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) |
hdmap1l6.fe | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) = 𝐸) |
Ref | Expression |
---|---|
hdmap1l6a | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmap1l6.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmap1l6.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hdmap1l6.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
4 | hdmap1l6.p | . . . 4 ⊢ + = (+g‘𝑈) | |
5 | hdmap1l6.s | . . . 4 ⊢ − = (-g‘𝑈) | |
6 | hdmap1l6c.o | . . . 4 ⊢ 0 = (0g‘𝑈) | |
7 | hdmap1l6.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑈) | |
8 | hdmap1l6.c | . . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
9 | hdmap1l6.d | . . . 4 ⊢ 𝐷 = (Base‘𝐶) | |
10 | hdmap1l6.a | . . . 4 ⊢ ✚ = (+g‘𝐶) | |
11 | hdmap1l6.r | . . . 4 ⊢ 𝑅 = (-g‘𝐶) | |
12 | hdmap1l6.q | . . . 4 ⊢ 𝑄 = (0g‘𝐶) | |
13 | hdmap1l6.l | . . . 4 ⊢ 𝐿 = (LSpan‘𝐶) | |
14 | hdmap1l6.m | . . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
15 | hdmap1l6.i | . . . 4 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) | |
16 | hdmap1l6.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
17 | hdmap1l6.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
18 | hdmap1l6cl.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
19 | hdmap1l6.mn | . . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹})) | |
20 | hdmap1l6e.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
21 | hdmap1l6e.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) | |
22 | hdmap1l6e.xn | . . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) | |
23 | hdmap1l6.yz | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) | |
24 | hdmap1l6.fg | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) | |
25 | hdmap1l6.fe | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) = 𝐸) | |
26 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25 | hdmap1l6lem2 41791 | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑌 + 𝑍)})) = (𝐿‘{(𝐺 ✚ 𝐸)})) |
27 | 24, 25 | oveq12d 7449 | . . . . 5 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) = (𝐺 ✚ 𝐸)) |
28 | 27 | sneqd 4643 | . . . 4 ⊢ (𝜑 → {((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))} = {(𝐺 ✚ 𝐸)}) |
29 | 28 | fveq2d 6911 | . . 3 ⊢ (𝜑 → (𝐿‘{((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))}) = (𝐿‘{(𝐺 ✚ 𝐸)})) |
30 | 26, 29 | eqtr4d 2778 | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑌 + 𝑍)})) = (𝐿‘{((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))})) |
31 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25 | hdmap1l6lem1 41790 | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − (𝑌 + 𝑍))})) = (𝐿‘{(𝐹𝑅(𝐺 ✚ 𝐸))})) |
32 | 27 | oveq2d 7447 | . . . . 5 ⊢ (𝜑 → (𝐹𝑅((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) = (𝐹𝑅(𝐺 ✚ 𝐸))) |
33 | 32 | sneqd 4643 | . . . 4 ⊢ (𝜑 → {(𝐹𝑅((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)))} = {(𝐹𝑅(𝐺 ✚ 𝐸))}) |
34 | 33 | fveq2d 6911 | . . 3 ⊢ (𝜑 → (𝐿‘{(𝐹𝑅((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)))}) = (𝐿‘{(𝐹𝑅(𝐺 ✚ 𝐸))})) |
35 | 31, 34 | eqtr4d 2778 | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − (𝑌 + 𝑍))})) = (𝐿‘{(𝐹𝑅((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)))})) |
36 | 1, 2, 16 | dvhlmod 41093 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
37 | 20 | eldifad 3975 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
38 | 21 | eldifad 3975 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
39 | 3, 4 | lmodvacl 20890 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑌 + 𝑍) ∈ 𝑉) |
40 | 36, 37, 38, 39 | syl3anc 1370 | . . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ 𝑉) |
41 | 3, 4, 6, 7, 36, 37, 38, 23 | lmodindp1 21030 | . . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) ≠ 0 ) |
42 | eldifsn 4791 | . . . 4 ⊢ ((𝑌 + 𝑍) ∈ (𝑉 ∖ { 0 }) ↔ ((𝑌 + 𝑍) ∈ 𝑉 ∧ (𝑌 + 𝑍) ≠ 0 )) | |
43 | 40, 41, 42 | sylanbrc 583 | . . 3 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ (𝑉 ∖ { 0 })) |
44 | 1, 8, 16 | lcdlmod 41575 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod) |
45 | 1, 2, 16 | dvhlvec 41092 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LVec) |
46 | 18 | eldifad 3975 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
47 | 3, 6, 7, 45, 37, 21, 46, 23, 22 | lspindp2 21155 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌}))) |
48 | 47 | simpld 494 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
49 | 1, 2, 3, 6, 7, 8, 9, 13, 14, 15, 16, 17, 19, 48, 18, 37 | hdmap1cl 41787 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) ∈ 𝐷) |
50 | 3, 6, 7, 45, 20, 38, 46, 23, 22 | lspindp1 21153 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}) ∧ ¬ 𝑌 ∈ (𝑁‘{𝑋, 𝑍}))) |
51 | 50 | simpld 494 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})) |
52 | 1, 2, 3, 6, 7, 8, 9, 13, 14, 15, 16, 17, 19, 51, 18, 38 | hdmap1cl 41787 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) ∈ 𝐷) |
53 | 9, 10 | lmodvacl 20890 | . . . 4 ⊢ ((𝐶 ∈ LMod ∧ (𝐼‘〈𝑋, 𝐹, 𝑌〉) ∈ 𝐷 ∧ (𝐼‘〈𝑋, 𝐹, 𝑍〉) ∈ 𝐷) → ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) ∈ 𝐷) |
54 | 44, 49, 52, 53 | syl3anc 1370 | . . 3 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) ∈ 𝐷) |
55 | eqid 2735 | . . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
56 | 3, 55, 7, 36, 37, 38 | lspprcl 20994 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) ∈ (LSubSp‘𝑈)) |
57 | 3, 4, 7, 36, 37, 38 | lspprvacl 21015 | . . . . . 6 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ (𝑁‘{𝑌, 𝑍})) |
58 | 55, 7, 36, 56, 57 | ellspsn5 21012 | . . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) ⊆ (𝑁‘{𝑌, 𝑍})) |
59 | 3, 55, 7, 36, 56, 46 | ellspsn5b 21011 | . . . . . 6 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍}))) |
60 | 22, 59 | mtbid 324 | . . . . 5 ⊢ (𝜑 → ¬ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍})) |
61 | nssne2 4059 | . . . . 5 ⊢ (((𝑁‘{(𝑌 + 𝑍)}) ⊆ (𝑁‘{𝑌, 𝑍}) ∧ ¬ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍})) → (𝑁‘{(𝑌 + 𝑍)}) ≠ (𝑁‘{𝑋})) | |
62 | 58, 60, 61 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) ≠ (𝑁‘{𝑋})) |
63 | 62 | necomd 2994 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑌 + 𝑍)})) |
64 | 1, 2, 3, 5, 6, 7, 8, 9, 11, 13, 14, 15, 16, 18, 17, 43, 54, 63, 19 | hdmap1eq 41784 | . 2 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) ↔ ((𝑀‘(𝑁‘{(𝑌 + 𝑍)})) = (𝐿‘{((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))}) ∧ (𝑀‘(𝑁‘{(𝑋 − (𝑌 + 𝑍))})) = (𝐿‘{(𝐹𝑅((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)))})))) |
65 | 30, 35, 64 | mpbir2and 713 | 1 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∖ cdif 3960 ⊆ wss 3963 {csn 4631 {cpr 4633 〈cotp 4639 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 0gc0g 17486 -gcsg 18966 LModclmod 20875 LSubSpclss 20947 LSpanclspn 20987 HLchlt 39332 LHypclh 39967 DVecHcdvh 41061 LCDualclcd 41569 mapdcmpd 41607 HDMap1chdma1 41774 |
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-ot 4640 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 df-hdmap1 41776 |
This theorem is referenced by: hdmap1l6d 41796 hdmap1l6e 41797 hdmap1l6f 41798 hdmap1l6j 41802 |
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