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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh6hN | Structured version Visualization version GIF version |
Description: Lemmma for mapdh6N 38882. Part (6) of [Baer] p. 48 line 2. (Contributed by NM, 1-May-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mapdh.q | ⊢ 𝑄 = (0g‘𝐶) |
mapdh.i | ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))))) |
mapdh.h | ⊢ 𝐻 = (LHyp‘𝐾) |
mapdh.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
mapdh.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
mapdh.v | ⊢ 𝑉 = (Base‘𝑈) |
mapdh.s | ⊢ − = (-g‘𝑈) |
mapdhc.o | ⊢ 0 = (0g‘𝑈) |
mapdh.n | ⊢ 𝑁 = (LSpan‘𝑈) |
mapdh.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
mapdh.d | ⊢ 𝐷 = (Base‘𝐶) |
mapdh.r | ⊢ 𝑅 = (-g‘𝐶) |
mapdh.j | ⊢ 𝐽 = (LSpan‘𝐶) |
mapdh.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
mapdhc.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) |
mapdh.mn | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
mapdhcl.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
mapdh.p | ⊢ + = (+g‘𝑈) |
mapdh.a | ⊢ ✚ = (+g‘𝐶) |
mapdh6d.xn | ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) |
mapdh6d.yz | ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) |
mapdh6d.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
mapdh6d.z | ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) |
mapdh6d.w | ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) |
mapdh6d.wn | ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) |
Ref | Expression |
---|---|
mapdh6hN | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdh.q | . . . 4 ⊢ 𝑄 = (0g‘𝐶) | |
2 | mapdh.i | . . . 4 ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))))) | |
3 | mapdh.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | mapdh.m | . . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
5 | mapdh.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
6 | mapdh.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
7 | mapdh.s | . . . 4 ⊢ − = (-g‘𝑈) | |
8 | mapdhc.o | . . . 4 ⊢ 0 = (0g‘𝑈) | |
9 | mapdh.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑈) | |
10 | mapdh.c | . . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
11 | mapdh.d | . . . 4 ⊢ 𝐷 = (Base‘𝐶) | |
12 | mapdh.r | . . . 4 ⊢ 𝑅 = (-g‘𝐶) | |
13 | mapdh.j | . . . 4 ⊢ 𝐽 = (LSpan‘𝐶) | |
14 | mapdh.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
15 | mapdhc.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
16 | mapdh.mn | . . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) | |
17 | mapdhcl.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
18 | mapdh.p | . . . 4 ⊢ + = (+g‘𝑈) | |
19 | mapdh.a | . . . 4 ⊢ ✚ = (+g‘𝐶) | |
20 | mapdh6d.xn | . . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) | |
21 | mapdh6d.yz | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) | |
22 | mapdh6d.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
23 | mapdh6d.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) | |
24 | mapdh6d.w | . . . 4 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) | |
25 | mapdh6d.wn | . . . 4 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) | |
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 | mapdh6gN 38877 | . . 3 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉)) = (((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑌〉)) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
27 | 3, 10, 14 | lcdlmod 38727 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod) |
28 | 24 | eldifad 3947 | . . . . 5 ⊢ (𝜑 → 𝑤 ∈ 𝑉) |
29 | 3, 5, 14 | dvhlvec 38244 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LVec) |
30 | 17 | eldifad 3947 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
31 | 22 | eldifad 3947 | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
32 | 6, 9, 29, 28, 30, 31, 25 | lspindpi 19903 | . . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}))) |
33 | 32 | simpld 497 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑋})) |
34 | 33 | necomd 3071 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑤})) |
35 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 28, 34 | mapdhcl 38862 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑤〉) ∈ 𝐷) |
36 | 23 | eldifad 3947 | . . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
37 | 6, 9, 29, 30, 31, 36, 20 | lspindpi 19903 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))) |
38 | 37 | simpld 497 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
39 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 31, 38 | mapdhcl 38862 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) ∈ 𝐷) |
40 | 37 | simprd 498 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})) |
41 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 36, 40 | mapdhcl 38862 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) ∈ 𝐷) |
42 | 11, 19 | lmodass 19648 | . . . 4 ⊢ ((𝐶 ∈ LMod ∧ ((𝐼‘〈𝑋, 𝐹, 𝑤〉) ∈ 𝐷 ∧ (𝐼‘〈𝑋, 𝐹, 𝑌〉) ∈ 𝐷 ∧ (𝐼‘〈𝑋, 𝐹, 𝑍〉) ∈ 𝐷)) → (((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑌〉)) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) = ((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)))) |
43 | 27, 35, 39, 41, 42 | syl13anc 1368 | . . 3 ⊢ (𝜑 → (((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑌〉)) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) = ((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)))) |
44 | 26, 43 | eqtrd 2856 | . 2 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉)) = ((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)))) |
45 | 3, 5, 14 | dvhlmod 38245 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
46 | 6, 18 | lmodvacl 19647 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑌 + 𝑍) ∈ 𝑉) |
47 | 45, 31, 36, 46 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ 𝑉) |
48 | 6, 18, 8, 9, 29, 17, 22, 23, 24, 21, 38, 25 | mapdindp1 38855 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑌 + 𝑍)})) |
49 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 47, 48 | mapdhcl 38862 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) ∈ 𝐷) |
50 | 11, 19 | lmodvacl 19647 | . . . 4 ⊢ ((𝐶 ∈ LMod ∧ (𝐼‘〈𝑋, 𝐹, 𝑌〉) ∈ 𝐷 ∧ (𝐼‘〈𝑋, 𝐹, 𝑍〉) ∈ 𝐷) → ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) ∈ 𝐷) |
51 | 27, 39, 41, 50 | syl3anc 1367 | . . 3 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) ∈ 𝐷) |
52 | 11, 19 | lmodlcan 19649 | . . 3 ⊢ ((𝐶 ∈ LMod ∧ ((𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) ∈ 𝐷 ∧ ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) ∈ 𝐷 ∧ (𝐼‘〈𝑋, 𝐹, 𝑤〉) ∈ 𝐷)) → (((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉)) = ((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) ↔ (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)))) |
53 | 27, 49, 51, 35, 52 | syl13anc 1368 | . 2 ⊢ (𝜑 → (((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉)) = ((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) ↔ (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)))) |
54 | 44, 53 | mpbid 234 | 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 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 Vcvv 3494 ∖ cdif 3932 ifcif 4466 {csn 4566 {cpr 4568 〈cotp 4574 ↦ cmpt 5145 ‘cfv 6354 ℩crio 7112 (class class class)co 7155 1st c1st 7686 2nd c2nd 7687 Basecbs 16482 +gcplusg 16564 0gc0g 16712 -gcsg 18104 LModclmod 19633 LSpanclspn 19742 HLchlt 36485 LHypclh 37119 DVecHcdvh 38213 LCDualclcd 38721 mapdcmpd 38759 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-riotaBAD 36088 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-ot 4575 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-tpos 7891 df-undef 7938 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-sca 16580 df-vsca 16581 df-0g 16714 df-mre 16856 df-mrc 16857 df-acs 16859 df-proset 17537 df-poset 17555 df-plt 17567 df-lub 17583 df-glb 17584 df-join 17585 df-meet 17586 df-p0 17648 df-p1 17649 df-lat 17655 df-clat 17717 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-submnd 17956 df-grp 18105 df-minusg 18106 df-sbg 18107 df-subg 18275 df-cntz 18446 df-oppg 18473 df-lsm 18760 df-cmn 18907 df-abl 18908 df-mgp 19239 df-ur 19251 df-ring 19298 df-oppr 19372 df-dvdsr 19390 df-unit 19391 df-invr 19421 df-dvr 19432 df-drng 19503 df-lmod 19635 df-lss 19703 df-lsp 19743 df-lvec 19874 df-lsatoms 36111 df-lshyp 36112 df-lcv 36154 df-lfl 36193 df-lkr 36221 df-ldual 36259 df-oposet 36311 df-ol 36313 df-oml 36314 df-covers 36401 df-ats 36402 df-atl 36433 df-cvlat 36457 df-hlat 36486 df-llines 36633 df-lplanes 36634 df-lvols 36635 df-lines 36636 df-psubsp 36638 df-pmap 36639 df-padd 36931 df-lhyp 37123 df-laut 37124 df-ldil 37239 df-ltrn 37240 df-trl 37294 df-tgrp 37878 df-tendo 37890 df-edring 37892 df-dveca 38138 df-disoa 38164 df-dvech 38214 df-dib 38274 df-dic 38308 df-dih 38364 df-doch 38483 df-djh 38530 df-lcdual 38722 df-mapd 38760 |
This theorem is referenced by: mapdh6iN 38879 |
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