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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh6hN | Structured version Visualization version GIF version |
Description: Lemmma for mapdh6N 41220. 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 41215 | . . 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 41065 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod) |
28 | 24 | eldifad 3959 | . . . . 5 ⊢ (𝜑 → 𝑤 ∈ 𝑉) |
29 | 3, 5, 14 | dvhlvec 40582 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LVec) |
30 | 17 | eldifad 3959 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
31 | 22 | eldifad 3959 | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
32 | 6, 9, 29, 28, 30, 31, 25 | lspindpi 21020 | . . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}))) |
33 | 32 | simpld 494 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑋})) |
34 | 33 | necomd 2993 | . . . . 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 41200 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑤〉) ∈ 𝐷) |
36 | 23 | eldifad 3959 | . . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
37 | 6, 9, 29, 30, 31, 36, 20 | lspindpi 21020 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))) |
38 | 37 | simpld 494 | . . . . 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 41200 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) ∈ 𝐷) |
40 | 37 | simprd 495 | . . . . 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 41200 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) ∈ 𝐷) |
42 | 11, 19 | lmodass 20759 | . . . 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 1370 | . . 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 2768 | . 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 40583 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
46 | 6, 18 | lmodvacl 20758 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑌 + 𝑍) ∈ 𝑉) |
47 | 45, 31, 36, 46 | syl3anc 1369 | . . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ 𝑉) |
48 | 6, 18, 8, 9, 29, 17, 22, 23, 24, 21, 38, 25 | mapdindp1 41193 | . . . 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 41200 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) ∈ 𝐷) |
50 | 11, 19 | lmodvacl 20758 | . . . 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 1369 | . . 3 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) ∈ 𝐷) |
52 | 11, 19 | lmodlcan 20760 | . . 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 1370 | . 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 231 | 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 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 Vcvv 3471 ∖ cdif 3944 ifcif 4529 {csn 4629 {cpr 4631 〈cotp 4637 ↦ cmpt 5231 ‘cfv 6548 ℩crio 7375 (class class class)co 7420 1st c1st 7991 2nd c2nd 7992 Basecbs 17180 +gcplusg 17233 0gc0g 17421 -gcsg 18892 LModclmod 20743 LSpanclspn 20855 HLchlt 38822 LHypclh 39457 DVecHcdvh 40551 LCDualclcd 41059 mapdcmpd 41097 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-riotaBAD 38425 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-ot 4638 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-tpos 8232 df-undef 8279 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-n0 12504 df-z 12590 df-uz 12854 df-fz 13518 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-sca 17249 df-vsca 17250 df-0g 17423 df-mre 17566 df-mrc 17567 df-acs 17569 df-proset 18287 df-poset 18305 df-plt 18322 df-lub 18338 df-glb 18339 df-join 18340 df-meet 18341 df-p0 18417 df-p1 18418 df-lat 18424 df-clat 18491 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18741 df-grp 18893 df-minusg 18894 df-sbg 18895 df-subg 19078 df-cntz 19268 df-oppg 19297 df-lsm 19591 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-oppr 20273 df-dvdsr 20296 df-unit 20297 df-invr 20327 df-dvr 20340 df-drng 20626 df-lmod 20745 df-lss 20816 df-lsp 20856 df-lvec 20988 df-lsatoms 38448 df-lshyp 38449 df-lcv 38491 df-lfl 38530 df-lkr 38558 df-ldual 38596 df-oposet 38648 df-ol 38650 df-oml 38651 df-covers 38738 df-ats 38739 df-atl 38770 df-cvlat 38794 df-hlat 38823 df-llines 38971 df-lplanes 38972 df-lvols 38973 df-lines 38974 df-psubsp 38976 df-pmap 38977 df-padd 39269 df-lhyp 39461 df-laut 39462 df-ldil 39577 df-ltrn 39578 df-trl 39632 df-tgrp 40216 df-tendo 40228 df-edring 40230 df-dveca 40476 df-disoa 40502 df-dvech 40552 df-dib 40612 df-dic 40646 df-dih 40702 df-doch 40821 df-djh 40868 df-lcdual 41060 df-mapd 41098 |
This theorem is referenced by: mapdh6iN 41217 |
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