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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh6hN | Structured version Visualization version GIF version |
Description: Lemmma for mapdh6N 41730. 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 41725 | . . 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 41575 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod) |
28 | 24 | eldifad 3975 | . . . . 5 ⊢ (𝜑 → 𝑤 ∈ 𝑉) |
29 | 3, 5, 14 | dvhlvec 41092 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LVec) |
30 | 17 | eldifad 3975 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
31 | 22 | eldifad 3975 | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
32 | 6, 9, 29, 28, 30, 31, 25 | lspindpi 21152 | . . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}))) |
33 | 32 | simpld 494 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑋})) |
34 | 33 | necomd 2994 | . . . . 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 41710 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑤〉) ∈ 𝐷) |
36 | 23 | eldifad 3975 | . . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
37 | 6, 9, 29, 30, 31, 36, 20 | lspindpi 21152 | . . . . . 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 41710 | . . . 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 41710 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) ∈ 𝐷) |
42 | 11, 19 | lmodass 20891 | . . . 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 1371 | . . 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 2775 | . 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 41093 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
46 | 6, 18 | lmodvacl 20890 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑌 + 𝑍) ∈ 𝑉) |
47 | 45, 31, 36, 46 | syl3anc 1370 | . . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ 𝑉) |
48 | 6, 18, 8, 9, 29, 17, 22, 23, 24, 21, 38, 25 | mapdindp1 41703 | . . . 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 41710 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) ∈ 𝐷) |
50 | 11, 19 | lmodvacl 20890 | . . . 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 1370 | . . 3 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) ∈ 𝐷) |
52 | 11, 19 | lmodlcan 20892 | . . 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 1371 | . 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 232 | 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 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ∖ cdif 3960 ifcif 4531 {csn 4631 {cpr 4633 〈cotp 4639 ↦ cmpt 5231 ‘cfv 6563 ℩crio 7387 (class class class)co 7431 1st c1st 8011 2nd c2nd 8012 Basecbs 17245 +gcplusg 17298 0gc0g 17486 -gcsg 18966 LModclmod 20875 LSpanclspn 20987 HLchlt 39332 LHypclh 39967 DVecHcdvh 41061 LCDualclcd 41569 mapdcmpd 41607 |
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 |
This theorem is referenced by: mapdh6iN 41727 |
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