hdmaprnlem6N - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmaprnlem6N		Structured version Visualization version GIF version

Theorem hdmaprnlem6N 41843

Description: Part of proof of part 12 in [Baer] p. 49 line 18, G(u'+s) = G(u'+t). (Contributed by NM, 27-May-2015.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
hdmaprnlem1.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmaprnlem1.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmaprnlem1.v	⊢ 𝑉 = (Base‘𝑈)
hdmaprnlem1.n	⊢ 𝑁 = (LSpan‘𝑈)
hdmaprnlem1.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmaprnlem1.l	⊢ 𝐿 = (LSpan‘𝐶)
hdmaprnlem1.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
hdmaprnlem1.s	⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmaprnlem1.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmaprnlem1.se	⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))
hdmaprnlem1.ve	⊢ (𝜑 → 𝑣 ∈ 𝑉)
hdmaprnlem1.e	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))
hdmaprnlem1.ue	⊢ (𝜑 → 𝑢 ∈ 𝑉)
hdmaprnlem1.un	⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))
hdmaprnlem1.d	⊢ 𝐷 = (Base‘𝐶)
hdmaprnlem1.q	⊢ 𝑄 = (0_g‘𝐶)
hdmaprnlem1.o	⊢ 0 = (0_g‘𝑈)
hdmaprnlem1.a	⊢ ✚ = (+_g‘𝐶)
hdmaprnlem1.t2	⊢ (𝜑 → 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }))
hdmaprnlem1.p	⊢ + = (+_g‘𝑈)
hdmaprnlem1.pt	⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))

**Assertion**
Ref	Expression
hdmaprnlem6N	⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝐿‘{((𝑆‘𝑢) ✚ (𝑆‘𝑡))}))

**Proof of Theorem hdmaprnlem6N**
Step	Hyp	Ref	Expression
1		hdmaprnlem1.pt	. 2 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))
2		hdmaprnlem1.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
3		hdmaprnlem1.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		hdmaprnlem1.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
5		hdmaprnlem1.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
6		hdmaprnlem1.c	. . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
7		hdmaprnlem1.l	. . 3 ⊢ 𝐿 = (LSpan‘𝐶)
8		hdmaprnlem1.m	. . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
9		hdmaprnlem1.s	. . 3 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
10		hdmaprnlem1.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11	2, 3, 10	dvhlmod 41099	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
12		hdmaprnlem1.ue	. . . 4 ⊢ (𝜑 → 𝑢 ∈ 𝑉)
13		hdmaprnlem1.se	. . . . 5 ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))
14		hdmaprnlem1.ve	. . . . 5 ⊢ (𝜑 → 𝑣 ∈ 𝑉)
15		hdmaprnlem1.e	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))
16		hdmaprnlem1.un	. . . . 5 ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))
17		hdmaprnlem1.d	. . . . 5 ⊢ 𝐷 = (Base‘𝐶)
18		hdmaprnlem1.q	. . . . 5 ⊢ 𝑄 = (0_g‘𝐶)
19		hdmaprnlem1.o	. . . . 5 ⊢ 0 = (0_g‘𝑈)
20		hdmaprnlem1.a	. . . . 5 ⊢ ✚ = (+_g‘𝐶)
21		hdmaprnlem1.t2	. . . . 5 ⊢ (𝜑 → 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }))
22	2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 15, 12, 16, 17, 18, 19, 20, 21	hdmaprnlem4tN 41841	. . . 4 ⊢ (𝜑 → 𝑡 ∈ 𝑉)
23		hdmaprnlem1.p	. . . . 5 ⊢ + = (+_g‘𝑈)
24	4, 23	lmodvacl 20787	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑢 ∈ 𝑉 ∧ 𝑡 ∈ 𝑉) → (𝑢 + 𝑡) ∈ 𝑉)
25	11, 12, 22, 24	syl3anc 1373	. . 3 ⊢ (𝜑 → (𝑢 + 𝑡) ∈ 𝑉)
26	2, 3, 4, 5, 6, 7, 8, 9, 10, 25	hdmap10 41829	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑢 + 𝑡)})) = (𝐿‘{(𝑆‘(𝑢 + 𝑡))}))
27	2, 3, 4, 23, 6, 20, 9, 10, 12, 22	hdmapadd 41832	. . . 4 ⊢ (𝜑 → (𝑆‘(𝑢 + 𝑡)) = ((𝑆‘𝑢) ✚ (𝑆‘𝑡)))
28	27	sneqd 4603	. . 3 ⊢ (𝜑 → {(𝑆‘(𝑢 + 𝑡))} = {((𝑆‘𝑢) ✚ (𝑆‘𝑡))})
29	28	fveq2d 6864	. 2 ⊢ (𝜑 → (𝐿‘{(𝑆‘(𝑢 + 𝑡))}) = (𝐿‘{((𝑆‘𝑢) ✚ (𝑆‘𝑡))}))
30	1, 26, 29	3eqtrd 2769	1 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝐿‘{((𝑆‘𝑢) ✚ (𝑆‘𝑡))}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∖ cdif 3913 {csn 4591 ‘cfv 6513 (class class class)co 7389 Basecbs 17185 +_gcplusg 17226 0_gc0g 17408 LModclmod 20772 LSpanclspn 20883 HLchlt 39338 LHypclh 39973 DVecHcdvh 41067 LCDualclcd 41575 mapdcmpd 41613 HDMapchdma 41781

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-riotaBAD 38941

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-ot 4600 df-uni 4874 df-int 4913 df-iun 4959 df-iin 4960 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-of 7655 df-om 7845 df-1st 7970 df-2nd 7971 df-tpos 8207 df-undef 8254 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-1o 8436 df-2o 8437 df-er 8673 df-map 8803 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-nn 12188 df-2 12250 df-3 12251 df-4 12252 df-5 12253 df-6 12254 df-n0 12449 df-z 12536 df-uz 12800 df-fz 13475 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17186 df-ress 17207 df-plusg 17239 df-mulr 17240 df-sca 17242 df-vsca 17243 df-0g 17410 df-mre 17553 df-mrc 17554 df-acs 17556 df-proset 18261 df-poset 18280 df-plt 18295 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-p0 18390 df-p1 18391 df-lat 18397 df-clat 18464 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18717 df-grp 18874 df-minusg 18875 df-sbg 18876 df-subg 19061 df-cntz 19255 df-oppg 19284 df-lsm 19572 df-cmn 19718 df-abl 19719 df-mgp 20056 df-rng 20068 df-ur 20097 df-ring 20150 df-oppr 20252 df-dvdsr 20272 df-unit 20273 df-invr 20303 df-dvr 20316 df-nzr 20428 df-rlreg 20609 df-domn 20610 df-drng 20646 df-lmod 20774 df-lss 20844 df-lsp 20884 df-lvec 21016 df-lsatoms 38964 df-lshyp 38965 df-lcv 39007 df-lfl 39046 df-lkr 39074 df-ldual 39112 df-oposet 39164 df-ol 39166 df-oml 39167 df-covers 39254 df-ats 39255 df-atl 39286 df-cvlat 39310 df-hlat 39339 df-llines 39487 df-lplanes 39488 df-lvols 39489 df-lines 39490 df-psubsp 39492 df-pmap 39493 df-padd 39785 df-lhyp 39977 df-laut 39978 df-ldil 40093 df-ltrn 40094 df-trl 40148 df-tgrp 40732 df-tendo 40744 df-edring 40746 df-dveca 40992 df-disoa 41018 df-dvech 41068 df-dib 41128 df-dic 41162 df-dih 41218 df-doch 41337 df-djh 41384 df-lcdual 41576 df-mapd 41614 df-hvmap 41746 df-hdmap1 41782 df-hdmap 41783

This theorem is referenced by: hdmaprnlem7N 41844

W3C validator