mapdh6jN - Mathbox for Norm Megill

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Theorem mapdh6jN 41733

Description: Lemmma for mapdh6N 41735. Eliminate (𝑁‘{𝑌}) = (𝑁‘{𝑍}) hypothesis. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
mapdh.q	⊢ 𝑄 = (0_g‘𝐶)
mapdh.i	⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘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 = (0_g‘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)
mapdh6i.xn	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
mapdh6i.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
mapdh6i.z	⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))

**Assertion**
Ref	Expression
mapdh6jN	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

Distinct variable groups: 𝑥,𝐷,ℎ ℎ,𝐹,𝑥 𝑥,𝐽 𝑥,𝑀 𝑥,𝑁 𝑥, 0 𝑥,𝑄 𝑥,𝑅 𝑥, − ℎ,𝑋,𝑥 ℎ,𝑌,𝑥 𝜑,ℎ 0 ,ℎ 𝐶,ℎ 𝐷,ℎ ℎ,𝐽 ℎ,𝑀 ℎ,𝑁 𝑅,ℎ 𝑈,ℎ − ,ℎ ℎ,𝑍,𝑥 ✚ ,ℎ ℎ,𝐼,𝑥 + ,ℎ,𝑥 ℎ,𝑉 Allowed substitution hints: 𝜑(𝑥) 𝐶(𝑥) ✚ (𝑥) 𝑄(ℎ) 𝑈(𝑥) 𝐻(𝑥,ℎ) 𝐾(𝑥,ℎ) 𝑉(𝑥) 𝑊(𝑥,ℎ)

**Proof of Theorem mapdh6jN**
Step	Hyp	Ref	Expression
1		mapdh.q	. . 3 ⊢ 𝑄 = (0_g‘𝐶)
2		mapdh.i	. . 3 ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)})))))
3		mapdh.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
4		mapdh.m	. . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
5		mapdh.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
6		mapdh.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
7		mapdh.s	. . 3 ⊢ − = (-_g‘𝑈)
8		mapdhc.o	. . 3 ⊢ 0 = (0_g‘𝑈)
9		mapdh.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
10		mapdh.c	. . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
11		mapdh.d	. . 3 ⊢ 𝐷 = (Base‘𝐶)
12		mapdh.r	. . 3 ⊢ 𝑅 = (-_g‘𝐶)
13		mapdh.j	. . 3 ⊢ 𝐽 = (LSpan‘𝐶)
14		mapdh.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
15	14	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) = (𝑁‘{𝑍})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
16		mapdhc.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
17	16	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) = (𝑁‘{𝑍})) → 𝐹 ∈ 𝐷)
18		mapdh.mn	. . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
19	18	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) = (𝑁‘{𝑍})) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
20		mapdhcl.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
21	20	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) = (𝑁‘{𝑍})) → 𝑋 ∈ (𝑉 ∖ { 0 }))
22		mapdh.p	. . 3 ⊢ + = (+_g‘𝑈)
23		mapdh.a	. . 3 ⊢ ✚ = (+_g‘𝐶)
24		mapdh6i.xn	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
25	24	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) = (𝑁‘{𝑍})) → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
26		mapdh6i.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
27	26	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) = (𝑁‘{𝑍})) → 𝑌 ∈ (𝑉 ∖ { 0 }))
28		mapdh6i.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
29	28	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) = (𝑁‘{𝑍})) → 𝑍 ∈ (𝑉 ∖ { 0 }))
30		simpr 484	. . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) = (𝑁‘{𝑍})) → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))
31	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 17, 19, 21, 22, 23, 25, 27, 29, 30	mapdh6iN 41732	. 2 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) = (𝑁‘{𝑍})) → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
32	14	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
33	16	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) → 𝐹 ∈ 𝐷)
34	18	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
35	20	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) → 𝑋 ∈ (𝑉 ∖ { 0 }))
36	26	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) → 𝑌 ∈ (𝑉 ∖ { 0 }))
37	28	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) → 𝑍 ∈ (𝑉 ∖ { 0 }))
38	24	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
39		simpr 484	. . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))
40		eqidd 2730	. . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑌⟩))
41		eqidd 2730	. . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))
42	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 32, 33, 34, 35, 22, 23, 36, 37, 38, 39, 40, 41	mapdh6aN 41723	. 2 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
43	31, 42	pm2.61dane 3012	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 1540 ∈ wcel 2109 ≠ wne 2925 Vcvv 3444 ∖ cdif 3908 ifcif 4484 {csn 4585 {cpr 4587 ⟨cotp 4593 ↦ cmpt 5183 ‘cfv 6499 ℩crio 7325 (class class class)co 7369 1^st c1st 7945 2^nd c2nd 7946 Basecbs 17156 +_gcplusg 17197 0_gc0g 17379 -_gcsg 18850 LSpanclspn 20910 HLchlt 39337 LHypclh 39972 DVecHcdvh 41066 LCDualclcd 41574 mapdcmpd 41612

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 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11102 ax-resscn 11103 ax-1cn 11104 ax-icn 11105 ax-addcl 11106 ax-addrcl 11107 ax-mulcl 11108 ax-mulrcl 11109 ax-mulcom 11110 ax-addass 11111 ax-mulass 11112 ax-distr 11113 ax-i2m1 11114 ax-1ne0 11115 ax-1rid 11116 ax-rnegex 11117 ax-rrecex 11118 ax-cnre 11119 ax-pre-lttri 11120 ax-pre-lttrn 11121 ax-pre-ltadd 11122 ax-pre-mulgt0 11123 ax-riotaBAD 38940

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-ot 4594 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-tpos 8182 df-undef 8229 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11385 df-neg 11386 df-nn 12165 df-2 12227 df-3 12228 df-4 12229 df-5 12230 df-6 12231 df-n0 12421 df-z 12508 df-uz 12772 df-fz 13447 df-struct 17094 df-sets 17111 df-slot 17129 df-ndx 17141 df-base 17157 df-ress 17178 df-plusg 17210 df-mulr 17211 df-sca 17213 df-vsca 17214 df-0g 17381 df-mre 17524 df-mrc 17525 df-acs 17527 df-proset 18236 df-poset 18255 df-plt 18270 df-lub 18286 df-glb 18287 df-join 18288 df-meet 18289 df-p0 18365 df-p1 18366 df-lat 18374 df-clat 18441 df-mgm 18550 df-sgrp 18629 df-mnd 18645 df-submnd 18694 df-grp 18851 df-minusg 18852 df-sbg 18853 df-subg 19038 df-cntz 19232 df-oppg 19261 df-lsm 19551 df-cmn 19697 df-abl 19698 df-mgp 20062 df-rng 20074 df-ur 20103 df-ring 20156 df-oppr 20258 df-dvdsr 20278 df-unit 20279 df-invr 20309 df-dvr 20322 df-nzr 20434 df-rlreg 20615 df-domn 20616 df-drng 20652 df-lmod 20801 df-lss 20871 df-lsp 20911 df-lvec 21043 df-lsatoms 38963 df-lshyp 38964 df-lcv 39006 df-lfl 39045 df-lkr 39073 df-ldual 39111 df-oposet 39163 df-ol 39165 df-oml 39166 df-covers 39253 df-ats 39254 df-atl 39285 df-cvlat 39309 df-hlat 39338 df-llines 39486 df-lplanes 39487 df-lvols 39488 df-lines 39489 df-psubsp 39491 df-pmap 39492 df-padd 39784 df-lhyp 39976 df-laut 39977 df-ldil 40092 df-ltrn 40093 df-trl 40147 df-tgrp 40731 df-tendo 40743 df-edring 40745 df-dveca 40991 df-disoa 41017 df-dvech 41067 df-dib 41127 df-dic 41161 df-dih 41217 df-doch 41336 df-djh 41383 df-lcdual 41575 df-mapd 41613

This theorem is referenced by: mapdh6kN 41734

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