mapdh8b - Mathbox for Norm Megill

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Theorem mapdh8b 42404

Description: Part of Part (8) in [Baer] p. 48. (Contributed by NM, 6-May-2015.)

**Hypotheses**
Ref	Expression
mapdh8a.h	⊢ 𝐻 = (LHyp‘𝐾)
mapdh8a.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
mapdh8a.v	⊢ 𝑉 = (Base‘𝑈)
mapdh8a.s	⊢ − = (-_g‘𝑈)
mapdh8a.o	⊢ 0 = (0_g‘𝑈)
mapdh8a.n	⊢ 𝑁 = (LSpan‘𝑈)
mapdh8a.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
mapdh8a.d	⊢ 𝐷 = (Base‘𝐶)
mapdh8a.r	⊢ 𝑅 = (-_g‘𝐶)
mapdh8a.q	⊢ 𝑄 = (0_g‘𝐶)
mapdh8a.j	⊢ 𝐽 = (LSpan‘𝐶)
mapdh8a.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
mapdh8a.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)})))))
mapdh8a.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
mapdh8b.f	⊢ (𝜑 → 𝐺 ∈ 𝐷)
mapdh8b.mn	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}))
mapdh8b.a	⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑤⟩) = 𝐸)
mapdh8b.x	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
mapdh8b.y	⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))
mapdh8b.yz	⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))
mapdh8b.xt	⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 }))
mapdh8b.vw	⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤}))
mapdh8b.e	⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑇}))
mapdh8b.xn	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤}))

**Assertion**
Ref	Expression
mapdh8b	⊢ (𝜑 → (𝐼‘⟨𝑤, 𝐸, 𝑇⟩) = (𝐼‘⟨𝑌, 𝐺, 𝑇⟩))

Distinct variable groups: 𝑥,ℎ, − 0 ,ℎ,𝑥 𝐶,ℎ 𝐷,ℎ,𝑥 ℎ,𝐼 ℎ,𝐺,𝑥 ℎ,𝐽,𝑥 ℎ,𝑀,𝑥 ℎ,𝑁,𝑥 𝜑,ℎ 𝑅,ℎ,𝑥 𝑥,𝑄 𝑇,ℎ,𝑥 𝑈,ℎ ℎ,𝑋,𝑥 ℎ,𝑌,𝑥 ℎ,𝐸,𝑥 𝑤,ℎ,𝑥 Allowed substitution hints: 𝜑(𝑥,𝑤) 𝐶(𝑥,𝑤) 𝐷(𝑤) 𝑄(𝑤,ℎ) 𝑅(𝑤) 𝑇(𝑤) 𝑈(𝑥,𝑤) 𝐸(𝑤) 𝐺(𝑤) 𝐻(𝑥,𝑤,ℎ) 𝐼(𝑥,𝑤) 𝐽(𝑤) 𝐾(𝑥,𝑤,ℎ) 𝑀(𝑤) − (𝑤) 𝑁(𝑤) 𝑉(𝑥,𝑤,ℎ) 𝑊(𝑥,𝑤,ℎ) 𝑋(𝑤) 𝑌(𝑤) 0 (𝑤)

**Proof of Theorem mapdh8b**
Step	Hyp	Ref	Expression
1		mapdh8a.h	. 2 ⊢ 𝐻 = (LHyp‘𝐾)
2		mapdh8a.u	. 2 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		mapdh8a.v	. 2 ⊢ 𝑉 = (Base‘𝑈)
4		mapdh8a.s	. 2 ⊢ − = (-_g‘𝑈)
5		mapdh8a.o	. 2 ⊢ 0 = (0_g‘𝑈)
6		mapdh8a.n	. 2 ⊢ 𝑁 = (LSpan‘𝑈)
7		mapdh8a.c	. 2 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
8		mapdh8a.d	. 2 ⊢ 𝐷 = (Base‘𝐶)
9		mapdh8a.r	. 2 ⊢ 𝑅 = (-_g‘𝐶)
10		mapdh8a.q	. 2 ⊢ 𝑄 = (0_g‘𝐶)
11		mapdh8a.j	. 2 ⊢ 𝐽 = (LSpan‘𝐶)
12		mapdh8a.m	. 2 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
13		mapdh8a.i	. 2 ⊢ 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)})))))
14		mapdh8a.k	. 2 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
15		mapdh8b.f	. 2 ⊢ (𝜑 → 𝐺 ∈ 𝐷)
16		mapdh8b.mn	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}))
17		mapdh8b.a	. 2 ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑤⟩) = 𝐸)
18		mapdh8b.x	. 2 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
19		mapdh8b.y	. 2 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))
20		mapdh8b.yz	. 2 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))
21		mapdh8b.xt	. 2 ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 }))
22	1, 2, 14	dvhlvec 41733	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LVec)
23	18	eldifad 3916	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
24	19	eldifad 3916	. . . 4 ⊢ (𝜑 → 𝑤 ∈ 𝑉)
25	21	eldifad 3916	. . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝑉)
26		mapdh8b.e	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑇}))
27		mapdh8b.xn	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤}))
28	3, 6, 22, 23, 24, 25, 26, 27	lspindp5 42394	. . 3 ⊢ (𝜑 → ¬ 𝑇 ∈ (𝑁‘{𝑌, 𝑤}))
29		prcom 4691	. . . . . 6 ⊢ {𝑤, 𝑇} = {𝑇, 𝑤}
30	29	fveq2i 6870	. . . . 5 ⊢ (𝑁‘{𝑤, 𝑇}) = (𝑁‘{𝑇, 𝑤})
31	30	eleq2i 2854	. . . 4 ⊢ (𝑌 ∈ (𝑁‘{𝑤, 𝑇}) ↔ 𝑌 ∈ (𝑁‘{𝑇, 𝑤}))
32	22	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → 𝑈 ∈ LVec)
33	18	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → 𝑌 ∈ (𝑉 ∖ { 0 }))
34	25	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → 𝑇 ∈ 𝑉)
35	24	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → 𝑤 ∈ 𝑉)
36		mapdh8b.vw	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤}))
37	36	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤}))
38		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → 𝑌 ∈ (𝑁‘{𝑇, 𝑤}))
39	3, 5, 6, 32, 33, 34, 35, 37, 38	lspexch 21199	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑇, 𝑤})) → 𝑇 ∈ (𝑁‘{𝑌, 𝑤}))
40	39	ex 416	. . . 4 ⊢ (𝜑 → (𝑌 ∈ (𝑁‘{𝑇, 𝑤}) → 𝑇 ∈ (𝑁‘{𝑌, 𝑤})))
41	31, 40	biimtrid 244	. . 3 ⊢ (𝜑 → (𝑌 ∈ (𝑁‘{𝑤, 𝑇}) → 𝑇 ∈ (𝑁‘{𝑌, 𝑤})))
42	28, 41	mtod 200	. 2 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑤, 𝑇}))
43	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 42	mapdh8a 42399	1 ⊢ (𝜑 → (𝐼‘⟨𝑤, 𝐸, 𝑇⟩) = (𝐼‘⟨𝑌, 𝐺, 𝑇⟩))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1560 ∈ wcel 2142 ≠ wne 2957 Vcvv 3454 ∖ cdif 3901 ifcif 4480 {csn 4582 {cpr 4584 ⟨cotp 4590 ↦ cmpt 5181 ‘cfv 6521 ℩crio 7352 (class class class)co 7396 1^st c1st 7968 2^nd c2nd 7969 Basecbs 17245 0_gc0g 17468 -_gcsg 18977 LSpanclspn 21038 LVecclvec 21169 HLchlt 39974 LHypclh 40608 DVecHcdvh 41702 LCDualclcd 42210 mapdcmpd 42248

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-riotaBAD 39577

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-ot 4591 df-uni 4866 df-int 4906 df-iun 4951 df-iin 4952 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7660 df-om 7847 df-1st 7970 df-2nd 7971 df-tpos 8206 df-undef 8253 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8678 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-n0 12482 df-z 12569 df-uz 12840 df-fz 13513 df-struct 17183 df-sets 17200 df-slot 17218 df-ndx 17230 df-base 17246 df-ress 17267 df-plusg 17299 df-mulr 17300 df-sca 17302 df-vsca 17303 df-0g 17470 df-mre 17614 df-mrc 17615 df-acs 17617 df-proset 18326 df-poset 18345 df-plt 18360 df-lub 18376 df-glb 18377 df-join 18378 df-meet 18379 df-p0 18455 df-p1 18456 df-lat 18464 df-clat 18531 df-mgm 18674 df-sgrp 18753 df-mnd 18769 df-submnd 18818 df-grp 18978 df-minusg 18979 df-sbg 18980 df-subg 19165 df-cntz 19357 df-oppg 19386 df-lsm 19676 df-cmn 19822 df-abl 19823 df-mgp 20187 df-rng 20199 df-ur 20232 df-ring 20285 df-oppr 20386 df-dvdsr 20406 df-unit 20407 df-invr 20437 df-dvr 20450 df-nzr 20563 df-rlreg 20744 df-domn 20745 df-drng 20781 df-lmod 20929 df-lss 20999 df-lsp 21039 df-lvec 21170 df-lsatoms 39600 df-lshyp 39601 df-lcv 39643 df-lfl 39682 df-lkr 39710 df-ldual 39748 df-oposet 39800 df-ol 39802 df-oml 39803 df-covers 39890 df-ats 39891 df-atl 39922 df-cvlat 39946 df-hlat 39975 df-llines 40122 df-lplanes 40123 df-lvols 40124 df-lines 40125 df-psubsp 40127 df-pmap 40128 df-padd 40420 df-lhyp 40612 df-laut 40613 df-ldil 40728 df-ltrn 40729 df-trl 40783 df-tgrp 41367 df-tendo 41379 df-edring 41381 df-dveca 41627 df-disoa 41653 df-dvech 41703 df-dib 41763 df-dic 41797 df-dih 41853 df-doch 41972 df-djh 42019 df-lcdual 42211 df-mapd 42249

This theorem is referenced by: mapdh8c 42405 mapdh8d0N 42406 mapdh8d 42407

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