mapdh8g - Mathbox for Norm Megill

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Theorem mapdh8g 41122

Description: Part of Part (8) in [Baer] p. 48. Eliminate 𝑋 ∈ (𝑁‘{𝑌, 𝑇}). TODO: break out 𝑇 ≠ 0 in mapdh8e 41121 so we can share hypotheses. Also, look at hypothesis sharing for earlier mapdh8* and mapdh75* stuff. (Contributed by NM, 10-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))
mapdh8e.f	⊢ (𝜑 → 𝐹 ∈ 𝐷)
mapdh8e.mn	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
mapdh8e.eg	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)
mapdh8e.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
mapdh8e.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
mapdh8e.t	⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 }))
mapdh8e.xy	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
mapdh8e.xt	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇}))
mapdh8e.yt	⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))

**Assertion**
Ref	Expression
mapdh8g	⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑇⟩))

Distinct variable groups: 𝑥,ℎ, − 0 ,ℎ,𝑥 𝐶,ℎ 𝐷,ℎ,𝑥 ℎ,𝐹,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 𝑥,𝐼 ℎ,𝑉 Allowed substitution hints: 𝜑(𝑥) 𝐶(𝑥) 𝑄(ℎ) 𝑈(𝑥) 𝐻(𝑥,ℎ) 𝐾(𝑥,ℎ) 𝑉(𝑥) 𝑊(𝑥,ℎ)

**Proof of Theorem mapdh8g**
Step	Hyp	Ref	Expression
1		mapdh8a.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		mapdh8a.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		mapdh8a.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
4		mapdh8a.s	. . 3 ⊢ − = (-_g‘𝑈)
5		mapdh8a.o	. . 3 ⊢ 0 = (0_g‘𝑈)
6		mapdh8a.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
7		mapdh8a.c	. . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
8		mapdh8a.d	. . 3 ⊢ 𝐷 = (Base‘𝐶)
9		mapdh8a.r	. . 3 ⊢ 𝑅 = (-_g‘𝐶)
10		mapdh8a.q	. . 3 ⊢ 𝑄 = (0_g‘𝐶)
11		mapdh8a.j	. . 3 ⊢ 𝐽 = (LSpan‘𝐶)
12		mapdh8a.m	. . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
13		mapdh8a.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)})))))
14		mapdh8a.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
15	14	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
16		mapdh8e.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
17	16	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → 𝐹 ∈ 𝐷)
18		mapdh8e.mn	. . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
19	18	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
20		mapdh8e.eg	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)
21	20	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)
22		mapdh8e.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
23	22	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → 𝑋 ∈ (𝑉 ∖ { 0 }))
24		mapdh8e.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
25	24	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → 𝑌 ∈ (𝑉 ∖ { 0 }))
26		mapdh8e.t	. . . 4 ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 }))
27	26	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → 𝑇 ∈ (𝑉 ∖ { 0 }))
28		mapdh8e.xy	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
29	28	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
30		mapdh8e.xt	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇}))
31	30	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇}))
32		mapdh8e.yt	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))
33	32	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))
34		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → 𝑋 ∈ (𝑁‘{𝑌, 𝑇}))
35	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 34	mapdh8e 41121	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑇⟩))
36	14	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
37	16	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → 𝐹 ∈ 𝐷)
38	18	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
39	20	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)
40	22	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → 𝑋 ∈ (𝑉 ∖ { 0 }))
41	24	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → 𝑌 ∈ (𝑉 ∖ { 0 }))
42	32	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))
43	26	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → 𝑇 ∈ (𝑉 ∖ { 0 }))
44		simpr 484	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇}))
45	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 36, 37, 38, 39, 40, 41, 42, 43, 44	mapdh8a 41112	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑇⟩))
46	35, 45	pm2.61dan 810	1 ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑇⟩))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 Vcvv 3473 ∖ cdif 3945 ifcif 4528 {csn 4628 {cpr 4630 ⟨cotp 4636 ↦ cmpt 5231 ‘cfv 6543 ℩crio 7367 (class class class)co 7412 1^st c1st 7977 2^nd c2nd 7978 Basecbs 17151 0_gc0g 17392 -_gcsg 18863 LSpanclspn 20814 HLchlt 38686 LHypclh 39321 DVecHcdvh 40415 LCDualclcd 40923 mapdcmpd 40961

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-riotaBAD 38289

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-ot 4637 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8217 df-undef 8264 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-n0 12480 df-z 12566 df-uz 12830 df-fz 13492 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-0g 17394 df-mre 17537 df-mrc 17538 df-acs 17540 df-proset 18258 df-poset 18276 df-plt 18293 df-lub 18309 df-glb 18310 df-join 18311 df-meet 18312 df-p0 18388 df-p1 18389 df-lat 18395 df-clat 18462 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19046 df-cntz 19229 df-oppg 19258 df-lsm 19552 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-dvr 20299 df-drng 20585 df-lmod 20704 df-lss 20775 df-lsp 20815 df-lvec 20947 df-lsatoms 38312 df-lshyp 38313 df-lcv 38355 df-lfl 38394 df-lkr 38422 df-ldual 38460 df-oposet 38512 df-ol 38514 df-oml 38515 df-covers 38602 df-ats 38603 df-atl 38634 df-cvlat 38658 df-hlat 38687 df-llines 38835 df-lplanes 38836 df-lvols 38837 df-lines 38838 df-psubsp 38840 df-pmap 38841 df-padd 39133 df-lhyp 39325 df-laut 39326 df-ldil 39441 df-ltrn 39442 df-trl 39496 df-tgrp 40080 df-tendo 40092 df-edring 40094 df-dveca 40340 df-disoa 40366 df-dvech 40416 df-dib 40476 df-dic 40510 df-dih 40566 df-doch 40685 df-djh 40732 df-lcdual 40924 df-mapd 40962

This theorem is referenced by: mapdh8i 41123

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