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Theorem odubas 17799

Description: Base set of an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.)

**Hypotheses**
Ref	Expression
oduval.d	⊢ 𝐷 = (ODual‘𝑂)
odubas.b	⊢ 𝐵 = (Base‘𝑂)

**Assertion**
Ref	Expression
odubas	⊢ 𝐵 = (Base‘𝐷)

**Proof of Theorem odubas**
Step	Hyp	Ref	Expression
1		baseid 16763	. . 3 ⊢ Base = Slot (Base‘ndx)
2		1re 10833	. . . . 5 ⊢ 1 ∈ ℝ
3		1lt10 12432	. . . . 5 ⊢ 1 < ;10
4	2, 3	ltneii 10945	. . . 4 ⊢ 1 ≠ ;10
5		basendx 16769	. . . . 5 ⊢ (Base‘ndx) = 1
6		plendx 16893	. . . . 5 ⊢ (le‘ndx) = ;10
7	5, 6	neeq12i 3007	. . . 4 ⊢ ((Base‘ndx) ≠ (le‘ndx) ↔ 1 ≠ ;10)
8	4, 7	mpbir 234	. . 3 ⊢ (Base‘ndx) ≠ (le‘ndx)
9	1, 8	setsnid 16759	. 2 ⊢ (Base‘𝑂) = (Base‘(𝑂 sSet ⟨(le‘ndx), ^◡(le‘𝑂)⟩))
10		odubas.b	. 2 ⊢ 𝐵 = (Base‘𝑂)
11		oduval.d	. . . 4 ⊢ 𝐷 = (ODual‘𝑂)
12		eqid 2737	. . . 4 ⊢ (le‘𝑂) = (le‘𝑂)
13	11, 12	oduval 17796	. . 3 ⊢ 𝐷 = (𝑂 sSet ⟨(le‘ndx), ^◡(le‘𝑂)⟩)
14	13	fveq2i 6720	. 2 ⊢ (Base‘𝐷) = (Base‘(𝑂 sSet ⟨(le‘ndx), ^◡(le‘𝑂)⟩))
15	9, 10, 14	3eqtr4i 2775	1 ⊢ 𝐵 = (Base‘𝐷)

Colors of variables: wff setvar class

Syntax hints: = wceq 1543 ≠ wne 2940 ⟨cop 4547 ^◡ccnv 5550 ‘cfv 6380 (class class class)co 7213 0cc0 10729 1c1 10730 cdc 12293 sSet csts 16716 ndxcnx 16744 Basecbs 16760 lecple 16809 ODualcodu 17794

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-dec 12294 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ple 16822 df-odu 17795

This theorem is referenced by: odupos 17834 oduposb 17835 odulub 17913 oduglb 17915 posglbdg 17921 odulatb 17940 latmass 18001 latdisd 18003 oduclatb 18013 odudlatb 18031 dlatjmdi 18032 oduprs 30961 odutos 30965 mgccnv 30996 ordtcnvNEW 31584 ordtrest2NEW 31587 glbprlem 45932

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