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Mirrors > Home > MPE Home > Th. List > znbaslemOLD | Structured version Visualization version GIF version |
Description: Obsolete version of znbaslem 19886 as of 28-Apr-2021. (Contributed by Mario Carneiro, 14-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
znval2.s | ⊢ 𝑆 = (RSpan‘ℤring) |
znval2.u | ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) |
znval2.y | ⊢ 𝑌 = (ℤ/nℤ‘𝑁) |
znbaslemOLD.e | ⊢ 𝐸 = Slot 𝐾 |
znbaslemOLD.k | ⊢ 𝐾 ∈ ℕ |
znbaslemOLD.l | ⊢ 𝐾 < 10 |
Ref | Expression |
---|---|
znbaslemOLD | ⊢ (𝑁 ∈ ℕ0 → (𝐸‘𝑈) = (𝐸‘𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | znval2.s | . . . 4 ⊢ 𝑆 = (RSpan‘ℤring) | |
2 | znval2.u | . . . 4 ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) | |
3 | znval2.y | . . . 4 ⊢ 𝑌 = (ℤ/nℤ‘𝑁) | |
4 | eqid 2622 | . . . 4 ⊢ (le‘𝑌) = (le‘𝑌) | |
5 | 1, 2, 3, 4 | znval2 19885 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑌 = (𝑈 sSet 〈(le‘ndx), (le‘𝑌)〉)) |
6 | 5 | fveq2d 6195 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝐸‘𝑌) = (𝐸‘(𝑈 sSet 〈(le‘ndx), (le‘𝑌)〉))) |
7 | znbaslemOLD.e | . . . 4 ⊢ 𝐸 = Slot 𝐾 | |
8 | znbaslemOLD.k | . . . 4 ⊢ 𝐾 ∈ ℕ | |
9 | 7, 8 | ndxid 15883 | . . 3 ⊢ 𝐸 = Slot (𝐸‘ndx) |
10 | 8 | nnrei 11029 | . . . . 5 ⊢ 𝐾 ∈ ℝ |
11 | znbaslemOLD.l | . . . . 5 ⊢ 𝐾 < 10 | |
12 | 10, 11 | ltneii 10150 | . . . 4 ⊢ 𝐾 ≠ 10 |
13 | 7, 8 | ndxarg 15882 | . . . . 5 ⊢ (𝐸‘ndx) = 𝐾 |
14 | plendxOLD 16048 | . . . . 5 ⊢ (le‘ndx) = 10 | |
15 | 13, 14 | neeq12i 2860 | . . . 4 ⊢ ((𝐸‘ndx) ≠ (le‘ndx) ↔ 𝐾 ≠ 10) |
16 | 12, 15 | mpbir 221 | . . 3 ⊢ (𝐸‘ndx) ≠ (le‘ndx) |
17 | 9, 16 | setsnid 15915 | . 2 ⊢ (𝐸‘𝑈) = (𝐸‘(𝑈 sSet 〈(le‘ndx), (le‘𝑌)〉)) |
18 | 6, 17 | syl6reqr 2675 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝐸‘𝑈) = (𝐸‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 {csn 4177 〈cop 4183 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 < clt 10074 ℕcn 11020 10c10 11078 ℕ0cn0 11292 ndxcnx 15854 sSet csts 15855 Slot cslot 15856 lecple 15948 /s cqus 16165 ~QG cqg 17590 RSpancrsp 19171 ℤringzring 19818 ℤ/nℤczn 19851 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-10OLD 11087 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-subg 17591 df-cmn 18195 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-subrg 18778 df-cnfld 19747 df-zring 19819 df-zn 19855 |
This theorem is referenced by: (None) |
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