Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > znbaslemOLD | Structured version Visualization version GIF version |
Description: Obsolete version of znbaslem 19705 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 2610 | . . . 4 ⊢ (le‘𝑌) = (le‘𝑌) | |
5 | 1, 2, 3, 4 | znval2 19704 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑌 = (𝑈 sSet 〈(le‘ndx), (le‘𝑌)〉)) |
6 | 5 | fveq2d 6107 | . 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 15716 | . . 3 ⊢ 𝐸 = Slot (𝐸‘ndx) |
10 | 8 | nnrei 10906 | . . . . 5 ⊢ 𝐾 ∈ ℝ |
11 | znbaslemOLD.l | . . . . 5 ⊢ 𝐾 < 10 | |
12 | 10, 11 | ltneii 10029 | . . . 4 ⊢ 𝐾 ≠ 10 |
13 | 7, 8 | ndxarg 15715 | . . . . 5 ⊢ (𝐸‘ndx) = 𝐾 |
14 | plendxOLD 15871 | . . . . 5 ⊢ (le‘ndx) = 10 | |
15 | 13, 14 | neeq12i 2848 | . . . 4 ⊢ ((𝐸‘ndx) ≠ (le‘ndx) ↔ 𝐾 ≠ 10) |
16 | 12, 15 | mpbir 220 | . . 3 ⊢ (𝐸‘ndx) ≠ (le‘ndx) |
17 | 9, 16 | setsnid 15743 | . 2 ⊢ (𝐸‘𝑈) = (𝐸‘(𝑈 sSet 〈(le‘ndx), (le‘𝑌)〉)) |
18 | 6, 17 | syl6reqr 2663 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝐸‘𝑈) = (𝐸‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 {csn 4125 〈cop 4131 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 < clt 9953 ℕcn 10897 10c10 10955 ℕ0cn0 11169 ndxcnx 15692 sSet csts 15693 Slot cslot 15694 lecple 15775 /s cqus 15988 ~QG cqg 17413 RSpancrsp 18992 ℤringzring 19637 ℤ/nℤczn 19670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-10OLD 10964 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-subg 17414 df-cmn 18018 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-subrg 18601 df-cnfld 19568 df-zring 19638 df-zn 19674 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |