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Mirrors > Home > MPE Home > Th. List > znbaslem | Structured version Visualization version GIF version |
Description: Lemma for znbas 20684. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 9-Sep-2021.) |
Ref | Expression |
---|---|
znval2.s | ⊢ 𝑆 = (RSpan‘ℤring) |
znval2.u | ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) |
znval2.y | ⊢ 𝑌 = (ℤ/nℤ‘𝑁) |
znbaslem.e | ⊢ 𝐸 = Slot 𝐾 |
znbaslem.k | ⊢ 𝐾 ∈ ℕ |
znbaslem.l | ⊢ 𝐾 < ;10 |
Ref | Expression |
---|---|
znbaslem | ⊢ (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 2821 | . . . 4 ⊢ (le‘𝑌) = (le‘𝑌) | |
5 | 1, 2, 3, 4 | znval2 20678 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑌 = (𝑈 sSet 〈(le‘ndx), (le‘𝑌)〉)) |
6 | 5 | fveq2d 6669 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝐸‘𝑌) = (𝐸‘(𝑈 sSet 〈(le‘ndx), (le‘𝑌)〉))) |
7 | znbaslem.e | . . . 4 ⊢ 𝐸 = Slot 𝐾 | |
8 | znbaslem.k | . . . 4 ⊢ 𝐾 ∈ ℕ | |
9 | 7, 8 | ndxid 16503 | . . 3 ⊢ 𝐸 = Slot (𝐸‘ndx) |
10 | 8 | nnrei 11641 | . . . . 5 ⊢ 𝐾 ∈ ℝ |
11 | znbaslem.l | . . . . 5 ⊢ 𝐾 < ;10 | |
12 | 10, 11 | ltneii 10747 | . . . 4 ⊢ 𝐾 ≠ ;10 |
13 | 7, 8 | ndxarg 16502 | . . . . 5 ⊢ (𝐸‘ndx) = 𝐾 |
14 | plendx 16660 | . . . . 5 ⊢ (le‘ndx) = ;10 | |
15 | 13, 14 | neeq12i 3082 | . . . 4 ⊢ ((𝐸‘ndx) ≠ (le‘ndx) ↔ 𝐾 ≠ ;10) |
16 | 12, 15 | mpbir 233 | . . 3 ⊢ (𝐸‘ndx) ≠ (le‘ndx) |
17 | 9, 16 | setsnid 16533 | . 2 ⊢ (𝐸‘𝑈) = (𝐸‘(𝑈 sSet 〈(le‘ndx), (le‘𝑌)〉)) |
18 | 6, 17 | syl6reqr 2875 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝐸‘𝑈) = (𝐸‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 {csn 4561 〈cop 4567 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 0cc0 10531 1c1 10532 < clt 10669 ℕcn 11632 ℕ0cn0 11891 ;cdc 12092 ndxcnx 16474 sSet csts 16475 Slot cslot 16476 lecple 16566 /s cqus 16772 ~QG cqg 18269 RSpancrsp 19937 ℤringzring 20611 ℤ/nℤczn 20644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-subg 18270 df-cmn 18902 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-subrg 19527 df-cnfld 20540 df-zring 20612 df-zn 20648 |
This theorem is referenced by: znbas2 20680 znadd 20681 znmul 20682 |
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