znzrh2 - Intuitionistic Logic Explorer

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Theorem znzrh2 14618

Description: The ℤ ring homomorphism maps elements to their equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.)

**Hypotheses**
Ref	Expression
znzrh2.s	⊢ 𝑆 = (RSpan‘ℤ_ring)
znzrh2.r	⊢ ∼ = (ℤ_ring ~_QG (𝑆‘{𝑁}))
znzrh2.y	⊢ 𝑌 = (ℤ/nℤ‘𝑁)
znzrh2.2	⊢ 𝐿 = (ℤRHom‘𝑌)

**Assertion**
Ref	Expression
znzrh2	⊢ (𝑁 ∈ ℕ₀ → 𝐿 = (𝑥 ∈ ℤ ↦ [𝑥] ∼ ))

Distinct variable groups: 𝑥,𝑁 𝑥, ∼ 𝑥,𝑆 Allowed substitution hints: 𝐿(𝑥) 𝑌(𝑥)

**Proof of Theorem znzrh2**
Step	Hyp	Ref	Expression
1		znzrh2.2	. 2 ⊢ 𝐿 = (ℤRHom‘𝑌)
2		zringring 14565	. . . . 5 ⊢ ℤ_ring ∈ Ring
3		nn0z 9474	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
4		znzrh2.s	. . . . . . 7 ⊢ 𝑆 = (RSpan‘ℤ_ring)
5	4	znlidl 14606	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑆‘{𝑁}) ∈ (LIdeal‘ℤ_ring))
6	3, 5	syl 14	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (𝑆‘{𝑁}) ∈ (LIdeal‘ℤ_ring))
7		znzrh2.r	. . . . . . 7 ⊢ ∼ = (ℤ_ring ~_QG (𝑆‘{𝑁}))
8	7	oveq2i 6018	. . . . . 6 ⊢ (ℤ_ring /_s ∼ ) = (ℤ_ring /_s (ℤ_ring ~_QG (𝑆‘{𝑁})))
9		zringcrng 14564	. . . . . . 7 ⊢ ℤ_ring ∈ CRing
10		eqid 2229	. . . . . . . 8 ⊢ (LIdeal‘ℤ_ring) = (LIdeal‘ℤ_ring)
11	10	crng2idl 14503	. . . . . . 7 ⊢ (ℤ_ring ∈ CRing → (LIdeal‘ℤ_ring) = (2Ideal‘ℤ_ring))
12	9, 11	ax-mp 5	. . . . . 6 ⊢ (LIdeal‘ℤ_ring) = (2Ideal‘ℤ_ring)
13		zringbas 14568	. . . . . 6 ⊢ ℤ = (Base‘ℤ_ring)
14		eceq2 6725	. . . . . . . 8 ⊢ ( ∼ = (ℤ_ring ~_QG (𝑆‘{𝑁})) → [𝑥] ∼ = [𝑥](ℤ_ring ~_QG (𝑆‘{𝑁})))
15	7, 14	ax-mp 5	. . . . . . 7 ⊢ [𝑥] ∼ = [𝑥](ℤ_ring ~_QG (𝑆‘{𝑁}))
16	15	mpteq2i 4171	. . . . . 6 ⊢ (𝑥 ∈ ℤ ↦ [𝑥] ∼ ) = (𝑥 ∈ ℤ ↦ [𝑥](ℤ_ring ~_QG (𝑆‘{𝑁})))
17	8, 12, 13, 16	qusrhm 14500	. . . . 5 ⊢ ((ℤ_ring ∈ Ring ∧ (𝑆‘{𝑁}) ∈ (LIdeal‘ℤ_ring)) → (𝑥 ∈ ℤ ↦ [𝑥] ∼ ) ∈ (ℤ_ring RingHom (ℤ_ring /_s ∼ )))
18	2, 6, 17	sylancr 414	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → (𝑥 ∈ ℤ ↦ [𝑥] ∼ ) ∈ (ℤ_ring RingHom (ℤ_ring /_s ∼ )))
19	4, 8	zncrng2 14607	. . . . 5 ⊢ (𝑁 ∈ ℤ → (ℤ_ring /_s ∼ ) ∈ CRing)
20		crngring 13979	. . . . 5 ⊢ ((ℤ_ring /_s ∼ ) ∈ CRing → (ℤ_ring /_s ∼ ) ∈ Ring)
21		eqid 2229	. . . . . 6 ⊢ (ℤRHom‘(ℤ_ring /_s ∼ )) = (ℤRHom‘(ℤ_ring /_s ∼ ))
22	21	zrhrhmb 14594	. . . . 5 ⊢ ((ℤ_ring /_s ∼ ) ∈ Ring → ((𝑥 ∈ ℤ ↦ [𝑥] ∼ ) ∈ (ℤ_ring RingHom (ℤ_ring /_s ∼ )) ↔ (𝑥 ∈ ℤ ↦ [𝑥] ∼ ) = (ℤRHom‘(ℤ_ring /_s ∼ ))))
23	3, 19, 20, 22	4syl 18	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → ((𝑥 ∈ ℤ ↦ [𝑥] ∼ ) ∈ (ℤ_ring RingHom (ℤ_ring /_s ∼ )) ↔ (𝑥 ∈ ℤ ↦ [𝑥] ∼ ) = (ℤRHom‘(ℤ_ring /_s ∼ ))))
24	18, 23	mpbid 147	. . 3 ⊢ (𝑁 ∈ ℕ₀ → (𝑥 ∈ ℤ ↦ [𝑥] ∼ ) = (ℤRHom‘(ℤ_ring /_s ∼ )))
25		znzrh2.y	. . . 4 ⊢ 𝑌 = (ℤ/nℤ‘𝑁)
26	4, 8, 25	znzrh 14615	. . 3 ⊢ (𝑁 ∈ ℕ₀ → (ℤRHom‘(ℤ_ring /_s ∼ )) = (ℤRHom‘𝑌))
27	24, 26	eqtr2d 2263	. 2 ⊢ (𝑁 ∈ ℕ₀ → (ℤRHom‘𝑌) = (𝑥 ∈ ℤ ↦ [𝑥] ∼ ))
28	1, 27	eqtrid 2274	1 ⊢ (𝑁 ∈ ℕ₀ → 𝐿 = (𝑥 ∈ ℤ ↦ [𝑥] ∼ ))

Colors of variables: wff set class

Syntax hints: → wi 4 ↔ wb 105 = wceq 1395 ∈ wcel 2200 {csn 3666 ↦ cmpt 4145 ‘cfv 5318 (class class class)co 6007 [cec 6686 ℕ₀cn0 9377 ℤcz 9454 /_s cqus 13341 ~_QG cqg 13714 Ringcrg 13967 CRingccrg 13968 RingHom crh 14122 LIdealclidl 14439 RSpancrsp 14440 2Idealc2idl 14471 ℤ_ringczring 14562 ℤRHomczrh 14583 ℤ/nℤczn 14585

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8098 ax-resscn 8099 ax-1cn 8100 ax-1re 8101 ax-icn 8102 ax-addcl 8103 ax-addrcl 8104 ax-mulcl 8105 ax-mulrcl 8106 ax-addcom 8107 ax-mulcom 8108 ax-addass 8109 ax-mulass 8110 ax-distr 8111 ax-i2m1 8112 ax-0lt1 8113 ax-1rid 8114 ax-0id 8115 ax-rnegex 8116 ax-precex 8117 ax-cnre 8118 ax-pre-ltirr 8119 ax-pre-ltwlin 8120 ax-pre-lttrn 8121 ax-pre-apti 8122 ax-pre-ltadd 8123 ax-pre-mulgt0 8124 ax-addf 8129 ax-mulf 8130

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-tp 3674 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-tpos 6397 df-recs 6457 df-frec 6543 df-er 6688 df-ec 6690 df-qs 6694 df-map 6805 df-pnf 8191 df-mnf 8192 df-xr 8193 df-ltxr 8194 df-le 8195 df-sub 8327 df-neg 8328 df-reap 8730 df-inn 9119 df-2 9177 df-3 9178 df-4 9179 df-5 9180 df-6 9181 df-7 9182 df-8 9183 df-9 9184 df-n0 9378 df-z 9455 df-dec 9587 df-uz 9731 df-rp 9858 df-fz 10213 df-fzo 10347 df-seqfrec 10678 df-cj 11361 df-abs 11518 df-struct 13042 df-ndx 13043 df-slot 13044 df-base 13046 df-sets 13047 df-iress 13048 df-plusg 13131 df-mulr 13132 df-starv 13133 df-sca 13134 df-vsca 13135 df-ip 13136 df-tset 13137 df-ple 13138 df-ds 13140 df-unif 13141 df-0g 13299 df-topgen 13301 df-iimas 13343 df-qus 13344 df-mgm 13397 df-sgrp 13443 df-mnd 13458 df-mhm 13500 df-grp 13544 df-minusg 13545 df-sbg 13546 df-mulg 13665 df-subg 13715 df-nsg 13716 df-eqg 13717 df-ghm 13786 df-cmn 13831 df-abl 13832 df-mgp 13892 df-rng 13904 df-ur 13931 df-srg 13935 df-ring 13969 df-cring 13970 df-oppr 14039 df-rhm 14124 df-subrg 14191 df-lmod 14261 df-lssm 14325 df-lsp 14359 df-sra 14407 df-rgmod 14408 df-lidl 14441 df-rsp 14442 df-2idl 14472 df-bl 14518 df-mopn 14519 df-fg 14521 df-metu 14522 df-cnfld 14529 df-zring 14563 df-zrh 14586 df-zn 14588

This theorem is referenced by: znzrhval 14619 znzrhfo 14620

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