znzrh2 - Intuitionistic Logic Explorer

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Theorem znzrh2 14653

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 14600	. . . . 5 ⊢ ℤ_ring ∈ Ring
3		nn0z 9492	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
4		znzrh2.s	. . . . . . 7 ⊢ 𝑆 = (RSpan‘ℤ_ring)
5	4	znlidl 14641	. . . . . 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 6024	. . . . . 6 ⊢ (ℤ_ring /_s ∼ ) = (ℤ_ring /_s (ℤ_ring ~_QG (𝑆‘{𝑁})))
9		zringcrng 14599	. . . . . . 7 ⊢ ℤ_ring ∈ CRing
10		eqid 2229	. . . . . . . 8 ⊢ (LIdeal‘ℤ_ring) = (LIdeal‘ℤ_ring)
11	10	crng2idl 14538	. . . . . . 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 14603	. . . . . 6 ⊢ ℤ = (Base‘ℤ_ring)
14		eceq2 6734	. . . . . . . 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 4174	. . . . . 6 ⊢ (𝑥 ∈ ℤ ↦ [𝑥] ∼ ) = (𝑥 ∈ ℤ ↦ [𝑥](ℤ_ring ~_QG (𝑆‘{𝑁})))
17	8, 12, 13, 16	qusrhm 14535	. . . . 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 14642	. . . . 5 ⊢ (𝑁 ∈ ℤ → (ℤ_ring /_s ∼ ) ∈ CRing)
20		crngring 14014	. . . . 5 ⊢ ((ℤ_ring /_s ∼ ) ∈ CRing → (ℤ_ring /_s ∼ ) ∈ Ring)
21		eqid 2229	. . . . . 6 ⊢ (ℤRHom‘(ℤ_ring /_s ∼ )) = (ℤRHom‘(ℤ_ring /_s ∼ ))
22	21	zrhrhmb 14629	. . . . 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 14650	. . 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 3667 ↦ cmpt 4148 ‘cfv 5324 (class class class)co 6013 [cec 6695 ℕ₀cn0 9395 ℤcz 9472 /_s cqus 13376 ~_QG cqg 13749 Ringcrg 14002 CRingccrg 14003 RingHom crh 14157 LIdealclidl 14474 RSpancrsp 14475 2Idealc2idl 14506 ℤ_ringczring 14597 ℤRHomczrh 14618 ℤ/nℤczn 14620

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 4202 ax-sep 4205 ax-nul 4213 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-iinf 4684 ax-cnex 8116 ax-resscn 8117 ax-1cn 8118 ax-1re 8119 ax-icn 8120 ax-addcl 8121 ax-addrcl 8122 ax-mulcl 8123 ax-mulrcl 8124 ax-addcom 8125 ax-mulcom 8126 ax-addass 8127 ax-mulass 8128 ax-distr 8129 ax-i2m1 8130 ax-0lt1 8131 ax-1rid 8132 ax-0id 8133 ax-rnegex 8134 ax-precex 8135 ax-cnre 8136 ax-pre-ltirr 8137 ax-pre-ltwlin 8138 ax-pre-lttrn 8139 ax-pre-apti 8140 ax-pre-ltadd 8141 ax-pre-mulgt0 8142 ax-addf 8147 ax-mulf 8148

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 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-if 3604 df-pw 3652 df-sn 3673 df-pr 3674 df-tp 3675 df-op 3676 df-uni 3892 df-int 3927 df-iun 3970 df-br 4087 df-opab 4149 df-mpt 4150 df-tr 4186 df-id 4388 df-iord 4461 df-on 4463 df-ilim 4464 df-suc 4466 df-iom 4687 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-1st 6298 df-2nd 6299 df-tpos 6406 df-recs 6466 df-frec 6552 df-er 6697 df-ec 6699 df-qs 6703 df-map 6814 df-pnf 8209 df-mnf 8210 df-xr 8211 df-ltxr 8212 df-le 8213 df-sub 8345 df-neg 8346 df-reap 8748 df-inn 9137 df-2 9195 df-3 9196 df-4 9197 df-5 9198 df-6 9199 df-7 9200 df-8 9201 df-9 9202 df-n0 9396 df-z 9473 df-dec 9605 df-uz 9749 df-rp 9882 df-fz 10237 df-fzo 10371 df-seqfrec 10703 df-cj 11396 df-abs 11553 df-struct 13077 df-ndx 13078 df-slot 13079 df-base 13081 df-sets 13082 df-iress 13083 df-plusg 13166 df-mulr 13167 df-starv 13168 df-sca 13169 df-vsca 13170 df-ip 13171 df-tset 13172 df-ple 13173 df-ds 13175 df-unif 13176 df-0g 13334 df-topgen 13336 df-iimas 13378 df-qus 13379 df-mgm 13432 df-sgrp 13478 df-mnd 13493 df-mhm 13535 df-grp 13579 df-minusg 13580 df-sbg 13581 df-mulg 13700 df-subg 13750 df-nsg 13751 df-eqg 13752 df-ghm 13821 df-cmn 13866 df-abl 13867 df-mgp 13927 df-rng 13939 df-ur 13966 df-srg 13970 df-ring 14004 df-cring 14005 df-oppr 14074 df-rhm 14159 df-subrg 14226 df-lmod 14296 df-lssm 14360 df-lsp 14394 df-sra 14442 df-rgmod 14443 df-lidl 14476 df-rsp 14477 df-2idl 14507 df-bl 14553 df-mopn 14554 df-fg 14556 df-metu 14557 df-cnfld 14564 df-zring 14598 df-zrh 14621 df-zn 14623

This theorem is referenced by: znzrhval 14654 znzrhfo 14655

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