znzrh2 - Intuitionistic Logic Explorer

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Theorem znzrh2 14924

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 14871	. . . . 5 ⊢ ℤ_ring ∈ Ring
3		nn0z 9617	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
4		znzrh2.s	. . . . . . 7 ⊢ 𝑆 = (RSpan‘ℤ_ring)
5	4	znlidl 14912	. . . . . 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 6069	. . . . . 6 ⊢ (ℤ_ring /_s ∼ ) = (ℤ_ring /_s (ℤ_ring ~_QG (𝑆‘{𝑁})))
9		zringcrng 14870	. . . . . . 7 ⊢ ℤ_ring ∈ CRing
10		eqid 2234	. . . . . . . 8 ⊢ (LIdeal‘ℤ_ring) = (LIdeal‘ℤ_ring)
11	10	crng2idl 14809	. . . . . . 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 14874	. . . . . 6 ⊢ ℤ = (Base‘ℤ_ring)
14		eceq2 6817	. . . . . . . 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 4202	. . . . . 6 ⊢ (𝑥 ∈ ℤ ↦ [𝑥] ∼ ) = (𝑥 ∈ ℤ ↦ [𝑥](ℤ_ring ~_QG (𝑆‘{𝑁})))
17	8, 12, 13, 16	qusrhm 14806	. . . . 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 14913	. . . . 5 ⊢ (𝑁 ∈ ℤ → (ℤ_ring /_s ∼ ) ∈ CRing)
20		crngring 14255	. . . . 5 ⊢ ((ℤ_ring /_s ∼ ) ∈ CRing → (ℤ_ring /_s ∼ ) ∈ Ring)
21		eqid 2234	. . . . . 6 ⊢ (ℤRHom‘(ℤ_ring /_s ∼ )) = (ℤRHom‘(ℤ_ring /_s ∼ ))
22	21	zrhrhmb 14900	. . . . 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 14921	. . 3 ⊢ (𝑁 ∈ ℕ₀ → (ℤRHom‘(ℤ_ring /_s ∼ )) = (ℤRHom‘𝑌))
27	24, 26	eqtr2d 2268	. 2 ⊢ (𝑁 ∈ ℕ₀ → (ℤRHom‘𝑌) = (𝑥 ∈ ℤ ↦ [𝑥] ∼ ))
28	1, 27	eqtrid 2279	1 ⊢ (𝑁 ∈ ℕ₀ → 𝐿 = (𝑥 ∈ ℤ ↦ [𝑥] ∼ ))

Colors of variables: wff set class

Syntax hints: → wi 4 ↔ wb 105 = wceq 1398 ∈ wcel 2205 {csn 3694 ↦ cmpt 4176 ‘cfv 5357 (class class class)co 6058 [cec 6778 ℕ₀cn0 9516 ℤcz 9597 /_s cqus 13570 ~_QG cqg 13926 Ringcrg 14243 CRingccrg 14244 RingHom crh 14399 LIdealclidl 14745 RSpancrsp 14746 2Idealc2idl 14777 ℤ_ringczring 14868 ℤRHomczrh 14889 ℤ/nℤczn 14891

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-addf 8265 ax-mulf 8266

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-tp 3702 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-tpos 6489 df-recs 6549 df-frec 6635 df-er 6780 df-ec 6782 df-qs 6786 df-map 6897 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8463 df-neg 8464 df-reap 8867 df-inn 9258 df-2 9316 df-3 9317 df-4 9318 df-5 9319 df-6 9320 df-7 9321 df-8 9322 df-9 9323 df-n0 9517 df-z 9598 df-dec 9731 df-uz 9875 df-rp 10008 df-fz 10365 df-fzo 10502 df-seqfrec 10837 df-cj 11555 df-abs 11713 df-struct 13302 df-ndx 13303 df-slot 13304 df-base 13306 df-sets 13307 df-iress 13308 df-plusg 13391 df-mulr 13392 df-starv 13393 df-sca 13394 df-vsca 13395 df-ip 13396 df-tset 13397 df-ple 13398 df-ds 13400 df-unif 13401 df-0g 13559 df-topgen 13561 df-iimas 13571 df-qus 13572 df-mgm 13623 df-sgrp 13669 df-mnd 13682 df-mhm 13718 df-grp 13762 df-minusg 13763 df-sbg 13764 df-mulg 13877 df-subg 13927 df-nsg 13928 df-eqg 13929 df-ghm 13998 df-cmn 14043 df-abl 14044 df-mgp 14164 df-rng 14176 df-ur 14207 df-srg 14211 df-ring 14245 df-cring 14246 df-oppr 14315 df-rhm 14401 df-subrg 14469 df-lmod 14567 df-lssm 14631 df-lsp 14665 df-sra 14713 df-rgmod 14714 df-lidl 14747 df-rsp 14748 df-2idl 14778 df-bl 14824 df-mopn 14825 df-fg 14827 df-metu 14828 df-cnfld 14835 df-zring 14869 df-zrh 14892 df-zn 14894

This theorem is referenced by: znzrhval 14925 znzrhfo 14926

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