znzrh2 - Intuitionistic Logic Explorer

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Theorem znzrh2 14786

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 14733	. . . . 5 ⊢ ℤ_ring ∈ Ring
3		nn0z 9596	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
4		znzrh2.s	. . . . . . 7 ⊢ 𝑆 = (RSpan‘ℤ_ring)
5	4	znlidl 14774	. . . . . 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 6060	. . . . . 6 ⊢ (ℤ_ring /_s ∼ ) = (ℤ_ring /_s (ℤ_ring ~_QG (𝑆‘{𝑁})))
9		zringcrng 14732	. . . . . . 7 ⊢ ℤ_ring ∈ CRing
10		eqid 2232	. . . . . . . 8 ⊢ (LIdeal‘ℤ_ring) = (LIdeal‘ℤ_ring)
11	10	crng2idl 14671	. . . . . . 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 14736	. . . . . 6 ⊢ ℤ = (Base‘ℤ_ring)
14		eceq2 6803	. . . . . . . 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 4196	. . . . . 6 ⊢ (𝑥 ∈ ℤ ↦ [𝑥] ∼ ) = (𝑥 ∈ ℤ ↦ [𝑥](ℤ_ring ~_QG (𝑆‘{𝑁})))
17	8, 12, 13, 16	qusrhm 14668	. . . . 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 14775	. . . . 5 ⊢ (𝑁 ∈ ℤ → (ℤ_ring /_s ∼ ) ∈ CRing)
20		crngring 14144	. . . . 5 ⊢ ((ℤ_ring /_s ∼ ) ∈ CRing → (ℤ_ring /_s ∼ ) ∈ Ring)
21		eqid 2232	. . . . . 6 ⊢ (ℤRHom‘(ℤ_ring /_s ∼ )) = (ℤRHom‘(ℤ_ring /_s ∼ ))
22	21	zrhrhmb 14762	. . . . 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 14783	. . 3 ⊢ (𝑁 ∈ ℕ₀ → (ℤRHom‘(ℤ_ring /_s ∼ )) = (ℤRHom‘𝑌))
27	24, 26	eqtr2d 2266	. 2 ⊢ (𝑁 ∈ ℕ₀ → (ℤRHom‘𝑌) = (𝑥 ∈ ℤ ↦ [𝑥] ∼ ))
28	1, 27	eqtrid 2277	1 ⊢ (𝑁 ∈ ℕ₀ → 𝐿 = (𝑥 ∈ ℤ ↦ [𝑥] ∼ ))

Colors of variables: wff set class

Syntax hints: → wi 4 ↔ wb 105 = wceq 1398 ∈ wcel 2203 {csn 3688 ↦ cmpt 4170 ‘cfv 5351 (class class class)co 6049 [cec 6764 ℕ₀cn0 9495 ℤcz 9576 /_s cqus 13505 ~_QG cqg 13878 Ringcrg 14132 CRingccrg 14133 RingHom crh 14287 LIdealclidl 14607 RSpancrsp 14608 2Idealc2idl 14639 ℤ_ringczring 14730 ℤRHomczrh 14751 ℤ/nℤczn 14753

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 2205 ax-14 2206 ax-ext 2214 ax-coll 4224 ax-sep 4227 ax-nul 4235 ax-pow 4286 ax-pr 4321 ax-un 4553 ax-setind 4658 ax-iinf 4709 ax-cnex 8217 ax-resscn 8218 ax-1cn 8219 ax-1re 8220 ax-icn 8221 ax-addcl 8222 ax-addrcl 8223 ax-mulcl 8224 ax-mulrcl 8225 ax-addcom 8226 ax-mulcom 8227 ax-addass 8228 ax-mulass 8229 ax-distr 8230 ax-i2m1 8231 ax-0lt1 8232 ax-1rid 8233 ax-0id 8234 ax-rnegex 8235 ax-precex 8236 ax-cnre 8237 ax-pre-ltirr 8238 ax-pre-ltwlin 8239 ax-pre-lttrn 8240 ax-pre-apti 8241 ax-pre-ltadd 8242 ax-pre-mulgt0 8243 ax-addf 8248 ax-mulf 8249

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 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2814 df-sbc 3042 df-csb 3138 df-dif 3212 df-un 3214 df-in 3216 df-ss 3223 df-nul 3508 df-if 3620 df-pw 3670 df-sn 3694 df-pr 3695 df-tp 3696 df-op 3697 df-uni 3914 df-int 3949 df-iun 3992 df-br 4109 df-opab 4171 df-mpt 4172 df-tr 4208 df-id 4413 df-iord 4486 df-on 4488 df-ilim 4489 df-suc 4491 df-iom 4712 df-xp 4754 df-rel 4755 df-cnv 4756 df-co 4757 df-dm 4758 df-rn 4759 df-res 4760 df-ima 4761 df-iota 5311 df-fun 5353 df-fn 5354 df-f 5355 df-f1 5356 df-fo 5357 df-f1o 5358 df-fv 5359 df-riota 6002 df-ov 6052 df-oprab 6053 df-mpo 6054 df-1st 6333 df-2nd 6334 df-tpos 6475 df-recs 6535 df-frec 6621 df-er 6766 df-ec 6768 df-qs 6772 df-map 6883 df-pnf 8309 df-mnf 8310 df-xr 8311 df-ltxr 8312 df-le 8313 df-sub 8445 df-neg 8446 df-reap 8848 df-inn 9237 df-2 9295 df-3 9296 df-4 9297 df-5 9298 df-6 9299 df-7 9300 df-8 9301 df-9 9302 df-n0 9496 df-z 9577 df-dec 9709 df-uz 9853 df-rp 9986 df-fz 10342 df-fzo 10476 df-seqfrec 10809 df-cj 11523 df-abs 11680 df-struct 13206 df-ndx 13207 df-slot 13208 df-base 13210 df-sets 13211 df-iress 13212 df-plusg 13295 df-mulr 13296 df-starv 13297 df-sca 13298 df-vsca 13299 df-ip 13300 df-tset 13301 df-ple 13302 df-ds 13304 df-unif 13305 df-0g 13463 df-topgen 13465 df-iimas 13507 df-qus 13508 df-mgm 13561 df-sgrp 13607 df-mnd 13622 df-mhm 13664 df-grp 13708 df-minusg 13709 df-sbg 13710 df-mulg 13829 df-subg 13879 df-nsg 13880 df-eqg 13881 df-ghm 13950 df-cmn 13995 df-abl 13996 df-mgp 14057 df-rng 14069 df-ur 14096 df-srg 14100 df-ring 14134 df-cring 14135 df-oppr 14204 df-rhm 14289 df-subrg 14356 df-lmod 14429 df-lssm 14493 df-lsp 14527 df-sra 14575 df-rgmod 14576 df-lidl 14609 df-rsp 14610 df-2idl 14640 df-bl 14686 df-mopn 14687 df-fg 14689 df-metu 14690 df-cnfld 14697 df-zring 14731 df-zrh 14754 df-zn 14756

This theorem is referenced by: znzrhval 14787 znzrhfo 14788

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