znzrh2 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > znzrh2		GIF version

Theorem znzrh2 14663

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 14610	. . . . 5 ⊢ ℤ_ring ∈ Ring
3		nn0z 9499	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
4		znzrh2.s	. . . . . . 7 ⊢ 𝑆 = (RSpan‘ℤ_ring)
5	4	znlidl 14651	. . . . . 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 6029	. . . . . 6 ⊢ (ℤ_ring /_s ∼ ) = (ℤ_ring /_s (ℤ_ring ~_QG (𝑆‘{𝑁})))
9		zringcrng 14609	. . . . . . 7 ⊢ ℤ_ring ∈ CRing
10		eqid 2231	. . . . . . . 8 ⊢ (LIdeal‘ℤ_ring) = (LIdeal‘ℤ_ring)
11	10	crng2idl 14548	. . . . . . 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 14613	. . . . . 6 ⊢ ℤ = (Base‘ℤ_ring)
14		eceq2 6739	. . . . . . . 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 4176	. . . . . 6 ⊢ (𝑥 ∈ ℤ ↦ [𝑥] ∼ ) = (𝑥 ∈ ℤ ↦ [𝑥](ℤ_ring ~_QG (𝑆‘{𝑁})))
17	8, 12, 13, 16	qusrhm 14545	. . . . 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 14652	. . . . 5 ⊢ (𝑁 ∈ ℤ → (ℤ_ring /_s ∼ ) ∈ CRing)
20		crngring 14024	. . . . 5 ⊢ ((ℤ_ring /_s ∼ ) ∈ CRing → (ℤ_ring /_s ∼ ) ∈ Ring)
21		eqid 2231	. . . . . 6 ⊢ (ℤRHom‘(ℤ_ring /_s ∼ )) = (ℤRHom‘(ℤ_ring /_s ∼ ))
22	21	zrhrhmb 14639	. . . . 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 14660	. . 3 ⊢ (𝑁 ∈ ℕ₀ → (ℤRHom‘(ℤ_ring /_s ∼ )) = (ℤRHom‘𝑌))
27	24, 26	eqtr2d 2265	. 2 ⊢ (𝑁 ∈ ℕ₀ → (ℤRHom‘𝑌) = (𝑥 ∈ ℤ ↦ [𝑥] ∼ ))
28	1, 27	eqtrid 2276	1 ⊢ (𝑁 ∈ ℕ₀ → 𝐿 = (𝑥 ∈ ℤ ↦ [𝑥] ∼ ))

Colors of variables: wff set class

Syntax hints: → wi 4 ↔ wb 105 = wceq 1397 ∈ wcel 2202 {csn 3669 ↦ cmpt 4150 ‘cfv 5326 (class class class)co 6018 [cec 6700 ℕ₀cn0 9402 ℤcz 9479 /_s cqus 13385 ~_QG cqg 13758 Ringcrg 14012 CRingccrg 14013 RingHom crh 14167 LIdealclidl 14484 RSpancrsp 14485 2Idealc2idl 14516 ℤ_ringczring 14607 ℤRHomczrh 14628 ℤ/nℤczn 14630

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-mulrcl 8131 ax-addcom 8132 ax-mulcom 8133 ax-addass 8134 ax-mulass 8135 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-1rid 8139 ax-0id 8140 ax-rnegex 8141 ax-precex 8142 ax-cnre 8143 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146 ax-pre-apti 8147 ax-pre-ltadd 8148 ax-pre-mulgt0 8149 ax-addf 8154 ax-mulf 8155

This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-tp 3677 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-1st 6303 df-2nd 6304 df-tpos 6411 df-recs 6471 df-frec 6557 df-er 6702 df-ec 6704 df-qs 6708 df-map 6819 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220 df-sub 8352 df-neg 8353 df-reap 8755 df-inn 9144 df-2 9202 df-3 9203 df-4 9204 df-5 9205 df-6 9206 df-7 9207 df-8 9208 df-9 9209 df-n0 9403 df-z 9480 df-dec 9612 df-uz 9756 df-rp 9889 df-fz 10244 df-fzo 10378 df-seqfrec 10711 df-cj 11404 df-abs 11561 df-struct 13086 df-ndx 13087 df-slot 13088 df-base 13090 df-sets 13091 df-iress 13092 df-plusg 13175 df-mulr 13176 df-starv 13177 df-sca 13178 df-vsca 13179 df-ip 13180 df-tset 13181 df-ple 13182 df-ds 13184 df-unif 13185 df-0g 13343 df-topgen 13345 df-iimas 13387 df-qus 13388 df-mgm 13441 df-sgrp 13487 df-mnd 13502 df-mhm 13544 df-grp 13588 df-minusg 13589 df-sbg 13590 df-mulg 13709 df-subg 13759 df-nsg 13760 df-eqg 13761 df-ghm 13830 df-cmn 13875 df-abl 13876 df-mgp 13937 df-rng 13949 df-ur 13976 df-srg 13980 df-ring 14014 df-cring 14015 df-oppr 14084 df-rhm 14169 df-subrg 14236 df-lmod 14306 df-lssm 14370 df-lsp 14404 df-sra 14452 df-rgmod 14453 df-lidl 14486 df-rsp 14487 df-2idl 14517 df-bl 14563 df-mopn 14564 df-fg 14566 df-metu 14567 df-cnfld 14574 df-zring 14608 df-zrh 14631 df-zn 14633

This theorem is referenced by: znzrhval 14664 znzrhfo 14665

W3C validator