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Mirrors > Home > MPE Home > Th. List > zrhval | Structured version Visualization version GIF version |
Description: Define the unique homomorphism from the integers to a ring or field. (Contributed by Mario Carneiro, 13-Jun-2015.) (Revised by AV, 12-Jun-2019.) |
Ref | Expression |
---|---|
zrhval.l | ⊢ 𝐿 = (ℤRHom‘𝑅) |
Ref | Expression |
---|---|
zrhval | ⊢ 𝐿 = ∪ (ℤring RingHom 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zrhval.l | . 2 ⊢ 𝐿 = (ℤRHom‘𝑅) | |
2 | oveq2 7153 | . . . . 5 ⊢ (𝑟 = 𝑅 → (ℤring RingHom 𝑟) = (ℤring RingHom 𝑅)) | |
3 | 2 | unieqd 4840 | . . . 4 ⊢ (𝑟 = 𝑅 → ∪ (ℤring RingHom 𝑟) = ∪ (ℤring RingHom 𝑅)) |
4 | df-zrh 20579 | . . . 4 ⊢ ℤRHom = (𝑟 ∈ V ↦ ∪ (ℤring RingHom 𝑟)) | |
5 | ovex 7178 | . . . . 5 ⊢ (ℤring RingHom 𝑅) ∈ V | |
6 | 5 | uniex 7454 | . . . 4 ⊢ ∪ (ℤring RingHom 𝑅) ∈ V |
7 | 3, 4, 6 | fvmpt 6761 | . . 3 ⊢ (𝑅 ∈ V → (ℤRHom‘𝑅) = ∪ (ℤring RingHom 𝑅)) |
8 | fvprc 6656 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (ℤRHom‘𝑅) = ∅) | |
9 | dfrhm2 19398 | . . . . . . . 8 ⊢ RingHom = (𝑟 ∈ Ring, 𝑠 ∈ Ring ↦ ((𝑟 GrpHom 𝑠) ∩ ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠)))) | |
10 | 9 | reldmmpo 7274 | . . . . . . 7 ⊢ Rel dom RingHom |
11 | 10 | ovprc2 7185 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (ℤring RingHom 𝑅) = ∅) |
12 | 11 | unieqd 4840 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → ∪ (ℤring RingHom 𝑅) = ∪ ∅) |
13 | uni0 4857 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
14 | 12, 13 | syl6eq 2869 | . . . 4 ⊢ (¬ 𝑅 ∈ V → ∪ (ℤring RingHom 𝑅) = ∅) |
15 | 8, 14 | eqtr4d 2856 | . . 3 ⊢ (¬ 𝑅 ∈ V → (ℤRHom‘𝑅) = ∪ (ℤring RingHom 𝑅)) |
16 | 7, 15 | pm2.61i 183 | . 2 ⊢ (ℤRHom‘𝑅) = ∪ (ℤring RingHom 𝑅) |
17 | 1, 16 | eqtri 2841 | 1 ⊢ 𝐿 = ∪ (ℤring RingHom 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∩ cin 3932 ∅c0 4288 ∪ cuni 4830 ‘cfv 6348 (class class class)co 7145 MndHom cmhm 17942 GrpHom cghm 18293 mulGrpcmgp 19168 Ringcrg 19226 RingHom crh 19393 ℤringzring 20545 ℤRHomczrh 20575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-plusg 16566 df-0g 16703 df-mhm 17944 df-ghm 18294 df-mgp 19169 df-ur 19181 df-ring 19228 df-rnghom 19396 df-zrh 20579 |
This theorem is referenced by: zrhval2 20584 zrhpropd 20590 |
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