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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 7439 | . . . . 5 ⊢ (𝑟 = 𝑅 → (ℤring RingHom 𝑟) = (ℤring RingHom 𝑅)) | |
| 3 | 2 | unieqd 4920 | . . . 4 ⊢ (𝑟 = 𝑅 → ∪ (ℤring RingHom 𝑟) = ∪ (ℤring RingHom 𝑅)) |
| 4 | df-zrh 21514 | . . . 4 ⊢ ℤRHom = (𝑟 ∈ V ↦ ∪ (ℤring RingHom 𝑟)) | |
| 5 | ovex 7464 | . . . . 5 ⊢ (ℤring RingHom 𝑅) ∈ V | |
| 6 | 5 | uniex 7761 | . . . 4 ⊢ ∪ (ℤring RingHom 𝑅) ∈ V |
| 7 | 3, 4, 6 | fvmpt 7016 | . . 3 ⊢ (𝑅 ∈ V → (ℤRHom‘𝑅) = ∪ (ℤring RingHom 𝑅)) |
| 8 | fvprc 6898 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (ℤRHom‘𝑅) = ∅) | |
| 9 | dfrhm2 20474 | . . . . . . . 8 ⊢ RingHom = (𝑟 ∈ Ring, 𝑠 ∈ Ring ↦ ((𝑟 GrpHom 𝑠) ∩ ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠)))) | |
| 10 | 9 | reldmmpo 7567 | . . . . . . 7 ⊢ Rel dom RingHom |
| 11 | 10 | ovprc2 7471 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (ℤring RingHom 𝑅) = ∅) |
| 12 | 11 | unieqd 4920 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → ∪ (ℤring RingHom 𝑅) = ∪ ∅) |
| 13 | uni0 4935 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
| 14 | 12, 13 | eqtrdi 2793 | . . . 4 ⊢ (¬ 𝑅 ∈ V → ∪ (ℤring RingHom 𝑅) = ∅) |
| 15 | 8, 14 | eqtr4d 2780 | . . 3 ⊢ (¬ 𝑅 ∈ V → (ℤRHom‘𝑅) = ∪ (ℤring RingHom 𝑅)) |
| 16 | 7, 15 | pm2.61i 182 | . 2 ⊢ (ℤRHom‘𝑅) = ∪ (ℤring RingHom 𝑅) |
| 17 | 1, 16 | eqtri 2765 | 1 ⊢ 𝐿 = ∪ (ℤring RingHom 𝑅) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 ∅c0 4333 ∪ cuni 4907 ‘cfv 6561 (class class class)co 7431 MndHom cmhm 18794 GrpHom cghm 19230 mulGrpcmgp 20137 Ringcrg 20230 RingHom crh 20469 ℤringczring 21457 ℤRHomczrh 21510 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-0g 17486 df-mhm 18796 df-ghm 19231 df-mgp 20138 df-ur 20179 df-ring 20232 df-rhm 20472 df-zrh 21514 |
| This theorem is referenced by: zrhval2 21519 zrhpropd 21525 |
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