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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 6557 | . . . . 5 ⊢ (𝑟 = 𝑅 → (ℤring RingHom 𝑟) = (ℤring RingHom 𝑅)) | |
3 | 2 | unieqd 4382 | . . . 4 ⊢ (𝑟 = 𝑅 → ∪ (ℤring RingHom 𝑟) = ∪ (ℤring RingHom 𝑅)) |
4 | df-zrh 19671 | . . . 4 ⊢ ℤRHom = (𝑟 ∈ V ↦ ∪ (ℤring RingHom 𝑟)) | |
5 | ovex 6577 | . . . . 5 ⊢ (ℤring RingHom 𝑅) ∈ V | |
6 | 5 | uniex 6851 | . . . 4 ⊢ ∪ (ℤring RingHom 𝑅) ∈ V |
7 | 3, 4, 6 | fvmpt 6191 | . . 3 ⊢ (𝑅 ∈ V → (ℤRHom‘𝑅) = ∪ (ℤring RingHom 𝑅)) |
8 | fvprc 6097 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (ℤRHom‘𝑅) = ∅) | |
9 | dfrhm2 18540 | . . . . . . . 8 ⊢ RingHom = (𝑟 ∈ Ring, 𝑠 ∈ Ring ↦ ((𝑟 GrpHom 𝑠) ∩ ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠)))) | |
10 | 9 | reldmmpt2 6669 | . . . . . . 7 ⊢ Rel dom RingHom |
11 | 10 | ovprc2 6583 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (ℤring RingHom 𝑅) = ∅) |
12 | 11 | unieqd 4382 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → ∪ (ℤring RingHom 𝑅) = ∪ ∅) |
13 | uni0 4401 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
14 | 12, 13 | syl6eq 2660 | . . . 4 ⊢ (¬ 𝑅 ∈ V → ∪ (ℤring RingHom 𝑅) = ∅) |
15 | 8, 14 | eqtr4d 2647 | . . 3 ⊢ (¬ 𝑅 ∈ V → (ℤRHom‘𝑅) = ∪ (ℤring RingHom 𝑅)) |
16 | 7, 15 | pm2.61i 175 | . 2 ⊢ (ℤRHom‘𝑅) = ∪ (ℤring RingHom 𝑅) |
17 | 1, 16 | eqtri 2632 | 1 ⊢ 𝐿 = ∪ (ℤring RingHom 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∩ cin 3539 ∅c0 3874 ∪ cuni 4372 ‘cfv 5804 (class class class)co 6549 MndHom cmhm 17156 GrpHom cghm 17480 mulGrpcmgp 18312 Ringcrg 18370 RingHom crh 18535 ℤringzring 19637 ℤRHomczrh 19667 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-0g 15925 df-mhm 17158 df-ghm 17481 df-mgp 18313 df-ur 18325 df-ring 18372 df-rnghom 18538 df-zrh 19671 |
This theorem is referenced by: zrhval2 19676 zrhpropd 19682 |
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