![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > zringbas | GIF version |
Description: The integers are the base of the ring of integers. (Contributed by Thierry Arnoux, 31-Oct-2017.) (Revised by AV, 9-Jun-2019.) |
Ref | Expression |
---|---|
zringbas | ⊢ ℤ = (Base‘ℤring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-zring 13707 | . . . 4 ⊢ ℤring = (ℂfld ↾s ℤ) | |
2 | 1 | a1i 9 | . . 3 ⊢ (⊤ → ℤring = (ℂfld ↾s ℤ)) |
3 | cnfldbas 13685 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
4 | 3 | a1i 9 | . . 3 ⊢ (⊤ → ℂ = (Base‘ℂfld)) |
5 | cnfldex 13684 | . . . 4 ⊢ ℂfld ∈ V | |
6 | 5 | a1i 9 | . . 3 ⊢ (⊤ → ℂfld ∈ V) |
7 | zsscn 9274 | . . . 4 ⊢ ℤ ⊆ ℂ | |
8 | 7 | a1i 9 | . . 3 ⊢ (⊤ → ℤ ⊆ ℂ) |
9 | 2, 4, 6, 8 | ressbas2d 12541 | . 2 ⊢ (⊤ → ℤ = (Base‘ℤring)) |
10 | 9 | mptru 1372 | 1 ⊢ ℤ = (Base‘ℤring) |
Colors of variables: wff set class |
Syntax hints: = wceq 1363 ⊤wtru 1364 ∈ wcel 2158 Vcvv 2749 ⊆ wss 3141 ‘cfv 5228 (class class class)co 5888 ℂcc 7822 ℤcz 9266 Basecbs 12475 ↾s cress 12476 ℂfldccnfld 13681 ℤringczring 13706 |
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 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-mulrcl 7923 ax-addcom 7924 ax-mulcom 7925 ax-addass 7926 ax-mulass 7927 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-1rid 7931 ax-0id 7932 ax-rnegex 7933 ax-precex 7934 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-apti 7939 ax-pre-ltadd 7940 ax-pre-mulgt0 7941 ax-addf 7946 ax-mulf 7947 |
This theorem depends on definitions: df-bi 117 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-tp 3612 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-reap 8545 df-inn 8933 df-2 8991 df-3 8992 df-4 8993 df-5 8994 df-6 8995 df-7 8996 df-8 8997 df-9 8998 df-n0 9190 df-z 9267 df-dec 9398 df-uz 9542 df-fz 10022 df-cj 10864 df-struct 12477 df-ndx 12478 df-slot 12479 df-base 12481 df-sets 12482 df-iress 12483 df-plusg 12563 df-mulr 12564 df-starv 12565 df-icnfld 13682 df-zring 13707 |
This theorem is referenced by: dvdsrzring 13719 zringinvg 13720 znlidl 13736 |
Copyright terms: Public domain | W3C validator |