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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > qqh1 | Structured version Visualization version GIF version |
Description: The image of 1 by the ℚHom homomorphism is the ring's unit. (Contributed by Thierry Arnoux, 22-Oct-2017.) |
Ref | Expression |
---|---|
qqhval2.0 | ⊢ 𝐵 = (Base‘𝑅) |
qqhval2.1 | ⊢ / = (/r‘𝑅) |
qqhval2.2 | ⊢ 𝐿 = (ℤRHom‘𝑅) |
Ref | Expression |
---|---|
qqh1 | ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘1) = (1r‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zssq 11997 | . . . 4 ⊢ ℤ ⊆ ℚ | |
2 | 1z 11608 | . . . 4 ⊢ 1 ∈ ℤ | |
3 | 1, 2 | sselii 3749 | . . 3 ⊢ 1 ∈ ℚ |
4 | qqhval2.0 | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
5 | qqhval2.1 | . . . 4 ⊢ / = (/r‘𝑅) | |
6 | qqhval2.2 | . . . 4 ⊢ 𝐿 = (ℤRHom‘𝑅) | |
7 | 4, 5, 6 | qqhvval 30364 | . . 3 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ 1 ∈ ℚ) → ((ℚHom‘𝑅)‘1) = ((𝐿‘(numer‘1)) / (𝐿‘(denom‘1)))) |
8 | 3, 7 | mpan2 663 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘1) = ((𝐿‘(numer‘1)) / (𝐿‘(denom‘1)))) |
9 | gcd1 15456 | . . . . . . . . . 10 ⊢ (1 ∈ ℤ → (1 gcd 1) = 1) | |
10 | 2, 9 | ax-mp 5 | . . . . . . . . 9 ⊢ (1 gcd 1) = 1 |
11 | 1div1e1 10918 | . . . . . . . . . 10 ⊢ (1 / 1) = 1 | |
12 | 11 | eqcomi 2780 | . . . . . . . . 9 ⊢ 1 = (1 / 1) |
13 | 10, 12 | pm3.2i 447 | . . . . . . . 8 ⊢ ((1 gcd 1) = 1 ∧ 1 = (1 / 1)) |
14 | 1nn 11232 | . . . . . . . . 9 ⊢ 1 ∈ ℕ | |
15 | qnumdenbi 15658 | . . . . . . . . 9 ⊢ ((1 ∈ ℚ ∧ 1 ∈ ℤ ∧ 1 ∈ ℕ) → (((1 gcd 1) = 1 ∧ 1 = (1 / 1)) ↔ ((numer‘1) = 1 ∧ (denom‘1) = 1))) | |
16 | 3, 2, 14, 15 | mp3an 1572 | . . . . . . . 8 ⊢ (((1 gcd 1) = 1 ∧ 1 = (1 / 1)) ↔ ((numer‘1) = 1 ∧ (denom‘1) = 1)) |
17 | 13, 16 | mpbi 220 | . . . . . . 7 ⊢ ((numer‘1) = 1 ∧ (denom‘1) = 1) |
18 | 17 | simpli 470 | . . . . . 6 ⊢ (numer‘1) = 1 |
19 | 18 | fveq2i 6335 | . . . . 5 ⊢ (𝐿‘(numer‘1)) = (𝐿‘1) |
20 | 17 | simpri 473 | . . . . . 6 ⊢ (denom‘1) = 1 |
21 | 20 | fveq2i 6335 | . . . . 5 ⊢ (𝐿‘(denom‘1)) = (𝐿‘1) |
22 | 19, 21 | oveq12i 6804 | . . . 4 ⊢ ((𝐿‘(numer‘1)) / (𝐿‘(denom‘1))) = ((𝐿‘1) / (𝐿‘1)) |
23 | drngring 18963 | . . . . . 6 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
24 | eqid 2771 | . . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
25 | 6, 24 | zrh1 20075 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (𝐿‘1) = (1r‘𝑅)) |
26 | 25, 25 | oveq12d 6810 | . . . . . 6 ⊢ (𝑅 ∈ Ring → ((𝐿‘1) / (𝐿‘1)) = ((1r‘𝑅) / (1r‘𝑅))) |
27 | 23, 26 | syl 17 | . . . . 5 ⊢ (𝑅 ∈ DivRing → ((𝐿‘1) / (𝐿‘1)) = ((1r‘𝑅) / (1r‘𝑅))) |
28 | 4, 24 | ringidcl 18775 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵) |
29 | 23, 28 | syl 17 | . . . . . 6 ⊢ (𝑅 ∈ DivRing → (1r‘𝑅) ∈ 𝐵) |
30 | 4, 5, 24 | dvr1 18896 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ∈ 𝐵) → ((1r‘𝑅) / (1r‘𝑅)) = (1r‘𝑅)) |
31 | 23, 29, 30 | syl2anc 565 | . . . . 5 ⊢ (𝑅 ∈ DivRing → ((1r‘𝑅) / (1r‘𝑅)) = (1r‘𝑅)) |
32 | 27, 31 | eqtrd 2805 | . . . 4 ⊢ (𝑅 ∈ DivRing → ((𝐿‘1) / (𝐿‘1)) = (1r‘𝑅)) |
33 | 22, 32 | syl5eq 2817 | . . 3 ⊢ (𝑅 ∈ DivRing → ((𝐿‘(numer‘1)) / (𝐿‘(denom‘1))) = (1r‘𝑅)) |
34 | 33 | adantr 466 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((𝐿‘(numer‘1)) / (𝐿‘(denom‘1))) = (1r‘𝑅)) |
35 | 8, 34 | eqtrd 2805 | 1 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘1) = (1r‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ‘cfv 6031 (class class class)co 6792 0cc0 10137 1c1 10138 / cdiv 10885 ℕcn 11221 ℤcz 11578 ℚcq 11990 gcd cgcd 15423 numercnumer 15647 denomcdenom 15648 Basecbs 16063 1rcur 18708 Ringcrg 18754 /rcdvr 18889 DivRingcdr 18956 ℤRHomczrh 20062 chrcchr 20064 ℚHomcqqh 30353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7095 ax-inf2 8701 ax-cnex 10193 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 ax-pre-mulgt0 10214 ax-pre-sup 10215 ax-addf 10216 ax-mulf 10217 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7212 df-1st 7314 df-2nd 7315 df-tpos 7503 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-1o 7712 df-oadd 7716 df-er 7895 df-map 8010 df-en 8109 df-dom 8110 df-sdom 8111 df-fin 8112 df-sup 8503 df-inf 8504 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-sub 10469 df-neg 10470 df-div 10886 df-nn 11222 df-2 11280 df-3 11281 df-4 11282 df-5 11283 df-6 11284 df-7 11285 df-8 11286 df-9 11287 df-n0 11494 df-z 11579 df-dec 11695 df-uz 11888 df-q 11991 df-rp 12035 df-fz 12533 df-fl 12800 df-mod 12876 df-seq 13008 df-exp 13067 df-cj 14046 df-re 14047 df-im 14048 df-sqrt 14182 df-abs 14183 df-dvds 15189 df-gcd 15424 df-numer 15649 df-denom 15650 df-gz 15840 df-struct 16065 df-ndx 16066 df-slot 16067 df-base 16069 df-sets 16070 df-ress 16071 df-plusg 16161 df-mulr 16162 df-starv 16163 df-tset 16167 df-ple 16168 df-ds 16171 df-unif 16172 df-0g 16309 df-mgm 17449 df-sgrp 17491 df-mnd 17502 df-mhm 17542 df-grp 17632 df-minusg 17633 df-sbg 17634 df-mulg 17748 df-subg 17798 df-ghm 17865 df-od 18154 df-cmn 18401 df-mgp 18697 df-ur 18709 df-ring 18756 df-cring 18757 df-oppr 18830 df-dvdsr 18848 df-unit 18849 df-invr 18879 df-dvr 18890 df-rnghom 18924 df-drng 18958 df-subrg 18987 df-cnfld 19961 df-zring 20033 df-zrh 20066 df-chr 20068 df-qqh 30354 |
This theorem is referenced by: qqhrhm 30370 |
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