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 12343 | . . . 4 ⊢ ℤ ⊆ ℚ | |
2 | 1z 12000 | . . . 4 ⊢ 1 ∈ ℤ | |
3 | 1, 2 | sselii 3961 | . . 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 31123 | . . 3 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ 1 ∈ ℚ) → ((ℚHom‘𝑅)‘1) = ((𝐿‘(numer‘1)) / (𝐿‘(denom‘1)))) |
8 | 3, 7 | mpan2 687 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘1) = ((𝐿‘(numer‘1)) / (𝐿‘(denom‘1)))) |
9 | gcd1 15864 | . . . . . . . . . 10 ⊢ (1 ∈ ℤ → (1 gcd 1) = 1) | |
10 | 2, 9 | ax-mp 5 | . . . . . . . . 9 ⊢ (1 gcd 1) = 1 |
11 | 1div1e1 11318 | . . . . . . . . . 10 ⊢ (1 / 1) = 1 | |
12 | 11 | eqcomi 2827 | . . . . . . . . 9 ⊢ 1 = (1 / 1) |
13 | 10, 12 | pm3.2i 471 | . . . . . . . 8 ⊢ ((1 gcd 1) = 1 ∧ 1 = (1 / 1)) |
14 | 1nn 11637 | . . . . . . . . 9 ⊢ 1 ∈ ℕ | |
15 | qnumdenbi 16072 | . . . . . . . . 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 1452 | . . . . . . . 8 ⊢ (((1 gcd 1) = 1 ∧ 1 = (1 / 1)) ↔ ((numer‘1) = 1 ∧ (denom‘1) = 1)) |
17 | 13, 16 | mpbi 231 | . . . . . . 7 ⊢ ((numer‘1) = 1 ∧ (denom‘1) = 1) |
18 | 17 | simpli 484 | . . . . . 6 ⊢ (numer‘1) = 1 |
19 | 18 | fveq2i 6666 | . . . . 5 ⊢ (𝐿‘(numer‘1)) = (𝐿‘1) |
20 | 17 | simpri 486 | . . . . . 6 ⊢ (denom‘1) = 1 |
21 | 20 | fveq2i 6666 | . . . . 5 ⊢ (𝐿‘(denom‘1)) = (𝐿‘1) |
22 | 19, 21 | oveq12i 7157 | . . . 4 ⊢ ((𝐿‘(numer‘1)) / (𝐿‘(denom‘1))) = ((𝐿‘1) / (𝐿‘1)) |
23 | drngring 19438 | . . . . . 6 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
24 | eqid 2818 | . . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
25 | 6, 24 | zrh1 20588 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (𝐿‘1) = (1r‘𝑅)) |
26 | 25, 25 | oveq12d 7163 | . . . . . 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 19247 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵) |
29 | 4, 5, 24 | dvr1 19368 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ∈ 𝐵) → ((1r‘𝑅) / (1r‘𝑅)) = (1r‘𝑅)) |
30 | 23, 28, 29 | syl2anc2 585 | . . . . 5 ⊢ (𝑅 ∈ DivRing → ((1r‘𝑅) / (1r‘𝑅)) = (1r‘𝑅)) |
31 | 27, 30 | eqtrd 2853 | . . . 4 ⊢ (𝑅 ∈ DivRing → ((𝐿‘1) / (𝐿‘1)) = (1r‘𝑅)) |
32 | 22, 31 | syl5eq 2865 | . . 3 ⊢ (𝑅 ∈ DivRing → ((𝐿‘(numer‘1)) / (𝐿‘(denom‘1))) = (1r‘𝑅)) |
33 | 32 | adantr 481 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((𝐿‘(numer‘1)) / (𝐿‘(denom‘1))) = (1r‘𝑅)) |
34 | 8, 33 | eqtrd 2853 | 1 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘1) = (1r‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 0cc0 10525 1c1 10526 / cdiv 11285 ℕcn 11626 ℤcz 11969 ℚcq 12336 gcd cgcd 15831 numercnumer 16061 denomcdenom 16062 Basecbs 16471 1rcur 19180 Ringcrg 19226 /rcdvr 19361 DivRingcdr 19431 ℤRHomczrh 20575 chrcchr 20577 ℚHomcqqh 31112 |
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 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
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-rmo 3143 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-int 4868 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-1st 7678 df-2nd 7679 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-fz 12881 df-fl 13150 df-mod 13226 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-dvds 15596 df-gcd 15832 df-numer 16063 df-denom 16064 df-gz 16254 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mulg 18163 df-subg 18214 df-ghm 18294 df-od 18585 df-cmn 18837 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-rnghom 19396 df-drng 19433 df-subrg 19462 df-cnfld 20474 df-zring 20546 df-zrh 20579 df-chr 20581 df-qqh 31113 |
This theorem is referenced by: qqhrhm 31129 |
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