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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 12343 | . . . 4 ⊢ ℤ ⊆ ℚ | |
2 | 1z 12000 | . . . 4 ⊢ 1 ∈ ℤ | |
3 | 1, 2 | sselii 3912 | . . 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 31334 | . . 3 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ 1 ∈ ℚ) → ((ℚHom‘𝑅)‘1) = ((𝐿‘(numer‘1)) / (𝐿‘(denom‘1)))) |
8 | 3, 7 | mpan2 690 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘1) = ((𝐿‘(numer‘1)) / (𝐿‘(denom‘1)))) |
9 | gcd1 15866 | . . . . . . . . . 10 ⊢ (1 ∈ ℤ → (1 gcd 1) = 1) | |
10 | 2, 9 | ax-mp 5 | . . . . . . . . 9 ⊢ (1 gcd 1) = 1 |
11 | 1div1e1 11319 | . . . . . . . . . 10 ⊢ (1 / 1) = 1 | |
12 | 11 | eqcomi 2807 | . . . . . . . . 9 ⊢ 1 = (1 / 1) |
13 | 10, 12 | pm3.2i 474 | . . . . . . . 8 ⊢ ((1 gcd 1) = 1 ∧ 1 = (1 / 1)) |
14 | 1nn 11636 | . . . . . . . . 9 ⊢ 1 ∈ ℕ | |
15 | qnumdenbi 16074 | . . . . . . . . 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 1458 | . . . . . . . 8 ⊢ (((1 gcd 1) = 1 ∧ 1 = (1 / 1)) ↔ ((numer‘1) = 1 ∧ (denom‘1) = 1)) |
17 | 13, 16 | mpbi 233 | . . . . . . 7 ⊢ ((numer‘1) = 1 ∧ (denom‘1) = 1) |
18 | 17 | simpli 487 | . . . . . 6 ⊢ (numer‘1) = 1 |
19 | 18 | fveq2i 6648 | . . . . 5 ⊢ (𝐿‘(numer‘1)) = (𝐿‘1) |
20 | 17 | simpri 489 | . . . . . 6 ⊢ (denom‘1) = 1 |
21 | 20 | fveq2i 6648 | . . . . 5 ⊢ (𝐿‘(denom‘1)) = (𝐿‘1) |
22 | 19, 21 | oveq12i 7147 | . . . 4 ⊢ ((𝐿‘(numer‘1)) / (𝐿‘(denom‘1))) = ((𝐿‘1) / (𝐿‘1)) |
23 | drngring 19502 | . . . . . 6 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
24 | eqid 2798 | . . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
25 | 6, 24 | zrh1 20206 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (𝐿‘1) = (1r‘𝑅)) |
26 | 25, 25 | oveq12d 7153 | . . . . . 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 19314 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵) |
29 | 4, 5, 24 | dvr1 19435 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ∈ 𝐵) → ((1r‘𝑅) / (1r‘𝑅)) = (1r‘𝑅)) |
30 | 23, 28, 29 | syl2anc2 588 | . . . . 5 ⊢ (𝑅 ∈ DivRing → ((1r‘𝑅) / (1r‘𝑅)) = (1r‘𝑅)) |
31 | 27, 30 | eqtrd 2833 | . . . 4 ⊢ (𝑅 ∈ DivRing → ((𝐿‘1) / (𝐿‘1)) = (1r‘𝑅)) |
32 | 22, 31 | syl5eq 2845 | . . 3 ⊢ (𝑅 ∈ DivRing → ((𝐿‘(numer‘1)) / (𝐿‘(denom‘1))) = (1r‘𝑅)) |
33 | 32 | adantr 484 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((𝐿‘(numer‘1)) / (𝐿‘(denom‘1))) = (1r‘𝑅)) |
34 | 8, 33 | eqtrd 2833 | 1 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘1) = (1r‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 0cc0 10526 1c1 10527 / cdiv 11286 ℕcn 11625 ℤcz 11969 ℚcq 12336 gcd cgcd 15833 numercnumer 16063 denomcdenom 16064 Basecbs 16475 1rcur 19244 Ringcrg 19290 /rcdvr 19428 DivRingcdr 19495 ℤRHomczrh 20193 chrcchr 20195 ℚHomcqqh 31323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 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 12886 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-dvds 15600 df-gcd 15834 df-numer 16065 df-denom 16066 df-gz 16256 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-ghm 18348 df-od 18648 df-cmn 18900 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-rnghom 19463 df-drng 19497 df-subrg 19526 df-cnfld 20092 df-zring 20164 df-zrh 20197 df-chr 20199 df-qqh 31324 |
This theorem is referenced by: qqhrhm 31340 |
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