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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rhmdvd | Structured version Visualization version GIF version |
Description: A ring homomorphism preserves ratios. (Contributed by Thierry Arnoux, 22-Oct-2017.) |
Ref | Expression |
---|---|
rhmdvd.u | ⊢ 𝑈 = (Unit‘𝑆) |
rhmdvd.x | ⊢ 𝑋 = (Base‘𝑅) |
rhmdvd.d | ⊢ / = (/r‘𝑆) |
rhmdvd.m | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
rhmdvd | ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ ((𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → ((𝐹‘𝐴) / (𝐹‘𝐵)) = ((𝐹‘(𝐴 · 𝐶)) / (𝐹‘(𝐵 · 𝐶)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1127 | . . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ ((𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → 𝐹 ∈ (𝑅 RingHom 𝑆)) | |
2 | simp21 1197 | . . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ ((𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → 𝐴 ∈ 𝑋) | |
3 | simp23 1199 | . . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ ((𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → 𝐶 ∈ 𝑋) | |
4 | rhmdvd.x | . . . . 5 ⊢ 𝑋 = (Base‘𝑅) | |
5 | rhmdvd.m | . . . . 5 ⊢ · = (.r‘𝑅) | |
6 | eqid 2793 | . . . . 5 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
7 | 4, 5, 6 | rhmmul 19157 | . . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐹‘(𝐴 · 𝐶)) = ((𝐹‘𝐴)(.r‘𝑆)(𝐹‘𝐶))) |
8 | 1, 2, 3, 7 | syl3anc 1362 | . . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ ((𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → (𝐹‘(𝐴 · 𝐶)) = ((𝐹‘𝐴)(.r‘𝑆)(𝐹‘𝐶))) |
9 | simp22 1198 | . . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ ((𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → 𝐵 ∈ 𝑋) | |
10 | 4, 5, 6 | rhmmul 19157 | . . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐹‘(𝐵 · 𝐶)) = ((𝐹‘𝐵)(.r‘𝑆)(𝐹‘𝐶))) |
11 | 1, 9, 3, 10 | syl3anc 1362 | . . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ ((𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → (𝐹‘(𝐵 · 𝐶)) = ((𝐹‘𝐵)(.r‘𝑆)(𝐹‘𝐶))) |
12 | 8, 11 | oveq12d 7025 | . 2 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ ((𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → ((𝐹‘(𝐴 · 𝐶)) / (𝐹‘(𝐵 · 𝐶))) = (((𝐹‘𝐴)(.r‘𝑆)(𝐹‘𝐶)) / ((𝐹‘𝐵)(.r‘𝑆)(𝐹‘𝐶)))) |
13 | rhmrcl2 19150 | . . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring) | |
14 | 13 | 3ad2ant1 1124 | . . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ ((𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → 𝑆 ∈ Ring) |
15 | eqid 2793 | . . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
16 | 4, 15 | rhmf 19156 | . . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:𝑋⟶(Base‘𝑆)) |
17 | 16 | 3ad2ant1 1124 | . . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ ((𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → 𝐹:𝑋⟶(Base‘𝑆)) |
18 | 17, 2 | ffvelrnd 6708 | . . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ ((𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → (𝐹‘𝐴) ∈ (Base‘𝑆)) |
19 | simp3l 1192 | . . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ ((𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → (𝐹‘𝐵) ∈ 𝑈) | |
20 | simp3r 1193 | . . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ ((𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → (𝐹‘𝐶) ∈ 𝑈) | |
21 | rhmdvd.u | . . . 4 ⊢ 𝑈 = (Unit‘𝑆) | |
22 | rhmdvd.d | . . . 4 ⊢ / = (/r‘𝑆) | |
23 | 15, 21, 22, 6 | dvrcan5 30473 | . . 3 ⊢ ((𝑆 ∈ Ring ∧ ((𝐹‘𝐴) ∈ (Base‘𝑆) ∧ (𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → (((𝐹‘𝐴)(.r‘𝑆)(𝐹‘𝐶)) / ((𝐹‘𝐵)(.r‘𝑆)(𝐹‘𝐶))) = ((𝐹‘𝐴) / (𝐹‘𝐵))) |
24 | 14, 18, 19, 20, 23 | syl13anc 1363 | . 2 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ ((𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → (((𝐹‘𝐴)(.r‘𝑆)(𝐹‘𝐶)) / ((𝐹‘𝐵)(.r‘𝑆)(𝐹‘𝐶))) = ((𝐹‘𝐴) / (𝐹‘𝐵))) |
25 | 12, 24 | eqtr2d 2830 | 1 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ ((𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → ((𝐹‘𝐴) / (𝐹‘𝐵)) = ((𝐹‘(𝐴 · 𝐶)) / (𝐹‘(𝐵 · 𝐶)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1078 = wceq 1520 ∈ wcel 2079 ⟶wf 6213 ‘cfv 6217 (class class class)co 7007 Basecbs 16300 .rcmulr 16383 Ringcrg 18975 Unitcui 19067 /rcdvr 19110 RingHom crh 19142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-1st 7536 df-2nd 7537 df-tpos 7734 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-er 8130 df-map 8249 df-en 8348 df-dom 8349 df-sdom 8350 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-nn 11476 df-2 11537 df-3 11538 df-ndx 16303 df-slot 16304 df-base 16306 df-sets 16307 df-ress 16308 df-plusg 16395 df-mulr 16396 df-0g 16532 df-mgm 17669 df-sgrp 17711 df-mnd 17722 df-mhm 17762 df-grp 17852 df-minusg 17853 df-ghm 18085 df-mgp 18918 df-ur 18930 df-ring 18977 df-oppr 19051 df-dvdsr 19069 df-unit 19070 df-invr 19100 df-dvr 19111 df-rnghom 19145 |
This theorem is referenced by: qqhval2lem 30795 qqhghm 30802 qqhrhm 30803 |
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