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 1132 | . . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ ((𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → 𝐹 ∈ (𝑅 RingHom 𝑆)) | |
2 | simp21 1202 | . . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ ((𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → 𝐴 ∈ 𝑋) | |
3 | simp23 1204 | . . . 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 2821 | . . . . 5 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
7 | 4, 5, 6 | rhmmul 19479 | . . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐹‘(𝐴 · 𝐶)) = ((𝐹‘𝐴)(.r‘𝑆)(𝐹‘𝐶))) |
8 | 1, 2, 3, 7 | syl3anc 1367 | . . 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 1203 | . . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ ((𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → 𝐵 ∈ 𝑋) | |
10 | 4, 5, 6 | rhmmul 19479 | . . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐹‘(𝐵 · 𝐶)) = ((𝐹‘𝐵)(.r‘𝑆)(𝐹‘𝐶))) |
11 | 1, 9, 3, 10 | syl3anc 1367 | . . 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 7174 | . 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 19472 | . . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring) | |
14 | 13 | 3ad2ant1 1129 | . . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ ((𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → 𝑆 ∈ Ring) |
15 | eqid 2821 | . . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
16 | 4, 15 | rhmf 19478 | . . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:𝑋⟶(Base‘𝑆)) |
17 | 16 | 3ad2ant1 1129 | . . . 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 6852 | . . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ ((𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → (𝐹‘𝐴) ∈ (Base‘𝑆)) |
19 | simp3l 1197 | . . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ ((𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → (𝐹‘𝐵) ∈ 𝑈) | |
20 | simp3r 1198 | . . 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 30864 | . . 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 1368 | . 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 2857 | 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 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 .rcmulr 16566 Ringcrg 19297 Unitcui 19389 /rcdvr 19432 RingHom crh 19464 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-grp 18106 df-minusg 18107 df-ghm 18356 df-mgp 19240 df-ur 19252 df-ring 19299 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-dvr 19433 df-rnghom 19467 |
This theorem is referenced by: qqhval2lem 31222 qqhghm 31229 qqhrhm 31230 |
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