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 1131 | . . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ ((𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → 𝐹 ∈ (𝑅 RingHom 𝑆)) | |
2 | simp21 1201 | . . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ ((𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → 𝐴 ∈ 𝑋) | |
3 | simp23 1203 | . . . 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 2820 | . . . . 5 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
7 | 4, 5, 6 | rhmmul 19474 | . . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐹‘(𝐴 · 𝐶)) = ((𝐹‘𝐴)(.r‘𝑆)(𝐹‘𝐶))) |
8 | 1, 2, 3, 7 | syl3anc 1366 | . . 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 1202 | . . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ ((𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → 𝐵 ∈ 𝑋) | |
10 | 4, 5, 6 | rhmmul 19474 | . . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐹‘(𝐵 · 𝐶)) = ((𝐹‘𝐵)(.r‘𝑆)(𝐹‘𝐶))) |
11 | 1, 9, 3, 10 | syl3anc 1366 | . . 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 7167 | . 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 19467 | . . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring) | |
14 | 13 | 3ad2ant1 1128 | . . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ ((𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → 𝑆 ∈ Ring) |
15 | eqid 2820 | . . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
16 | 4, 15 | rhmf 19473 | . . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:𝑋⟶(Base‘𝑆)) |
17 | 16 | 3ad2ant1 1128 | . . . 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 6845 | . . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ ((𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → (𝐹‘𝐴) ∈ (Base‘𝑆)) |
19 | simp3l 1196 | . . 3 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ ((𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → (𝐹‘𝐵) ∈ 𝑈) | |
20 | simp3r 1197 | . . 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 30885 | . . 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 1367 | . 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 2856 | 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 1082 = wceq 1536 ∈ wcel 2113 ⟶wf 6344 ‘cfv 6348 (class class class)co 7149 Basecbs 16478 .rcmulr 16561 Ringcrg 19292 Unitcui 19384 /rcdvr 19427 RingHom crh 19459 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 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 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-tpos 7885 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-3 11695 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-ress 16486 df-plusg 16573 df-mulr 16574 df-0g 16710 df-mgm 17847 df-sgrp 17896 df-mnd 17907 df-mhm 17951 df-grp 18101 df-minusg 18102 df-ghm 18351 df-mgp 19235 df-ur 19247 df-ring 19294 df-oppr 19368 df-dvdsr 19386 df-unit 19387 df-invr 19417 df-dvr 19428 df-rnghom 19462 |
This theorem is referenced by: qqhval2lem 31243 qqhghm 31250 qqhrhm 31251 |
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