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Mirrors > Home > MPE Home > Th. List > abvrec | Structured version Visualization version GIF version |
Description: The absolute value distributes under reciprocal. (Contributed by Mario Carneiro, 10-Sep-2014.) |
Ref | Expression |
---|---|
abv0.a | ⊢ 𝐴 = (AbsVal‘𝑅) |
abvneg.b | ⊢ 𝐵 = (Base‘𝑅) |
abvrec.z | ⊢ 0 = (0g‘𝑅) |
abvrec.p | ⊢ 𝐼 = (invr‘𝑅) |
Ref | Expression |
---|---|
abvrec | ⊢ (((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )) → (𝐹‘(𝐼‘𝑋)) = (1 / (𝐹‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplr 767 | . . . 4 ⊢ (((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )) → 𝐹 ∈ 𝐴) | |
2 | simprl 769 | . . . 4 ⊢ (((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )) → 𝑋 ∈ 𝐵) | |
3 | abv0.a | . . . . 5 ⊢ 𝐴 = (AbsVal‘𝑅) | |
4 | abvneg.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
5 | 3, 4 | abvcl 20711 | . . . 4 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ ℝ) |
6 | 1, 2, 5 | syl2anc 582 | . . 3 ⊢ (((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )) → (𝐹‘𝑋) ∈ ℝ) |
7 | 6 | recnd 11280 | . 2 ⊢ (((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )) → (𝐹‘𝑋) ∈ ℂ) |
8 | simpll 765 | . . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )) → 𝑅 ∈ DivRing) | |
9 | simprr 771 | . . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )) → 𝑋 ≠ 0 ) | |
10 | abvrec.z | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
11 | abvrec.p | . . . . . 6 ⊢ 𝐼 = (invr‘𝑅) | |
12 | 4, 10, 11 | drnginvrcl 20653 | . . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐼‘𝑋) ∈ 𝐵) |
13 | 8, 2, 9, 12 | syl3anc 1368 | . . . 4 ⊢ (((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )) → (𝐼‘𝑋) ∈ 𝐵) |
14 | 3, 4 | abvcl 20711 | . . . 4 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐵) → (𝐹‘(𝐼‘𝑋)) ∈ ℝ) |
15 | 1, 13, 14 | syl2anc 582 | . . 3 ⊢ (((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )) → (𝐹‘(𝐼‘𝑋)) ∈ ℝ) |
16 | 15 | recnd 11280 | . 2 ⊢ (((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )) → (𝐹‘(𝐼‘𝑋)) ∈ ℂ) |
17 | 3, 4, 10 | abvne0 20714 | . . 3 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐹‘𝑋) ≠ 0) |
18 | 1, 2, 9, 17 | syl3anc 1368 | . 2 ⊢ (((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )) → (𝐹‘𝑋) ≠ 0) |
19 | eqid 2728 | . . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
20 | eqid 2728 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
21 | 4, 10, 19, 20, 11 | drnginvrr 20657 | . . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝑋(.r‘𝑅)(𝐼‘𝑋)) = (1r‘𝑅)) |
22 | 8, 2, 9, 21 | syl3anc 1368 | . . . 4 ⊢ (((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )) → (𝑋(.r‘𝑅)(𝐼‘𝑋)) = (1r‘𝑅)) |
23 | 22 | fveq2d 6906 | . . 3 ⊢ (((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )) → (𝐹‘(𝑋(.r‘𝑅)(𝐼‘𝑋))) = (𝐹‘(1r‘𝑅))) |
24 | 3, 4, 19 | abvmul 20716 | . . . 4 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ (𝐼‘𝑋) ∈ 𝐵) → (𝐹‘(𝑋(.r‘𝑅)(𝐼‘𝑋))) = ((𝐹‘𝑋) · (𝐹‘(𝐼‘𝑋)))) |
25 | 1, 2, 13, 24 | syl3anc 1368 | . . 3 ⊢ (((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )) → (𝐹‘(𝑋(.r‘𝑅)(𝐼‘𝑋))) = ((𝐹‘𝑋) · (𝐹‘(𝐼‘𝑋)))) |
26 | 3, 20 | abv1 20720 | . . . 4 ⊢ ((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐴) → (𝐹‘(1r‘𝑅)) = 1) |
27 | 26 | adantr 479 | . . 3 ⊢ (((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )) → (𝐹‘(1r‘𝑅)) = 1) |
28 | 23, 25, 27 | 3eqtr3d 2776 | . 2 ⊢ (((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )) → ((𝐹‘𝑋) · (𝐹‘(𝐼‘𝑋))) = 1) |
29 | 7, 16, 18, 28 | mvllmuld 12084 | 1 ⊢ (((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )) → (𝐹‘(𝐼‘𝑋)) = (1 / (𝐹‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ‘cfv 6553 (class class class)co 7426 ℝcr 11145 0cc0 11146 1c1 11147 · cmul 11151 / cdiv 11909 Basecbs 17187 .rcmulr 17241 0gc0g 17428 1rcur 20128 invrcinvr 20333 DivRingcdr 20631 AbsValcabv 20703 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-tpos 8238 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-ico 13370 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-oppr 20280 df-dvdsr 20303 df-unit 20304 df-invr 20334 df-drng 20633 df-abv 20704 |
This theorem is referenced by: abvdiv 20724 |
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