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Mirrors > Home > MPE Home > Th. List > abv1z | Structured version Visualization version GIF version |
Description: The absolute value of one is one in a non-trivial ring. (Contributed by Mario Carneiro, 8-Sep-2014.) |
Ref | Expression |
---|---|
abv0.a | ⊢ 𝐴 = (AbsVal‘𝑅) |
abv1.p | ⊢ 1 = (1r‘𝑅) |
abv1z.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
abv1z | ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → (𝐹‘ 1 ) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abv0.a | . . . . . . . 8 ⊢ 𝐴 = (AbsVal‘𝑅) | |
2 | 1 | abvrcl 20831 | . . . . . . 7 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring) |
3 | eqid 2735 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
4 | abv1.p | . . . . . . . 8 ⊢ 1 = (1r‘𝑅) | |
5 | 3, 4 | ringidcl 20280 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅)) |
6 | 2, 5 | syl 17 | . . . . . 6 ⊢ (𝐹 ∈ 𝐴 → 1 ∈ (Base‘𝑅)) |
7 | 1, 3 | abvcl 20834 | . . . . . 6 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ∈ (Base‘𝑅)) → (𝐹‘ 1 ) ∈ ℝ) |
8 | 6, 7 | mpdan 687 | . . . . 5 ⊢ (𝐹 ∈ 𝐴 → (𝐹‘ 1 ) ∈ ℝ) |
9 | 8 | adantr 480 | . . . 4 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → (𝐹‘ 1 ) ∈ ℝ) |
10 | 9 | recnd 11287 | . . 3 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → (𝐹‘ 1 ) ∈ ℂ) |
11 | simpl 482 | . . . 4 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → 𝐹 ∈ 𝐴) | |
12 | 6 | adantr 480 | . . . 4 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → 1 ∈ (Base‘𝑅)) |
13 | simpr 484 | . . . 4 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → 1 ≠ 0 ) | |
14 | abv1z.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
15 | 1, 3, 14 | abvne0 20837 | . . . 4 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ∈ (Base‘𝑅) ∧ 1 ≠ 0 ) → (𝐹‘ 1 ) ≠ 0) |
16 | 11, 12, 13, 15 | syl3anc 1370 | . . 3 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → (𝐹‘ 1 ) ≠ 0) |
17 | 10, 10, 16 | divcan3d 12046 | . 2 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → (((𝐹‘ 1 ) · (𝐹‘ 1 )) / (𝐹‘ 1 )) = (𝐹‘ 1 )) |
18 | eqid 2735 | . . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
19 | 3, 18, 4 | ringlidm 20283 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 1 ∈ (Base‘𝑅)) → ( 1 (.r‘𝑅) 1 ) = 1 ) |
20 | 2, 12, 19 | syl2an2r 685 | . . . . . 6 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → ( 1 (.r‘𝑅) 1 ) = 1 ) |
21 | 20 | fveq2d 6911 | . . . . 5 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → (𝐹‘( 1 (.r‘𝑅) 1 )) = (𝐹‘ 1 )) |
22 | 1, 3, 18 | abvmul 20839 | . . . . . 6 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ∈ (Base‘𝑅) ∧ 1 ∈ (Base‘𝑅)) → (𝐹‘( 1 (.r‘𝑅) 1 )) = ((𝐹‘ 1 ) · (𝐹‘ 1 ))) |
23 | 11, 12, 12, 22 | syl3anc 1370 | . . . . 5 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → (𝐹‘( 1 (.r‘𝑅) 1 )) = ((𝐹‘ 1 ) · (𝐹‘ 1 ))) |
24 | 21, 23 | eqtr3d 2777 | . . . 4 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → (𝐹‘ 1 ) = ((𝐹‘ 1 ) · (𝐹‘ 1 ))) |
25 | 24 | oveq1d 7446 | . . 3 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → ((𝐹‘ 1 ) / (𝐹‘ 1 )) = (((𝐹‘ 1 ) · (𝐹‘ 1 )) / (𝐹‘ 1 ))) |
26 | 10, 16 | dividd 12039 | . . 3 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → ((𝐹‘ 1 ) / (𝐹‘ 1 )) = 1) |
27 | 25, 26 | eqtr3d 2777 | . 2 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → (((𝐹‘ 1 ) · (𝐹‘ 1 )) / (𝐹‘ 1 )) = 1) |
28 | 17, 27 | eqtr3d 2777 | 1 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → (𝐹‘ 1 ) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 0cc0 11153 1c1 11154 · cmul 11158 / cdiv 11918 Basecbs 17245 .rcmulr 17299 0gc0g 17486 1rcur 20199 Ringcrg 20251 AbsValcabv 20826 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-ico 13390 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mgp 20153 df-ur 20200 df-ring 20253 df-abv 20827 |
This theorem is referenced by: abv1 20843 abvneg 20844 nm1 24704 qabvle 27684 qabvexp 27685 ostthlem2 27687 ostth3 27697 ostth 27698 abvexp 42519 fiabv 42523 |
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