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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 18644 | . . . . . . 7 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring) |
3 | eqid 2610 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
4 | abv1.p | . . . . . . . 8 ⊢ 1 = (1r‘𝑅) | |
5 | 3, 4 | ringidcl 18391 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅)) |
6 | 2, 5 | syl 17 | . . . . . 6 ⊢ (𝐹 ∈ 𝐴 → 1 ∈ (Base‘𝑅)) |
7 | 1, 3 | abvcl 18647 | . . . . . 6 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ∈ (Base‘𝑅)) → (𝐹‘ 1 ) ∈ ℝ) |
8 | 6, 7 | mpdan 699 | . . . . 5 ⊢ (𝐹 ∈ 𝐴 → (𝐹‘ 1 ) ∈ ℝ) |
9 | 8 | adantr 480 | . . . 4 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → (𝐹‘ 1 ) ∈ ℝ) |
10 | 9 | recnd 9947 | . . 3 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → (𝐹‘ 1 ) ∈ ℂ) |
11 | simpl 472 | . . . 4 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → 𝐹 ∈ 𝐴) | |
12 | 6 | adantr 480 | . . . 4 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → 1 ∈ (Base‘𝑅)) |
13 | simpr 476 | . . . 4 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → 1 ≠ 0 ) | |
14 | abv1z.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
15 | 1, 3, 14 | abvne0 18650 | . . . 4 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ∈ (Base‘𝑅) ∧ 1 ≠ 0 ) → (𝐹‘ 1 ) ≠ 0) |
16 | 11, 12, 13, 15 | syl3anc 1318 | . . 3 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → (𝐹‘ 1 ) ≠ 0) |
17 | 10, 10, 16 | divcan3d 10685 | . 2 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → (((𝐹‘ 1 ) · (𝐹‘ 1 )) / (𝐹‘ 1 )) = (𝐹‘ 1 )) |
18 | 2 | adantr 480 | . . . . . . 7 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → 𝑅 ∈ Ring) |
19 | eqid 2610 | . . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
20 | 3, 19, 4 | ringlidm 18394 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 1 ∈ (Base‘𝑅)) → ( 1 (.r‘𝑅) 1 ) = 1 ) |
21 | 18, 12, 20 | syl2anc 691 | . . . . . 6 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → ( 1 (.r‘𝑅) 1 ) = 1 ) |
22 | 21 | fveq2d 6107 | . . . . 5 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → (𝐹‘( 1 (.r‘𝑅) 1 )) = (𝐹‘ 1 )) |
23 | 1, 3, 19 | abvmul 18652 | . . . . . 6 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ∈ (Base‘𝑅) ∧ 1 ∈ (Base‘𝑅)) → (𝐹‘( 1 (.r‘𝑅) 1 )) = ((𝐹‘ 1 ) · (𝐹‘ 1 ))) |
24 | 11, 12, 12, 23 | syl3anc 1318 | . . . . 5 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → (𝐹‘( 1 (.r‘𝑅) 1 )) = ((𝐹‘ 1 ) · (𝐹‘ 1 ))) |
25 | 22, 24 | eqtr3d 2646 | . . . 4 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → (𝐹‘ 1 ) = ((𝐹‘ 1 ) · (𝐹‘ 1 ))) |
26 | 25 | oveq1d 6564 | . . 3 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → ((𝐹‘ 1 ) / (𝐹‘ 1 )) = (((𝐹‘ 1 ) · (𝐹‘ 1 )) / (𝐹‘ 1 ))) |
27 | 10, 16 | dividd 10678 | . . 3 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → ((𝐹‘ 1 ) / (𝐹‘ 1 )) = 1) |
28 | 26, 27 | eqtr3d 2646 | . 2 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → (((𝐹‘ 1 ) · (𝐹‘ 1 )) / (𝐹‘ 1 )) = 1) |
29 | 17, 28 | eqtr3d 2646 | 1 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → (𝐹‘ 1 ) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 · cmul 9820 / cdiv 10563 Basecbs 15695 .rcmulr 15769 0gc0g 15923 1rcur 18324 Ringcrg 18370 AbsValcabv 18639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-ico 12052 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mgp 18313 df-ur 18325 df-ring 18372 df-abv 18640 |
This theorem is referenced by: abv1 18656 abvneg 18657 nm1 22281 qabvle 25114 qabvexp 25115 ostthlem2 25117 ostth3 25127 ostth 25128 |
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