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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 20814 | . . . . . . 7 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring) |
| 3 | eqid 2737 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 4 | abv1.p | . . . . . . . 8 ⊢ 1 = (1r‘𝑅) | |
| 5 | 3, 4 | ringidcl 20262 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅)) |
| 6 | 2, 5 | syl 17 | . . . . . 6 ⊢ (𝐹 ∈ 𝐴 → 1 ∈ (Base‘𝑅)) |
| 7 | 1, 3 | abvcl 20817 | . . . . . 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 11289 | . . 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 20820 | . . . 4 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ∈ (Base‘𝑅) ∧ 1 ≠ 0 ) → (𝐹‘ 1 ) ≠ 0) |
| 16 | 11, 12, 13, 15 | syl3anc 1373 | . . 3 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → (𝐹‘ 1 ) ≠ 0) |
| 17 | 10, 10, 16 | divcan3d 12048 | . 2 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → (((𝐹‘ 1 ) · (𝐹‘ 1 )) / (𝐹‘ 1 )) = (𝐹‘ 1 )) |
| 18 | eqid 2737 | . . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
| 19 | 3, 18, 4 | ringlidm 20266 | . . . . . . 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 6910 | . . . . 5 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → (𝐹‘( 1 (.r‘𝑅) 1 )) = (𝐹‘ 1 )) |
| 22 | 1, 3, 18 | abvmul 20822 | . . . . . 6 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ∈ (Base‘𝑅) ∧ 1 ∈ (Base‘𝑅)) → (𝐹‘( 1 (.r‘𝑅) 1 )) = ((𝐹‘ 1 ) · (𝐹‘ 1 ))) |
| 23 | 11, 12, 12, 22 | syl3anc 1373 | . . . . 5 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → (𝐹‘( 1 (.r‘𝑅) 1 )) = ((𝐹‘ 1 ) · (𝐹‘ 1 ))) |
| 24 | 21, 23 | eqtr3d 2779 | . . . 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 12041 | . . 3 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → ((𝐹‘ 1 ) / (𝐹‘ 1 )) = 1) |
| 27 | 25, 26 | eqtr3d 2779 | . 2 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → (((𝐹‘ 1 ) · (𝐹‘ 1 )) / (𝐹‘ 1 )) = 1) |
| 28 | 17, 27 | eqtr3d 2779 | 1 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → (𝐹‘ 1 ) = 1) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 0cc0 11155 1c1 11156 · cmul 11160 / cdiv 11920 Basecbs 17247 .rcmulr 17298 0gc0g 17484 1rcur 20178 Ringcrg 20230 AbsValcabv 20809 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-ico 13393 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mgp 20138 df-ur 20179 df-ring 20232 df-abv 20810 |
| This theorem is referenced by: abv1 20826 abvneg 20827 nm1 24688 qabvle 27669 qabvexp 27670 ostthlem2 27672 ostth3 27682 ostth 27683 abvexp 42542 fiabv 42546 |
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