![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > abvpropd2 | Structured version Visualization version GIF version |
Description: Weaker version of abvpropd 19325. (Contributed by Thierry Arnoux, 8-Nov-2017.) |
Ref | Expression |
---|---|
abvpropd2.1 | ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) |
abvpropd2.2 | ⊢ (𝜑 → (+g‘𝐾) = (+g‘𝐿)) |
abvpropd2.3 | ⊢ (𝜑 → (.r‘𝐾) = (.r‘𝐿)) |
Ref | Expression |
---|---|
abvpropd2 | ⊢ (𝜑 → (AbsVal‘𝐾) = (AbsVal‘𝐿)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2773 | . 2 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐾)) | |
2 | abvpropd2.1 | . 2 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) | |
3 | abvpropd2.2 | . . 3 ⊢ (𝜑 → (+g‘𝐾) = (+g‘𝐿)) | |
4 | 3 | oveqdr 6998 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) |
5 | abvpropd2.3 | . . 3 ⊢ (𝜑 → (.r‘𝐾) = (.r‘𝐿)) | |
6 | 5 | oveqdr 6998 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) |
7 | 1, 2, 4, 6 | abvpropd 19325 | 1 ⊢ (𝜑 → (AbsVal‘𝐾) = (AbsVal‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ‘cfv 6182 Basecbs 16329 +gcplusg 16411 .rcmulr 16412 AbsValcabv 19299 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-er 8081 df-map 8200 df-en 8299 df-dom 8300 df-sdom 8301 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-2 11496 df-ndx 16332 df-slot 16333 df-base 16335 df-sets 16336 df-plusg 16424 df-0g 16561 df-mgm 17700 df-sgrp 17742 df-mnd 17753 df-grp 17884 df-mgp 18953 df-ring 19012 df-abv 19300 |
This theorem is referenced by: zhmnrg 30809 |
Copyright terms: Public domain | W3C validator |