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Mirrors > Home > MPE Home > Th. List > Mathboxes > abvpropd2 | Structured version Visualization version GIF version |
Description: Weaker version of abvpropd 19681. (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 2759 | . 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 7178 | . 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 7178 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) |
7 | 1, 2, 4, 6 | abvpropd 19681 | 1 ⊢ (𝜑 → (AbsVal‘𝐾) = (AbsVal‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ‘cfv 6335 Basecbs 16541 +gcplusg 16623 .rcmulr 16624 AbsValcabv 19655 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-sep 5169 ax-nul 5176 ax-pow 5234 ax-pr 5298 ax-un 7459 ax-cnex 10631 ax-resscn 10632 ax-1cn 10633 ax-icn 10634 ax-addcl 10635 ax-addrcl 10636 ax-mulcl 10637 ax-mulrcl 10638 ax-mulcom 10639 ax-addass 10640 ax-mulass 10641 ax-distr 10642 ax-i2m1 10643 ax-1ne0 10644 ax-1rid 10645 ax-rnegex 10646 ax-rrecex 10647 ax-cnre 10648 ax-pre-lttri 10649 ax-pre-lttrn 10650 ax-pre-ltadd 10651 ax-pre-mulgt0 10652 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rab 3079 df-v 3411 df-sbc 3697 df-csb 3806 df-dif 3861 df-un 3863 df-in 3865 df-ss 3875 df-pss 3877 df-nul 4226 df-if 4421 df-pw 4496 df-sn 4523 df-pr 4525 df-tp 4527 df-op 4529 df-uni 4799 df-iun 4885 df-br 5033 df-opab 5095 df-mpt 5113 df-tr 5139 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5483 df-we 5485 df-xp 5530 df-rel 5531 df-cnv 5532 df-co 5533 df-dm 5534 df-rn 5535 df-res 5536 df-ima 5537 df-pred 6126 df-ord 6172 df-on 6173 df-lim 6174 df-suc 6175 df-iota 6294 df-fun 6337 df-fn 6338 df-f 6339 df-f1 6340 df-fo 6341 df-f1o 6342 df-fv 6343 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7580 df-wrecs 7957 df-recs 8018 df-rdg 8056 df-er 8299 df-map 8418 df-en 8528 df-dom 8529 df-sdom 8530 df-pnf 10715 df-mnf 10716 df-xr 10717 df-ltxr 10718 df-le 10719 df-sub 10910 df-neg 10911 df-nn 11675 df-2 11737 df-ndx 16544 df-slot 16545 df-base 16547 df-sets 16548 df-plusg 16636 df-0g 16773 df-mgm 17918 df-sgrp 17967 df-mnd 17978 df-grp 18172 df-mgp 19308 df-ring 19367 df-abv 19656 |
This theorem is referenced by: zhmnrg 31436 |
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