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Mirrors > Home > MPE Home > Th. List > prdsinvgd2 | Structured version Visualization version GIF version |
Description: Negation of a single coordinate in a structure product. (Contributed by Stefan O'Rear, 11-Jan-2015.) |
Ref | Expression |
---|---|
prdsinvgd2.y | ⊢ 𝑌 = (𝑆Xs𝑅) |
prdsinvgd2.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
prdsinvgd2.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
prdsinvgd2.r | ⊢ (𝜑 → 𝑅:𝐼⟶Grp) |
prdsinvgd2.b | ⊢ 𝐵 = (Base‘𝑌) |
prdsinvgd2.n | ⊢ 𝑁 = (invg‘𝑌) |
prdsinvgd2.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
prdsinvgd2.j | ⊢ (𝜑 → 𝐽 ∈ 𝐼) |
Ref | Expression |
---|---|
prdsinvgd2 | ⊢ (𝜑 → ((𝑁‘𝑋)‘𝐽) = ((invg‘(𝑅‘𝐽))‘(𝑋‘𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prdsinvgd2.y | . . . 4 ⊢ 𝑌 = (𝑆Xs𝑅) | |
2 | prdsinvgd2.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
3 | prdsinvgd2.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
4 | prdsinvgd2.r | . . . 4 ⊢ (𝜑 → 𝑅:𝐼⟶Grp) | |
5 | prdsinvgd2.b | . . . 4 ⊢ 𝐵 = (Base‘𝑌) | |
6 | prdsinvgd2.n | . . . 4 ⊢ 𝑁 = (invg‘𝑌) | |
7 | prdsinvgd2.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
8 | 1, 2, 3, 4, 5, 6, 7 | prdsinvgd 17526 | . . 3 ⊢ (𝜑 → (𝑁‘𝑋) = (𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥)))) |
9 | 8 | fveq1d 6193 | . 2 ⊢ (𝜑 → ((𝑁‘𝑋)‘𝐽) = ((𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥)))‘𝐽)) |
10 | prdsinvgd2.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝐼) | |
11 | fveq2 6191 | . . . . . 6 ⊢ (𝑥 = 𝐽 → (𝑅‘𝑥) = (𝑅‘𝐽)) | |
12 | 11 | fveq2d 6195 | . . . . 5 ⊢ (𝑥 = 𝐽 → (invg‘(𝑅‘𝑥)) = (invg‘(𝑅‘𝐽))) |
13 | fveq2 6191 | . . . . 5 ⊢ (𝑥 = 𝐽 → (𝑋‘𝑥) = (𝑋‘𝐽)) | |
14 | 12, 13 | fveq12d 6197 | . . . 4 ⊢ (𝑥 = 𝐽 → ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥)) = ((invg‘(𝑅‘𝐽))‘(𝑋‘𝐽))) |
15 | eqid 2622 | . . . 4 ⊢ (𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥))) | |
16 | fvex 6201 | . . . 4 ⊢ ((invg‘(𝑅‘𝐽))‘(𝑋‘𝐽)) ∈ V | |
17 | 14, 15, 16 | fvmpt 6282 | . . 3 ⊢ (𝐽 ∈ 𝐼 → ((𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥)))‘𝐽) = ((invg‘(𝑅‘𝐽))‘(𝑋‘𝐽))) |
18 | 10, 17 | syl 17 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥)))‘𝐽) = ((invg‘(𝑅‘𝐽))‘(𝑋‘𝐽))) |
19 | 9, 18 | eqtrd 2656 | 1 ⊢ (𝜑 → ((𝑁‘𝑋)‘𝐽) = ((invg‘(𝑅‘𝐽))‘(𝑋‘𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ↦ cmpt 4729 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 Xscprds 16106 Grpcgrp 17422 invgcminusg 17423 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-0g 16102 df-prds 16108 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 |
This theorem is referenced by: dsmmsubg 20087 |
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