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Mirrors > Home > MPE Home > Th. List > prdsinvgd | Structured version Visualization version GIF version |
Description: Negation in a product of groups. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
Ref | Expression |
---|---|
prdsgrpd.y | ⊢ 𝑌 = (𝑆Xs𝑅) |
prdsgrpd.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
prdsgrpd.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
prdsgrpd.r | ⊢ (𝜑 → 𝑅:𝐼⟶Grp) |
prdsinvgd.b | ⊢ 𝐵 = (Base‘𝑌) |
prdsinvgd.n | ⊢ 𝑁 = (invg‘𝑌) |
prdsinvgd.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
prdsinvgd | ⊢ (𝜑 → (𝑁‘𝑋) = (𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prdsgrpd.y | . . . . 5 ⊢ 𝑌 = (𝑆Xs𝑅) | |
2 | prdsinvgd.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑌) | |
3 | eqid 2651 | . . . . 5 ⊢ (+g‘𝑌) = (+g‘𝑌) | |
4 | prdsgrpd.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
5 | elex 3243 | . . . . . 6 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ V) |
7 | prdsgrpd.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
8 | elex 3243 | . . . . . 6 ⊢ (𝐼 ∈ 𝑊 → 𝐼 ∈ V) | |
9 | 7, 8 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ V) |
10 | prdsgrpd.r | . . . . 5 ⊢ (𝜑 → 𝑅:𝐼⟶Grp) | |
11 | prdsinvgd.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
12 | eqid 2651 | . . . . 5 ⊢ (0g ∘ 𝑅) = (0g ∘ 𝑅) | |
13 | eqid 2651 | . . . . 5 ⊢ (𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥))) | |
14 | 1, 2, 3, 6, 9, 10, 11, 12, 13 | prdsinvlem 17571 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥))) ∈ 𝐵 ∧ ((𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥)))(+g‘𝑌)𝑋) = (0g ∘ 𝑅))) |
15 | 14 | simprd 478 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥)))(+g‘𝑌)𝑋) = (0g ∘ 𝑅)) |
16 | grpmnd 17476 | . . . . . 6 ⊢ (𝑎 ∈ Grp → 𝑎 ∈ Mnd) | |
17 | 16 | ssriv 3640 | . . . . 5 ⊢ Grp ⊆ Mnd |
18 | fss 6094 | . . . . 5 ⊢ ((𝑅:𝐼⟶Grp ∧ Grp ⊆ Mnd) → 𝑅:𝐼⟶Mnd) | |
19 | 10, 17, 18 | sylancl 695 | . . . 4 ⊢ (𝜑 → 𝑅:𝐼⟶Mnd) |
20 | 1, 7, 4, 19 | prds0g 17371 | . . 3 ⊢ (𝜑 → (0g ∘ 𝑅) = (0g‘𝑌)) |
21 | 15, 20 | eqtrd 2685 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥)))(+g‘𝑌)𝑋) = (0g‘𝑌)) |
22 | 1, 7, 4, 10 | prdsgrpd 17572 | . . 3 ⊢ (𝜑 → 𝑌 ∈ Grp) |
23 | 14 | simpld 474 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥))) ∈ 𝐵) |
24 | eqid 2651 | . . . 4 ⊢ (0g‘𝑌) = (0g‘𝑌) | |
25 | prdsinvgd.n | . . . 4 ⊢ 𝑁 = (invg‘𝑌) | |
26 | 2, 3, 24, 25 | grpinvid2 17518 | . . 3 ⊢ ((𝑌 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ (𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥))) ∈ 𝐵) → ((𝑁‘𝑋) = (𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥))) ↔ ((𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥)))(+g‘𝑌)𝑋) = (0g‘𝑌))) |
27 | 22, 11, 23, 26 | syl3anc 1366 | . 2 ⊢ (𝜑 → ((𝑁‘𝑋) = (𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥))) ↔ ((𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥)))(+g‘𝑌)𝑋) = (0g‘𝑌))) |
28 | 21, 27 | mpbird 247 | 1 ⊢ (𝜑 → (𝑁‘𝑋) = (𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1523 ∈ wcel 2030 Vcvv 3231 ⊆ wss 3607 ↦ cmpt 4762 ∘ ccom 5147 ⟶wf 5922 ‘cfv 5926 (class class class)co 6690 Basecbs 15904 +gcplusg 15988 0gc0g 16147 Xscprds 16153 Mndcmnd 17341 Grpcgrp 17469 invgcminusg 17470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-map 7901 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-sup 8389 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-fz 12365 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-plusg 16001 df-mulr 16002 df-sca 16004 df-vsca 16005 df-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-hom 16013 df-cco 16014 df-0g 16149 df-prds 16155 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-grp 17472 df-minusg 17473 |
This theorem is referenced by: pwsinvg 17575 prdsinvgd2 20134 prdstgpd 21975 |
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