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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 2821 | . . . . 5 ⊢ (+g‘𝑌) = (+g‘𝑌) | |
4 | prdsgrpd.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
5 | 4 | elexd 3515 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ V) |
6 | prdsgrpd.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
7 | 6 | elexd 3515 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ V) |
8 | prdsgrpd.r | . . . . 5 ⊢ (𝜑 → 𝑅:𝐼⟶Grp) | |
9 | prdsinvgd.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
10 | eqid 2821 | . . . . 5 ⊢ (0g ∘ 𝑅) = (0g ∘ 𝑅) | |
11 | eqid 2821 | . . . . 5 ⊢ (𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥))) | |
12 | 1, 2, 3, 5, 7, 8, 9, 10, 11 | prdsinvlem 18202 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥))) ∈ 𝐵 ∧ ((𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥)))(+g‘𝑌)𝑋) = (0g ∘ 𝑅))) |
13 | 12 | simprd 498 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥)))(+g‘𝑌)𝑋) = (0g ∘ 𝑅)) |
14 | grpmnd 18104 | . . . . . 6 ⊢ (𝑎 ∈ Grp → 𝑎 ∈ Mnd) | |
15 | 14 | ssriv 3971 | . . . . 5 ⊢ Grp ⊆ Mnd |
16 | fss 6522 | . . . . 5 ⊢ ((𝑅:𝐼⟶Grp ∧ Grp ⊆ Mnd) → 𝑅:𝐼⟶Mnd) | |
17 | 8, 15, 16 | sylancl 588 | . . . 4 ⊢ (𝜑 → 𝑅:𝐼⟶Mnd) |
18 | 1, 6, 4, 17 | prds0g 17939 | . . 3 ⊢ (𝜑 → (0g ∘ 𝑅) = (0g‘𝑌)) |
19 | 13, 18 | eqtrd 2856 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥)))(+g‘𝑌)𝑋) = (0g‘𝑌)) |
20 | 1, 6, 4, 8 | prdsgrpd 18203 | . . 3 ⊢ (𝜑 → 𝑌 ∈ Grp) |
21 | 12 | simpld 497 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥))) ∈ 𝐵) |
22 | eqid 2821 | . . . 4 ⊢ (0g‘𝑌) = (0g‘𝑌) | |
23 | prdsinvgd.n | . . . 4 ⊢ 𝑁 = (invg‘𝑌) | |
24 | 2, 3, 22, 23 | grpinvid2 18149 | . . 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))) |
25 | 20, 9, 21, 24 | syl3anc 1367 | . 2 ⊢ (𝜑 → ((𝑁‘𝑋) = (𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥))) ↔ ((𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥)))(+g‘𝑌)𝑋) = (0g‘𝑌))) |
26 | 19, 25 | mpbird 259 | 1 ⊢ (𝜑 → (𝑁‘𝑋) = (𝑥 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑥))‘(𝑋‘𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 Vcvv 3495 ⊆ wss 3936 ↦ cmpt 5139 ∘ ccom 5554 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 0gc0g 16707 Xscprds 16713 Mndcmnd 17905 Grpcgrp 18097 invgcminusg 18098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-hom 16583 df-cco 16584 df-0g 16709 df-prds 16715 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 |
This theorem is referenced by: pwsinvg 18206 prdsinvgd2 20880 prdstgpd 22727 |
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