![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > xpsinv | Structured version Visualization version GIF version |
Description: Value of the negation operation in a binary structure product. (Contributed by AV, 18-Mar-2025.) |
Ref | Expression |
---|---|
xpsinv.t | ⊢ 𝑇 = (𝑅 ×s 𝑆) |
xpsinv.x | ⊢ 𝑋 = (Base‘𝑅) |
xpsinv.y | ⊢ 𝑌 = (Base‘𝑆) |
xpsinv.r | ⊢ (𝜑 → 𝑅 ∈ Grp) |
xpsinv.s | ⊢ (𝜑 → 𝑆 ∈ Grp) |
xpsinv.a | ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
xpsinv.b | ⊢ (𝜑 → 𝐵 ∈ 𝑌) |
xpsinv.m | ⊢ 𝑀 = (invg‘𝑅) |
xpsinv.n | ⊢ 𝑁 = (invg‘𝑆) |
xpsinv.i | ⊢ 𝐼 = (invg‘𝑇) |
Ref | Expression |
---|---|
xpsinv | ⊢ (𝜑 → (𝐼‘〈𝐴, 𝐵〉) = 〈(𝑀‘𝐴), (𝑁‘𝐵)〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpsinv.x | . . . . 5 ⊢ 𝑋 = (Base‘𝑅) | |
2 | eqid 2734 | . . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
3 | eqid 2734 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
4 | xpsinv.m | . . . . 5 ⊢ 𝑀 = (invg‘𝑅) | |
5 | xpsinv.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Grp) | |
6 | xpsinv.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
7 | 1, 2, 3, 4, 5, 6 | grplinvd 19024 | . . . 4 ⊢ (𝜑 → ((𝑀‘𝐴)(+g‘𝑅)𝐴) = (0g‘𝑅)) |
8 | xpsinv.y | . . . . 5 ⊢ 𝑌 = (Base‘𝑆) | |
9 | eqid 2734 | . . . . 5 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
10 | eqid 2734 | . . . . 5 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
11 | xpsinv.n | . . . . 5 ⊢ 𝑁 = (invg‘𝑆) | |
12 | xpsinv.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ Grp) | |
13 | xpsinv.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑌) | |
14 | 8, 9, 10, 11, 12, 13 | grplinvd 19024 | . . . 4 ⊢ (𝜑 → ((𝑁‘𝐵)(+g‘𝑆)𝐵) = (0g‘𝑆)) |
15 | 7, 14 | opeq12d 4885 | . . 3 ⊢ (𝜑 → 〈((𝑀‘𝐴)(+g‘𝑅)𝐴), ((𝑁‘𝐵)(+g‘𝑆)𝐵)〉 = 〈(0g‘𝑅), (0g‘𝑆)〉) |
16 | xpsinv.t | . . . 4 ⊢ 𝑇 = (𝑅 ×s 𝑆) | |
17 | 1, 4, 5, 6 | grpinvcld 19018 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑋) |
18 | 8, 11, 12, 13 | grpinvcld 19018 | . . . 4 ⊢ (𝜑 → (𝑁‘𝐵) ∈ 𝑌) |
19 | 1, 2, 5, 17, 6 | grpcld 18977 | . . . 4 ⊢ (𝜑 → ((𝑀‘𝐴)(+g‘𝑅)𝐴) ∈ 𝑋) |
20 | 8, 9, 12, 18, 13 | grpcld 18977 | . . . 4 ⊢ (𝜑 → ((𝑁‘𝐵)(+g‘𝑆)𝐵) ∈ 𝑌) |
21 | eqid 2734 | . . . 4 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
22 | 16, 1, 8, 5, 12, 17, 18, 6, 13, 19, 20, 2, 9, 21 | xpsadd 17620 | . . 3 ⊢ (𝜑 → (〈(𝑀‘𝐴), (𝑁‘𝐵)〉(+g‘𝑇)〈𝐴, 𝐵〉) = 〈((𝑀‘𝐴)(+g‘𝑅)𝐴), ((𝑁‘𝐵)(+g‘𝑆)𝐵)〉) |
23 | 5 | grpmndd 18976 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Mnd) |
24 | 12 | grpmndd 18976 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ Mnd) |
25 | 16 | xpsmnd0 18803 | . . . 4 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (0g‘𝑇) = 〈(0g‘𝑅), (0g‘𝑆)〉) |
26 | 23, 24, 25 | syl2anc 584 | . . 3 ⊢ (𝜑 → (0g‘𝑇) = 〈(0g‘𝑅), (0g‘𝑆)〉) |
27 | 15, 22, 26 | 3eqtr4d 2784 | . 2 ⊢ (𝜑 → (〈(𝑀‘𝐴), (𝑁‘𝐵)〉(+g‘𝑇)〈𝐴, 𝐵〉) = (0g‘𝑇)) |
28 | 16 | xpsgrp 19089 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) → 𝑇 ∈ Grp) |
29 | 5, 12, 28 | syl2anc 584 | . . 3 ⊢ (𝜑 → 𝑇 ∈ Grp) |
30 | 6, 13 | opelxpd 5727 | . . . 4 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑌)) |
31 | 16, 1, 8, 5, 12 | xpsbas 17618 | . . . 4 ⊢ (𝜑 → (𝑋 × 𝑌) = (Base‘𝑇)) |
32 | 30, 31 | eleqtrd 2840 | . . 3 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ (Base‘𝑇)) |
33 | 17, 18 | opelxpd 5727 | . . . 4 ⊢ (𝜑 → 〈(𝑀‘𝐴), (𝑁‘𝐵)〉 ∈ (𝑋 × 𝑌)) |
34 | 33, 31 | eleqtrd 2840 | . . 3 ⊢ (𝜑 → 〈(𝑀‘𝐴), (𝑁‘𝐵)〉 ∈ (Base‘𝑇)) |
35 | eqid 2734 | . . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
36 | eqid 2734 | . . . 4 ⊢ (0g‘𝑇) = (0g‘𝑇) | |
37 | xpsinv.i | . . . 4 ⊢ 𝐼 = (invg‘𝑇) | |
38 | 35, 21, 36, 37 | grpinvid2 19022 | . . 3 ⊢ ((𝑇 ∈ Grp ∧ 〈𝐴, 𝐵〉 ∈ (Base‘𝑇) ∧ 〈(𝑀‘𝐴), (𝑁‘𝐵)〉 ∈ (Base‘𝑇)) → ((𝐼‘〈𝐴, 𝐵〉) = 〈(𝑀‘𝐴), (𝑁‘𝐵)〉 ↔ (〈(𝑀‘𝐴), (𝑁‘𝐵)〉(+g‘𝑇)〈𝐴, 𝐵〉) = (0g‘𝑇))) |
39 | 29, 32, 34, 38 | syl3anc 1370 | . 2 ⊢ (𝜑 → ((𝐼‘〈𝐴, 𝐵〉) = 〈(𝑀‘𝐴), (𝑁‘𝐵)〉 ↔ (〈(𝑀‘𝐴), (𝑁‘𝐵)〉(+g‘𝑇)〈𝐴, 𝐵〉) = (0g‘𝑇))) |
40 | 27, 39 | mpbird 257 | 1 ⊢ (𝜑 → (𝐼‘〈𝐴, 𝐵〉) = 〈(𝑀‘𝐴), (𝑁‘𝐵)〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1536 ∈ wcel 2105 〈cop 4636 × cxp 5686 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 +gcplusg 17297 0gc0g 17485 ×s cxps 17552 Mndcmnd 18759 Grpcgrp 18963 invgcminusg 18964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-inf 9480 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-fz 13544 df-struct 17180 df-slot 17215 df-ndx 17227 df-base 17245 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17487 df-prds 17493 df-imas 17554 df-xps 17556 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-minusg 18967 |
This theorem is referenced by: pzriprnglem4 21512 |
Copyright terms: Public domain | W3C validator |