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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 2724 | . . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
3 | eqid 2724 | . . . . 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 18914 | . . . 4 ⊢ (𝜑 → ((𝑀‘𝐴)(+g‘𝑅)𝐴) = (0g‘𝑅)) |
8 | xpsinv.y | . . . . 5 ⊢ 𝑌 = (Base‘𝑆) | |
9 | eqid 2724 | . . . . 5 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
10 | eqid 2724 | . . . . 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 18914 | . . . 4 ⊢ (𝜑 → ((𝑁‘𝐵)(+g‘𝑆)𝐵) = (0g‘𝑆)) |
15 | 7, 14 | opeq12d 4873 | . . 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 18908 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑋) |
18 | 8, 11, 12, 13 | grpinvcld 18908 | . . . 4 ⊢ (𝜑 → (𝑁‘𝐵) ∈ 𝑌) |
19 | 1, 2, 5, 17, 6 | grpcld 18867 | . . . 4 ⊢ (𝜑 → ((𝑀‘𝐴)(+g‘𝑅)𝐴) ∈ 𝑋) |
20 | 8, 9, 12, 18, 13 | grpcld 18867 | . . . 4 ⊢ (𝜑 → ((𝑁‘𝐵)(+g‘𝑆)𝐵) ∈ 𝑌) |
21 | eqid 2724 | . . . 4 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
22 | 16, 1, 8, 5, 12, 17, 18, 6, 13, 19, 20, 2, 9, 21 | xpsadd 17519 | . . 3 ⊢ (𝜑 → (⟨(𝑀‘𝐴), (𝑁‘𝐵)⟩(+g‘𝑇)⟨𝐴, 𝐵⟩) = ⟨((𝑀‘𝐴)(+g‘𝑅)𝐴), ((𝑁‘𝐵)(+g‘𝑆)𝐵)⟩) |
23 | 5 | grpmndd 18866 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Mnd) |
24 | 12 | grpmndd 18866 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ Mnd) |
25 | 16 | xpsmnd0 18698 | . . . 4 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (0g‘𝑇) = ⟨(0g‘𝑅), (0g‘𝑆)⟩) |
26 | 23, 24, 25 | syl2anc 583 | . . 3 ⊢ (𝜑 → (0g‘𝑇) = ⟨(0g‘𝑅), (0g‘𝑆)⟩) |
27 | 15, 22, 26 | 3eqtr4d 2774 | . 2 ⊢ (𝜑 → (⟨(𝑀‘𝐴), (𝑁‘𝐵)⟩(+g‘𝑇)⟨𝐴, 𝐵⟩) = (0g‘𝑇)) |
28 | 16 | xpsgrp 18977 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) → 𝑇 ∈ Grp) |
29 | 5, 12, 28 | syl2anc 583 | . . 3 ⊢ (𝜑 → 𝑇 ∈ Grp) |
30 | 6, 13 | opelxpd 5705 | . . . 4 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌)) |
31 | 16, 1, 8, 5, 12 | xpsbas 17517 | . . . 4 ⊢ (𝜑 → (𝑋 × 𝑌) = (Base‘𝑇)) |
32 | 30, 31 | eleqtrd 2827 | . . 3 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (Base‘𝑇)) |
33 | 17, 18 | opelxpd 5705 | . . . 4 ⊢ (𝜑 → ⟨(𝑀‘𝐴), (𝑁‘𝐵)⟩ ∈ (𝑋 × 𝑌)) |
34 | 33, 31 | eleqtrd 2827 | . . 3 ⊢ (𝜑 → ⟨(𝑀‘𝐴), (𝑁‘𝐵)⟩ ∈ (Base‘𝑇)) |
35 | eqid 2724 | . . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
36 | eqid 2724 | . . . 4 ⊢ (0g‘𝑇) = (0g‘𝑇) | |
37 | xpsinv.i | . . . 4 ⊢ 𝐼 = (invg‘𝑇) | |
38 | 35, 21, 36, 37 | grpinvid2 18912 | . . 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 1368 | . 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 205 = wceq 1533 ∈ wcel 2098 ⟨cop 4626 × cxp 5664 ‘cfv 6533 (class class class)co 7401 Basecbs 17143 +gcplusg 17196 0gc0g 17384 ×s cxps 17451 Mndcmnd 18657 Grpcgrp 18853 invgcminusg 18854 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8699 df-map 8818 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-struct 17079 df-slot 17114 df-ndx 17126 df-base 17144 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-hom 17220 df-cco 17221 df-0g 17386 df-prds 17392 df-imas 17453 df-xps 17455 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-grp 18856 df-minusg 18857 |
This theorem is referenced by: pzriprnglem4 21339 |
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