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| Mirrors > Home > MPE Home > Th. List > xpsgrpsub | Structured version Visualization version GIF version | ||
| Description: Value of the subtraction operation in a binary structure product of groups. (Contributed by AV, 24-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) | 
| xpsgrpsub.c | ⊢ (𝜑 → 𝐶 ∈ 𝑋) | 
| xpsgrpsub.d | ⊢ (𝜑 → 𝐷 ∈ 𝑌) | 
| xpsgrpsub.m | ⊢ · = (-g‘𝑅) | 
| xpsgrpsub.n | ⊢ × = (-g‘𝑆) | 
| xpsgrpsub.o | ⊢ − = (-g‘𝑇) | 
| Ref | Expression | 
|---|---|
| xpsgrpsub | ⊢ (𝜑 → (〈𝐴, 𝐵〉 − 〈𝐶, 𝐷〉) = 〈(𝐴 · 𝐶), (𝐵 × 𝐷)〉) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | xpsinv.t | . . . 4 ⊢ 𝑇 = (𝑅 ×s 𝑆) | |
| 2 | xpsinv.x | . . . 4 ⊢ 𝑋 = (Base‘𝑅) | |
| 3 | xpsinv.y | . . . 4 ⊢ 𝑌 = (Base‘𝑆) | |
| 4 | xpsinv.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp) | |
| 5 | xpsinv.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ Grp) | |
| 6 | xpsinv.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
| 7 | xpsgrpsub.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑋) | |
| 8 | xpsgrpsub.m | . . . . . 6 ⊢ · = (-g‘𝑅) | |
| 9 | 2, 8 | grpsubcl 19038 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴 · 𝐶) ∈ 𝑋) | 
| 10 | 4, 6, 7, 9 | syl3anc 1373 | . . . 4 ⊢ (𝜑 → (𝐴 · 𝐶) ∈ 𝑋) | 
| 11 | xpsinv.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑌) | |
| 12 | xpsgrpsub.d | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑌) | |
| 13 | xpsgrpsub.n | . . . . . 6 ⊢ × = (-g‘𝑆) | |
| 14 | 3, 13 | grpsubcl 19038 | . . . . 5 ⊢ ((𝑆 ∈ Grp ∧ 𝐵 ∈ 𝑌 ∧ 𝐷 ∈ 𝑌) → (𝐵 × 𝐷) ∈ 𝑌) | 
| 15 | 5, 11, 12, 14 | syl3anc 1373 | . . . 4 ⊢ (𝜑 → (𝐵 × 𝐷) ∈ 𝑌) | 
| 16 | eqid 2737 | . . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
| 17 | 2, 16, 4, 10, 7 | grpcld 18965 | . . . 4 ⊢ (𝜑 → ((𝐴 · 𝐶)(+g‘𝑅)𝐶) ∈ 𝑋) | 
| 18 | eqid 2737 | . . . . 5 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
| 19 | 3, 18, 5, 15, 12 | grpcld 18965 | . . . 4 ⊢ (𝜑 → ((𝐵 × 𝐷)(+g‘𝑆)𝐷) ∈ 𝑌) | 
| 20 | eqid 2737 | . . . 4 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
| 21 | 1, 2, 3, 4, 5, 10, 15, 7, 12, 17, 19, 16, 18, 20 | xpsadd 17619 | . . 3 ⊢ (𝜑 → (〈(𝐴 · 𝐶), (𝐵 × 𝐷)〉(+g‘𝑇)〈𝐶, 𝐷〉) = 〈((𝐴 · 𝐶)(+g‘𝑅)𝐶), ((𝐵 × 𝐷)(+g‘𝑆)𝐷)〉) | 
| 22 | 2, 16, 8 | grpnpcan 19050 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐴 · 𝐶)(+g‘𝑅)𝐶) = 𝐴) | 
| 23 | 4, 6, 7, 22 | syl3anc 1373 | . . . 4 ⊢ (𝜑 → ((𝐴 · 𝐶)(+g‘𝑅)𝐶) = 𝐴) | 
| 24 | 3, 18, 13 | grpnpcan 19050 | . . . . 5 ⊢ ((𝑆 ∈ Grp ∧ 𝐵 ∈ 𝑌 ∧ 𝐷 ∈ 𝑌) → ((𝐵 × 𝐷)(+g‘𝑆)𝐷) = 𝐵) | 
| 25 | 5, 11, 12, 24 | syl3anc 1373 | . . . 4 ⊢ (𝜑 → ((𝐵 × 𝐷)(+g‘𝑆)𝐷) = 𝐵) | 
| 26 | 23, 25 | opeq12d 4881 | . . 3 ⊢ (𝜑 → 〈((𝐴 · 𝐶)(+g‘𝑅)𝐶), ((𝐵 × 𝐷)(+g‘𝑆)𝐷)〉 = 〈𝐴, 𝐵〉) | 
| 27 | 21, 26 | eqtrd 2777 | . 2 ⊢ (𝜑 → (〈(𝐴 · 𝐶), (𝐵 × 𝐷)〉(+g‘𝑇)〈𝐶, 𝐷〉) = 〈𝐴, 𝐵〉) | 
| 28 | 1 | xpsgrp 19077 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) → 𝑇 ∈ Grp) | 
| 29 | 4, 5, 28 | syl2anc 584 | . . 3 ⊢ (𝜑 → 𝑇 ∈ Grp) | 
| 30 | 6, 11 | opelxpd 5724 | . . . 4 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑌)) | 
| 31 | 1, 2, 3, 4, 5 | xpsbas 17617 | . . . 4 ⊢ (𝜑 → (𝑋 × 𝑌) = (Base‘𝑇)) | 
| 32 | 30, 31 | eleqtrd 2843 | . . 3 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ (Base‘𝑇)) | 
| 33 | 7, 12 | opelxpd 5724 | . . . 4 ⊢ (𝜑 → 〈𝐶, 𝐷〉 ∈ (𝑋 × 𝑌)) | 
| 34 | 33, 31 | eleqtrd 2843 | . . 3 ⊢ (𝜑 → 〈𝐶, 𝐷〉 ∈ (Base‘𝑇)) | 
| 35 | 10, 15 | opelxpd 5724 | . . . 4 ⊢ (𝜑 → 〈(𝐴 · 𝐶), (𝐵 × 𝐷)〉 ∈ (𝑋 × 𝑌)) | 
| 36 | 35, 31 | eleqtrd 2843 | . . 3 ⊢ (𝜑 → 〈(𝐴 · 𝐶), (𝐵 × 𝐷)〉 ∈ (Base‘𝑇)) | 
| 37 | eqid 2737 | . . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
| 38 | xpsgrpsub.o | . . . 4 ⊢ − = (-g‘𝑇) | |
| 39 | 37, 20, 38 | grpsubadd 19046 | . . 3 ⊢ ((𝑇 ∈ Grp ∧ (〈𝐴, 𝐵〉 ∈ (Base‘𝑇) ∧ 〈𝐶, 𝐷〉 ∈ (Base‘𝑇) ∧ 〈(𝐴 · 𝐶), (𝐵 × 𝐷)〉 ∈ (Base‘𝑇))) → ((〈𝐴, 𝐵〉 − 〈𝐶, 𝐷〉) = 〈(𝐴 · 𝐶), (𝐵 × 𝐷)〉 ↔ (〈(𝐴 · 𝐶), (𝐵 × 𝐷)〉(+g‘𝑇)〈𝐶, 𝐷〉) = 〈𝐴, 𝐵〉)) | 
| 40 | 29, 32, 34, 36, 39 | syl13anc 1374 | . 2 ⊢ (𝜑 → ((〈𝐴, 𝐵〉 − 〈𝐶, 𝐷〉) = 〈(𝐴 · 𝐶), (𝐵 × 𝐷)〉 ↔ (〈(𝐴 · 𝐶), (𝐵 × 𝐷)〉(+g‘𝑇)〈𝐶, 𝐷〉) = 〈𝐴, 𝐵〉)) | 
| 41 | 27, 40 | mpbird 257 | 1 ⊢ (𝜑 → (〈𝐴, 𝐵〉 − 〈𝐶, 𝐷〉) = 〈(𝐴 · 𝐶), (𝐵 × 𝐷)〉) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 〈cop 4632 × cxp 5683 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 ×s cxps 17551 Grpcgrp 18951 -gcsg 18953 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 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 17486 df-prds 17492 df-imas 17553 df-xps 17555 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 | 
| This theorem is referenced by: pzriprng1ALT 21507 | 
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