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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 18943 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴 · 𝐶) ∈ 𝑋) |
10 | 4, 6, 7, 9 | syl3anc 1370 | . . . 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 18943 | . . . . 5 ⊢ ((𝑆 ∈ Grp ∧ 𝐵 ∈ 𝑌 ∧ 𝐷 ∈ 𝑌) → (𝐵 × 𝐷) ∈ 𝑌) |
15 | 5, 11, 12, 14 | syl3anc 1370 | . . . 4 ⊢ (𝜑 → (𝐵 × 𝐷) ∈ 𝑌) |
16 | eqid 2731 | . . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
17 | 2, 16, 4, 10, 7 | grpcld 18872 | . . . 4 ⊢ (𝜑 → ((𝐴 · 𝐶)(+g‘𝑅)𝐶) ∈ 𝑋) |
18 | eqid 2731 | . . . . 5 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
19 | 3, 18, 5, 15, 12 | grpcld 18872 | . . . 4 ⊢ (𝜑 → ((𝐵 × 𝐷)(+g‘𝑆)𝐷) ∈ 𝑌) |
20 | eqid 2731 | . . . 4 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
21 | 1, 2, 3, 4, 5, 10, 15, 7, 12, 17, 19, 16, 18, 20 | xpsadd 17527 | . . 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 18955 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐴 · 𝐶)(+g‘𝑅)𝐶) = 𝐴) |
23 | 4, 6, 7, 22 | syl3anc 1370 | . . . 4 ⊢ (𝜑 → ((𝐴 · 𝐶)(+g‘𝑅)𝐶) = 𝐴) |
24 | 3, 18, 13 | grpnpcan 18955 | . . . . 5 ⊢ ((𝑆 ∈ Grp ∧ 𝐵 ∈ 𝑌 ∧ 𝐷 ∈ 𝑌) → ((𝐵 × 𝐷)(+g‘𝑆)𝐷) = 𝐵) |
25 | 5, 11, 12, 24 | syl3anc 1370 | . . . 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 2771 | . 2 ⊢ (𝜑 → (〈(𝐴 · 𝐶), (𝐵 × 𝐷)〉(+g‘𝑇)〈𝐶, 𝐷〉) = 〈𝐴, 𝐵〉) |
28 | 1 | xpsgrp 18982 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) → 𝑇 ∈ Grp) |
29 | 4, 5, 28 | syl2anc 583 | . . 3 ⊢ (𝜑 → 𝑇 ∈ Grp) |
30 | 6, 11 | opelxpd 5715 | . . . 4 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑌)) |
31 | 1, 2, 3, 4, 5 | xpsbas 17525 | . . . 4 ⊢ (𝜑 → (𝑋 × 𝑌) = (Base‘𝑇)) |
32 | 30, 31 | eleqtrd 2834 | . . 3 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ (Base‘𝑇)) |
33 | 7, 12 | opelxpd 5715 | . . . 4 ⊢ (𝜑 → 〈𝐶, 𝐷〉 ∈ (𝑋 × 𝑌)) |
34 | 33, 31 | eleqtrd 2834 | . . 3 ⊢ (𝜑 → 〈𝐶, 𝐷〉 ∈ (Base‘𝑇)) |
35 | 10, 15 | opelxpd 5715 | . . . 4 ⊢ (𝜑 → 〈(𝐴 · 𝐶), (𝐵 × 𝐷)〉 ∈ (𝑋 × 𝑌)) |
36 | 35, 31 | eleqtrd 2834 | . . 3 ⊢ (𝜑 → 〈(𝐴 · 𝐶), (𝐵 × 𝐷)〉 ∈ (Base‘𝑇)) |
37 | eqid 2731 | . . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
38 | xpsgrpsub.o | . . . 4 ⊢ − = (-g‘𝑇) | |
39 | 37, 20, 38 | grpsubadd 18951 | . . 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 1371 | . 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 205 = wceq 1540 ∈ wcel 2105 〈cop 4634 × cxp 5674 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 +gcplusg 17204 ×s cxps 17459 Grpcgrp 18858 -gcsg 18860 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-2o 8473 df-er 8709 df-map 8828 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9443 df-inf 9444 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-fz 13492 df-struct 17087 df-slot 17122 df-ndx 17134 df-base 17152 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-hom 17228 df-cco 17229 df-0g 17394 df-prds 17400 df-imas 17461 df-xps 17463 df-mgm 18568 df-sgrp 18647 df-mnd 18663 df-grp 18861 df-minusg 18862 df-sbg 18863 |
This theorem is referenced by: pzriprng1ALT 21269 |
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