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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 19051 | . . . . 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 19051 | . . . . 5 ⊢ ((𝑆 ∈ Grp ∧ 𝐵 ∈ 𝑌 ∧ 𝐷 ∈ 𝑌) → (𝐵 × 𝐷) ∈ 𝑌) |
15 | 5, 11, 12, 14 | syl3anc 1370 | . . . 4 ⊢ (𝜑 → (𝐵 × 𝐷) ∈ 𝑌) |
16 | eqid 2735 | . . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
17 | 2, 16, 4, 10, 7 | grpcld 18978 | . . . 4 ⊢ (𝜑 → ((𝐴 · 𝐶)(+g‘𝑅)𝐶) ∈ 𝑋) |
18 | eqid 2735 | . . . . 5 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
19 | 3, 18, 5, 15, 12 | grpcld 18978 | . . . 4 ⊢ (𝜑 → ((𝐵 × 𝐷)(+g‘𝑆)𝐷) ∈ 𝑌) |
20 | eqid 2735 | . . . 4 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
21 | 1, 2, 3, 4, 5, 10, 15, 7, 12, 17, 19, 16, 18, 20 | xpsadd 17621 | . . 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 19063 | . . . . 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 19063 | . . . . 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 4886 | . . 3 ⊢ (𝜑 → 〈((𝐴 · 𝐶)(+g‘𝑅)𝐶), ((𝐵 × 𝐷)(+g‘𝑆)𝐷)〉 = 〈𝐴, 𝐵〉) |
27 | 21, 26 | eqtrd 2775 | . 2 ⊢ (𝜑 → (〈(𝐴 · 𝐶), (𝐵 × 𝐷)〉(+g‘𝑇)〈𝐶, 𝐷〉) = 〈𝐴, 𝐵〉) |
28 | 1 | xpsgrp 19090 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) → 𝑇 ∈ Grp) |
29 | 4, 5, 28 | syl2anc 584 | . . 3 ⊢ (𝜑 → 𝑇 ∈ Grp) |
30 | 6, 11 | opelxpd 5728 | . . . 4 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑌)) |
31 | 1, 2, 3, 4, 5 | xpsbas 17619 | . . . 4 ⊢ (𝜑 → (𝑋 × 𝑌) = (Base‘𝑇)) |
32 | 30, 31 | eleqtrd 2841 | . . 3 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ (Base‘𝑇)) |
33 | 7, 12 | opelxpd 5728 | . . . 4 ⊢ (𝜑 → 〈𝐶, 𝐷〉 ∈ (𝑋 × 𝑌)) |
34 | 33, 31 | eleqtrd 2841 | . . 3 ⊢ (𝜑 → 〈𝐶, 𝐷〉 ∈ (Base‘𝑇)) |
35 | 10, 15 | opelxpd 5728 | . . . 4 ⊢ (𝜑 → 〈(𝐴 · 𝐶), (𝐵 × 𝐷)〉 ∈ (𝑋 × 𝑌)) |
36 | 35, 31 | eleqtrd 2841 | . . 3 ⊢ (𝜑 → 〈(𝐴 · 𝐶), (𝐵 × 𝐷)〉 ∈ (Base‘𝑇)) |
37 | eqid 2735 | . . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
38 | xpsgrpsub.o | . . . 4 ⊢ − = (-g‘𝑇) | |
39 | 37, 20, 38 | grpsubadd 19059 | . . 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 206 = wceq 1537 ∈ wcel 2106 〈cop 4637 × cxp 5687 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 ×s cxps 17553 Grpcgrp 18964 -gcsg 18966 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17488 df-prds 17494 df-imas 17555 df-xps 17557 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-sbg 18969 |
This theorem is referenced by: pzriprng1ALT 21525 |
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