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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > grpsubcld | Structured version Visualization version GIF version |
Description: Closure of group subtraction. (Contributed by Thierry Arnoux, 3-Aug-2025.) |
Ref | Expression |
---|---|
grpsubcld.b | ⊢ 𝐵 = (Base‘𝐺) |
grpsubcld.m | ⊢ − = (-g‘𝐺) |
grpsubcld.g | ⊢ (𝜑 → 𝐺 ∈ Grp) |
grpsubcld.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
grpsubcld.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
grpsubcld | ⊢ (𝜑 → (𝑋 − 𝑌) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpsubcld.g | . 2 ⊢ (𝜑 → 𝐺 ∈ Grp) | |
2 | grpsubcld.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
3 | grpsubcld.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
4 | grpsubcld.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
5 | grpsubcld.m | . . 3 ⊢ − = (-g‘𝐺) | |
6 | 4, 5 | grpsubcl 19037 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) ∈ 𝐵) |
7 | 1, 2, 3, 6 | syl3anc 1369 | 1 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1535 ∈ wcel 2104 ‘cfv 6559 (class class class)co 7426 Basecbs 17235 Grpcgrp 18950 -gcsg 18952 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7748 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4916 df-iun 5001 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6511 df-fun 6561 df-fn 6562 df-f 6563 df-fv 6567 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 8008 df-2nd 8009 df-0g 17478 df-mgm 18655 df-sgrp 18734 df-mnd 18750 df-grp 18953 df-minusg 18954 df-sbg 18955 |
This theorem is referenced by: assalactf1o 33626 |
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