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Mirrors > Home > MPE Home > Th. List > subgsub | Structured version Visualization version GIF version |
Description: The subtraction of elements in a subgroup is the same as subtraction in the group. (Contributed by Mario Carneiro, 15-Jun-2015.) |
Ref | Expression |
---|---|
subgsubcl.p | ⊢ − = (-g‘𝐺) |
subgsub.h | ⊢ 𝐻 = (𝐺 ↾s 𝑆) |
subgsub.n | ⊢ 𝑁 = (-g‘𝐻) |
Ref | Expression |
---|---|
subgsub | ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 − 𝑌) = (𝑋𝑁𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subgsub.h | . . . . 5 ⊢ 𝐻 = (𝐺 ↾s 𝑆) | |
2 | eqid 2821 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | 1, 2 | ressplusg 16612 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (+g‘𝐺) = (+g‘𝐻)) |
4 | 3 | 3ad2ant1 1129 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (+g‘𝐺) = (+g‘𝐻)) |
5 | eqidd 2822 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑋 = 𝑋) | |
6 | eqid 2821 | . . . . 5 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
7 | eqid 2821 | . . . . 5 ⊢ (invg‘𝐻) = (invg‘𝐻) | |
8 | 1, 6, 7 | subginv 18286 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑌 ∈ 𝑆) → ((invg‘𝐺)‘𝑌) = ((invg‘𝐻)‘𝑌)) |
9 | 8 | 3adant2 1127 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → ((invg‘𝐺)‘𝑌) = ((invg‘𝐻)‘𝑌)) |
10 | 4, 5, 9 | oveq123d 7177 | . 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌)) = (𝑋(+g‘𝐻)((invg‘𝐻)‘𝑌))) |
11 | eqid 2821 | . . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
12 | 11 | subgss 18280 | . . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺)) |
13 | 12 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑆 ⊆ (Base‘𝐺)) |
14 | simp2 1133 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑋 ∈ 𝑆) | |
15 | 13, 14 | sseldd 3968 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑋 ∈ (Base‘𝐺)) |
16 | simp3 1134 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑌 ∈ 𝑆) | |
17 | 13, 16 | sseldd 3968 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑌 ∈ (Base‘𝐺)) |
18 | subgsubcl.p | . . . 4 ⊢ − = (-g‘𝐺) | |
19 | 11, 2, 6, 18 | grpsubval 18149 | . . 3 ⊢ ((𝑋 ∈ (Base‘𝐺) ∧ 𝑌 ∈ (Base‘𝐺)) → (𝑋 − 𝑌) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌))) |
20 | 15, 17, 19 | syl2anc 586 | . 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 − 𝑌) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌))) |
21 | 1 | subgbas 18283 | . . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻)) |
22 | 21 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑆 = (Base‘𝐻)) |
23 | 14, 22 | eleqtrd 2915 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑋 ∈ (Base‘𝐻)) |
24 | 16, 22 | eleqtrd 2915 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑌 ∈ (Base‘𝐻)) |
25 | eqid 2821 | . . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
26 | eqid 2821 | . . . 4 ⊢ (+g‘𝐻) = (+g‘𝐻) | |
27 | subgsub.n | . . . 4 ⊢ 𝑁 = (-g‘𝐻) | |
28 | 25, 26, 7, 27 | grpsubval 18149 | . . 3 ⊢ ((𝑋 ∈ (Base‘𝐻) ∧ 𝑌 ∈ (Base‘𝐻)) → (𝑋𝑁𝑌) = (𝑋(+g‘𝐻)((invg‘𝐻)‘𝑌))) |
29 | 23, 24, 28 | syl2anc 586 | . 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋𝑁𝑌) = (𝑋(+g‘𝐻)((invg‘𝐻)‘𝑌))) |
30 | 10, 20, 29 | 3eqtr4d 2866 | 1 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 − 𝑌) = (𝑋𝑁𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 ↾s cress 16484 +gcplusg 16565 invgcminusg 18104 -gcsg 18105 SubGrpcsubg 18273 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 |
This theorem is referenced by: zndvds 20696 resubgval 20753 frlmsubgval 20909 scmatsgrp1 21131 subgngp 23244 clmsub 23684 qqhucn 31233 zringsubgval 44469 |
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