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Mirrors > Home > MPE Home > Th. List > ngpsubcan | Structured version Visualization version GIF version |
Description: Cancel right subtraction inside a distance calculation. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
ngpsubcan.x | ⊢ 𝑋 = (Base‘𝐺) |
ngpsubcan.m | ⊢ − = (-g‘𝐺) |
ngpsubcan.d | ⊢ 𝐷 = (dist‘𝐺) |
Ref | Expression |
---|---|
ngpsubcan | ⊢ ((𝐺 ∈ NrmGrp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴 − 𝐶)𝐷(𝐵 − 𝐶)) = (𝐴𝐷𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr1 1194 | . . . 4 ⊢ ((𝐺 ∈ NrmGrp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐴 ∈ 𝑋) | |
2 | simpr3 1196 | . . . 4 ⊢ ((𝐺 ∈ NrmGrp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐶 ∈ 𝑋) | |
3 | ngpsubcan.x | . . . . 5 ⊢ 𝑋 = (Base‘𝐺) | |
4 | eqid 2736 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
5 | eqid 2736 | . . . . 5 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
6 | ngpsubcan.m | . . . . 5 ⊢ − = (-g‘𝐺) | |
7 | 3, 4, 5, 6 | grpsubval 18793 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴 − 𝐶) = (𝐴(+g‘𝐺)((invg‘𝐺)‘𝐶))) |
8 | 1, 2, 7 | syl2anc 584 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴 − 𝐶) = (𝐴(+g‘𝐺)((invg‘𝐺)‘𝐶))) |
9 | simpr2 1195 | . . . 4 ⊢ ((𝐺 ∈ NrmGrp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐵 ∈ 𝑋) | |
10 | 3, 4, 5, 6 | grpsubval 18793 | . . . 4 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵 − 𝐶) = (𝐵(+g‘𝐺)((invg‘𝐺)‘𝐶))) |
11 | 9, 2, 10 | syl2anc 584 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵 − 𝐶) = (𝐵(+g‘𝐺)((invg‘𝐺)‘𝐶))) |
12 | 8, 11 | oveq12d 7372 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴 − 𝐶)𝐷(𝐵 − 𝐶)) = ((𝐴(+g‘𝐺)((invg‘𝐺)‘𝐶))𝐷(𝐵(+g‘𝐺)((invg‘𝐺)‘𝐶)))) |
13 | simpl 483 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐺 ∈ NrmGrp) | |
14 | ngpgrp 23951 | . . . 4 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) | |
15 | 3, 5 | grpinvcl 18795 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐶 ∈ 𝑋) → ((invg‘𝐺)‘𝐶) ∈ 𝑋) |
16 | 14, 2, 15 | syl2an2r 683 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((invg‘𝐺)‘𝐶) ∈ 𝑋) |
17 | ngpsubcan.d | . . . 4 ⊢ 𝐷 = (dist‘𝐺) | |
18 | 3, 4, 17 | ngprcan 23962 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((invg‘𝐺)‘𝐶) ∈ 𝑋)) → ((𝐴(+g‘𝐺)((invg‘𝐺)‘𝐶))𝐷(𝐵(+g‘𝐺)((invg‘𝐺)‘𝐶))) = (𝐴𝐷𝐵)) |
19 | 13, 1, 9, 16, 18 | syl13anc 1372 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴(+g‘𝐺)((invg‘𝐺)‘𝐶))𝐷(𝐵(+g‘𝐺)((invg‘𝐺)‘𝐶))) = (𝐴𝐷𝐵)) |
20 | 12, 19 | eqtrd 2776 | 1 ⊢ ((𝐺 ∈ NrmGrp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴 − 𝐶)𝐷(𝐵 − 𝐶)) = (𝐴𝐷𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ‘cfv 6494 (class class class)co 7354 Basecbs 17080 +gcplusg 17130 distcds 17139 Grpcgrp 18745 invgcminusg 18746 -gcsg 18747 NrmGrpcngp 23929 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-cnex 11104 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 ax-pre-mulgt0 11125 ax-pre-sup 11126 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-om 7800 df-1st 7918 df-2nd 7919 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-er 8645 df-map 8764 df-en 8881 df-dom 8882 df-sdom 8883 df-sup 9375 df-inf 9376 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11384 df-neg 11385 df-div 11810 df-nn 12151 df-2 12213 df-n0 12411 df-z 12497 df-uz 12761 df-q 12871 df-rp 12913 df-xneg 13030 df-xadd 13031 df-xmul 13032 df-0g 17320 df-topgen 17322 df-mgm 18494 df-sgrp 18543 df-mnd 18554 df-grp 18748 df-minusg 18749 df-sbg 18750 df-psmet 20784 df-xmet 20785 df-met 20786 df-bl 20787 df-mopn 20788 df-top 22239 df-topon 22256 df-topsp 22278 df-bases 22292 df-xms 23669 df-ms 23670 df-nm 23934 df-ngp 23935 |
This theorem is referenced by: ngptgp 23988 |
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