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Theorem grpsubadd 17724
Description: Relationship between group subtraction and addition. (Contributed by NM, 31-Mar-2014.)
Hypotheses
Ref Expression
grpsubadd.b 𝐵 = (Base‘𝐺)
grpsubadd.p + = (+g𝐺)
grpsubadd.m = (-g𝐺)
Assertion
Ref Expression
grpsubadd ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) = 𝑍 ↔ (𝑍 + 𝑌) = 𝑋))

Proof of Theorem grpsubadd
StepHypRef Expression
1 grpsubadd.b . . . . . . 7 𝐵 = (Base‘𝐺)
2 grpsubadd.p . . . . . . 7 + = (+g𝐺)
3 eqid 2760 . . . . . . 7 (invg𝐺) = (invg𝐺)
4 grpsubadd.m . . . . . . 7 = (-g𝐺)
51, 2, 3, 4grpsubval 17686 . . . . . 6 ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋 + ((invg𝐺)‘𝑌)))
653adant3 1127 . . . . 5 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝑋 𝑌) = (𝑋 + ((invg𝐺)‘𝑌)))
76adantl 473 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌) = (𝑋 + ((invg𝐺)‘𝑌)))
87eqeq1d 2762 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) = 𝑍 ↔ (𝑋 + ((invg𝐺)‘𝑌)) = 𝑍))
9 simpl 474 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐺 ∈ Grp)
10 simpr1 1234 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
111, 3grpinvcl 17688 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ((invg𝐺)‘𝑌) ∈ 𝐵)
12113ad2antr2 1205 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((invg𝐺)‘𝑌) ∈ 𝐵)
131, 2grpcl 17651 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵 ∧ ((invg𝐺)‘𝑌) ∈ 𝐵) → (𝑋 + ((invg𝐺)‘𝑌)) ∈ 𝐵)
149, 10, 12, 13syl3anc 1477 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + ((invg𝐺)‘𝑌)) ∈ 𝐵)
15 simpr3 1238 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
16 simpr2 1236 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
171, 2grprcan 17676 . . . 4 ((𝐺 ∈ Grp ∧ ((𝑋 + ((invg𝐺)‘𝑌)) ∈ 𝐵𝑍𝐵𝑌𝐵)) → (((𝑋 + ((invg𝐺)‘𝑌)) + 𝑌) = (𝑍 + 𝑌) ↔ (𝑋 + ((invg𝐺)‘𝑌)) = 𝑍))
189, 14, 15, 16, 17syl13anc 1479 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 + ((invg𝐺)‘𝑌)) + 𝑌) = (𝑍 + 𝑌) ↔ (𝑋 + ((invg𝐺)‘𝑌)) = 𝑍))
191, 2grpass 17652 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵 ∧ ((invg𝐺)‘𝑌) ∈ 𝐵𝑌𝐵)) → ((𝑋 + ((invg𝐺)‘𝑌)) + 𝑌) = (𝑋 + (((invg𝐺)‘𝑌) + 𝑌)))
209, 10, 12, 16, 19syl13anc 1479 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + ((invg𝐺)‘𝑌)) + 𝑌) = (𝑋 + (((invg𝐺)‘𝑌) + 𝑌)))
21 eqid 2760 . . . . . . . 8 (0g𝐺) = (0g𝐺)
221, 2, 21, 3grplinv 17689 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (((invg𝐺)‘𝑌) + 𝑌) = (0g𝐺))
23223ad2antr2 1205 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((invg𝐺)‘𝑌) + 𝑌) = (0g𝐺))
2423oveq2d 6830 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + (((invg𝐺)‘𝑌) + 𝑌)) = (𝑋 + (0g𝐺)))
251, 2, 21grprid 17674 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (0g𝐺)) = 𝑋)
26253ad2antr1 1204 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + (0g𝐺)) = 𝑋)
2720, 24, 263eqtrd 2798 . . . 4 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + ((invg𝐺)‘𝑌)) + 𝑌) = 𝑋)
2827eqeq1d 2762 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 + ((invg𝐺)‘𝑌)) + 𝑌) = (𝑍 + 𝑌) ↔ 𝑋 = (𝑍 + 𝑌)))
298, 18, 283bitr2d 296 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) = 𝑍𝑋 = (𝑍 + 𝑌)))
30 eqcom 2767 . 2 (𝑋 = (𝑍 + 𝑌) ↔ (𝑍 + 𝑌) = 𝑋)
3129, 30syl6bb 276 1 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) = 𝑍 ↔ (𝑍 + 𝑌) = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  cfv 6049  (class class class)co 6814  Basecbs 16079  +gcplusg 16163  0gc0g 16322  Grpcgrp 17643  invgcminusg 17644  -gcsg 17645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-0g 16324  df-mgm 17463  df-sgrp 17505  df-mnd 17516  df-grp 17646  df-minusg 17647  df-sbg 17648
This theorem is referenced by:  grpsubsub4  17729  conjghm  17912  conjnmzb  17916  sylow3lem2  18263  ablsubadd  18437  ablsubsub23  18450  pgpfac1lem2  18694  pgpfac1lem4  18697  lspexch  19351  coe1subfv  19858  ipsubdir  20209  ipsubdi  20210  zlmodzxzsub  42666
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