![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > subgdisjb | Structured version Visualization version GIF version |
Description: Vectors belonging to disjoint commuting subgroups are uniquely determined by their sum. Analogous to opth 5165, this theorem shows a way of representing a pair of vectors. (Contributed by NM, 5-Jul-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
Ref | Expression |
---|---|
subgdisj.p | ⊢ + = (+g‘𝐺) |
subgdisj.o | ⊢ 0 = (0g‘𝐺) |
subgdisj.z | ⊢ 𝑍 = (Cntz‘𝐺) |
subgdisj.t | ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) |
subgdisj.u | ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) |
subgdisj.i | ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) |
subgdisj.s | ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) |
subgdisj.a | ⊢ (𝜑 → 𝐴 ∈ 𝑇) |
subgdisj.c | ⊢ (𝜑 → 𝐶 ∈ 𝑇) |
subgdisj.b | ⊢ (𝜑 → 𝐵 ∈ 𝑈) |
subgdisj.d | ⊢ (𝜑 → 𝐷 ∈ 𝑈) |
Ref | Expression |
---|---|
subgdisjb | ⊢ (𝜑 → ((𝐴 + 𝐵) = (𝐶 + 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subgdisj.p | . . . . 5 ⊢ + = (+g‘𝐺) | |
2 | subgdisj.o | . . . . 5 ⊢ 0 = (0g‘𝐺) | |
3 | subgdisj.z | . . . . 5 ⊢ 𝑍 = (Cntz‘𝐺) | |
4 | subgdisj.t | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) | |
5 | 4 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → 𝑇 ∈ (SubGrp‘𝐺)) |
6 | subgdisj.u | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) | |
7 | 6 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → 𝑈 ∈ (SubGrp‘𝐺)) |
8 | subgdisj.i | . . . . . 6 ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) | |
9 | 8 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → (𝑇 ∩ 𝑈) = { 0 }) |
10 | subgdisj.s | . . . . . 6 ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) | |
11 | 10 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → 𝑇 ⊆ (𝑍‘𝑈)) |
12 | subgdisj.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑇) | |
13 | 12 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → 𝐴 ∈ 𝑇) |
14 | subgdisj.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑇) | |
15 | 14 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → 𝐶 ∈ 𝑇) |
16 | subgdisj.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑈) | |
17 | 16 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → 𝐵 ∈ 𝑈) |
18 | subgdisj.d | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑈) | |
19 | 18 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → 𝐷 ∈ 𝑈) |
20 | simpr 479 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → (𝐴 + 𝐵) = (𝐶 + 𝐷)) | |
21 | 1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 20 | subgdisj1 18455 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → 𝐴 = 𝐶) |
22 | 1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 20 | subgdisj2 18456 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → 𝐵 = 𝐷) |
23 | 21, 22 | jca 509 | . . 3 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
24 | 23 | ex 403 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐵) = (𝐶 + 𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
25 | oveq12 6914 | . 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐴 + 𝐵) = (𝐶 + 𝐷)) | |
26 | 24, 25 | impbid1 217 | 1 ⊢ (𝜑 → ((𝐴 + 𝐵) = (𝐶 + 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∩ cin 3797 ⊆ wss 3798 {csn 4397 ‘cfv 6123 (class class class)co 6905 +gcplusg 16305 0gc0g 16453 SubGrpcsubg 17939 Cntzccntz 18098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-0g 16455 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-grp 17779 df-minusg 17780 df-sbg 17781 df-subg 17942 df-cntz 18100 |
This theorem is referenced by: pj1eu 18460 pj1eq 18464 lvecindp2 19499 |
Copyright terms: Public domain | W3C validator |