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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 5370, 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 483 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → 𝑇 ∈ (SubGrp‘𝐺)) |
6 | subgdisj.u | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) | |
7 | 6 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → 𝑈 ∈ (SubGrp‘𝐺)) |
8 | subgdisj.i | . . . . . 6 ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) | |
9 | 8 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → (𝑇 ∩ 𝑈) = { 0 }) |
10 | subgdisj.s | . . . . . 6 ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) | |
11 | 10 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → 𝑇 ⊆ (𝑍‘𝑈)) |
12 | subgdisj.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑇) | |
13 | 12 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → 𝐴 ∈ 𝑇) |
14 | subgdisj.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑇) | |
15 | 14 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → 𝐶 ∈ 𝑇) |
16 | subgdisj.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑈) | |
17 | 16 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → 𝐵 ∈ 𝑈) |
18 | subgdisj.d | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑈) | |
19 | 18 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → 𝐷 ∈ 𝑈) |
20 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → (𝐴 + 𝐵) = (𝐶 + 𝐷)) | |
21 | 1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 20 | subgdisj1 18819 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → 𝐴 = 𝐶) |
22 | 1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 20 | subgdisj2 18820 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → 𝐵 = 𝐷) |
23 | 21, 22 | jca 514 | . . 3 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
24 | 23 | ex 415 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐵) = (𝐶 + 𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
25 | oveq12 7167 | . 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐴 + 𝐵) = (𝐶 + 𝐷)) | |
26 | 24, 25 | impbid1 227 | 1 ⊢ (𝜑 → ((𝐴 + 𝐵) = (𝐶 + 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∩ cin 3937 ⊆ wss 3938 {csn 4569 ‘cfv 6357 (class class class)co 7158 +gcplusg 16567 0gc0g 16715 SubGrpcsubg 18275 Cntzccntz 18447 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-sbg 18110 df-subg 18278 df-cntz 18449 |
This theorem is referenced by: pj1eu 18824 pj1eq 18828 lvecindp2 19913 |
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