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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 4871, 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 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → 𝑇 ∈ (SubGrp‘𝐺)) |
6 | subgdisj.u | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) | |
7 | 6 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → 𝑈 ∈ (SubGrp‘𝐺)) |
8 | subgdisj.i | . . . . . 6 ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) | |
9 | 8 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → (𝑇 ∩ 𝑈) = { 0 }) |
10 | subgdisj.s | . . . . . 6 ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) | |
11 | 10 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → 𝑇 ⊆ (𝑍‘𝑈)) |
12 | subgdisj.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑇) | |
13 | 12 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → 𝐴 ∈ 𝑇) |
14 | subgdisj.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑇) | |
15 | 14 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → 𝐶 ∈ 𝑇) |
16 | subgdisj.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑈) | |
17 | 16 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → 𝐵 ∈ 𝑈) |
18 | subgdisj.d | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑈) | |
19 | 18 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → 𝐷 ∈ 𝑈) |
20 | simpr 476 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → (𝐴 + 𝐵) = (𝐶 + 𝐷)) | |
21 | 1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 20 | subgdisj1 17927 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → 𝐴 = 𝐶) |
22 | 1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 20 | subgdisj2 17928 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → 𝐵 = 𝐷) |
23 | 21, 22 | jca 553 | . . 3 ⊢ ((𝜑 ∧ (𝐴 + 𝐵) = (𝐶 + 𝐷)) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
24 | 23 | ex 449 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐵) = (𝐶 + 𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
25 | oveq12 6558 | . 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐴 + 𝐵) = (𝐶 + 𝐷)) | |
26 | 24, 25 | impbid1 214 | 1 ⊢ (𝜑 → ((𝐴 + 𝐵) = (𝐶 + 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 ⊆ wss 3540 {csn 4125 ‘cfv 5804 (class class class)co 6549 +gcplusg 15768 0gc0g 15923 SubGrpcsubg 17411 Cntzccntz 17571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-cntz 17573 |
This theorem is referenced by: pj1eu 17932 pj1eq 17936 lvecindp2 18960 |
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