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Mirrors > Home > MPE Home > Th. List > lssnle | Structured version Visualization version GIF version |
Description: Equivalent expressions for "not less than". (chnlei 30325 analog.) (Contributed by NM, 10-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.) |
Ref | Expression |
---|---|
lssnle.p | ⊢ ⊕ = (LSSum‘𝐺) |
lssnle.t | ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) |
lssnle.u | ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) |
Ref | Expression |
---|---|
lssnle | ⊢ (𝜑 → (¬ 𝑈 ⊆ 𝑇 ↔ 𝑇 ⊊ (𝑇 ⊕ 𝑈))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lssnle.t | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) | |
2 | lssnle.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) | |
3 | lssnle.p | . . . . . 6 ⊢ ⊕ = (LSSum‘𝐺) | |
4 | 3 | lsmss2b 19446 | . . . . 5 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑈 ⊆ 𝑇 ↔ (𝑇 ⊕ 𝑈) = 𝑇)) |
5 | 1, 2, 4 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝑈 ⊆ 𝑇 ↔ (𝑇 ⊕ 𝑈) = 𝑇)) |
6 | eqcom 2743 | . . . 4 ⊢ ((𝑇 ⊕ 𝑈) = 𝑇 ↔ 𝑇 = (𝑇 ⊕ 𝑈)) | |
7 | 5, 6 | bitrdi 286 | . . 3 ⊢ (𝜑 → (𝑈 ⊆ 𝑇 ↔ 𝑇 = (𝑇 ⊕ 𝑈))) |
8 | 7 | necon3bbid 2980 | . 2 ⊢ (𝜑 → (¬ 𝑈 ⊆ 𝑇 ↔ 𝑇 ≠ (𝑇 ⊕ 𝑈))) |
9 | 3 | lsmub1 19435 | . . . 4 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝑇 ⊆ (𝑇 ⊕ 𝑈)) |
10 | 1, 2, 9 | syl2anc 584 | . . 3 ⊢ (𝜑 → 𝑇 ⊆ (𝑇 ⊕ 𝑈)) |
11 | df-pss 3928 | . . . 4 ⊢ (𝑇 ⊊ (𝑇 ⊕ 𝑈) ↔ (𝑇 ⊆ (𝑇 ⊕ 𝑈) ∧ 𝑇 ≠ (𝑇 ⊕ 𝑈))) | |
12 | 11 | baib 536 | . . 3 ⊢ (𝑇 ⊆ (𝑇 ⊕ 𝑈) → (𝑇 ⊊ (𝑇 ⊕ 𝑈) ↔ 𝑇 ≠ (𝑇 ⊕ 𝑈))) |
13 | 10, 12 | syl 17 | . 2 ⊢ (𝜑 → (𝑇 ⊊ (𝑇 ⊕ 𝑈) ↔ 𝑇 ≠ (𝑇 ⊕ 𝑈))) |
14 | 8, 13 | bitr4d 281 | 1 ⊢ (𝜑 → (¬ 𝑈 ⊆ 𝑇 ↔ 𝑇 ⊊ (𝑇 ⊕ 𝑈))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 = wceq 1541 ∈ wcel 2106 ≠ wne 2942 ⊆ wss 3909 ⊊ wpss 3910 ‘cfv 6494 (class class class)co 7354 SubGrpcsubg 18918 LSSumclsm 19412 |
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-rep 5241 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 |
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-en 8881 df-dom 8882 df-sdom 8883 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11384 df-neg 11385 df-nn 12151 df-2 12213 df-sets 17033 df-slot 17051 df-ndx 17063 df-base 17081 df-ress 17110 df-plusg 17143 df-0g 17320 df-mgm 18494 df-sgrp 18543 df-mnd 18554 df-submnd 18599 df-grp 18748 df-minusg 18749 df-subg 18921 df-lsm 19414 |
This theorem is referenced by: lrelat 37465 lsmcv2 37480 lcv1 37492 lcv2 37493 |
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