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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatcmp2 | Structured version Visualization version GIF version |
Description: If an atom is included in at-most an atom, they are equal. More general version of lsatcmp 38607. TODO: can lspsncmp 21021 shorten this? (Contributed by NM, 3-Feb-2015.) |
Ref | Expression |
---|---|
lsatcmp2.o | ⊢ 0 = (0g‘𝑊) |
lsatcmp2.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lsatcmp2.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lsatcmp2.t | ⊢ (𝜑 → 𝑇 ∈ 𝐴) |
lsatcmp2.u | ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∨ 𝑈 = { 0 })) |
Ref | Expression |
---|---|
lsatcmp2 | ⊢ (𝜑 → (𝑇 ⊆ 𝑈 ↔ 𝑇 = 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑇 ⊆ 𝑈) → 𝑇 ⊆ 𝑈) | |
2 | lsatcmp2.a | . . . . 5 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
3 | lsatcmp2.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
4 | 3 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑇 ⊆ 𝑈) → 𝑊 ∈ LVec) |
5 | lsatcmp2.t | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝐴) | |
6 | 5 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑇 ⊆ 𝑈) → 𝑇 ∈ 𝐴) |
7 | lsatcmp2.o | . . . . . . 7 ⊢ 0 = (0g‘𝑊) | |
8 | lveclmod 21008 | . . . . . . . . 9 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
9 | 3, 8 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ LMod) |
10 | 9 | adantr 479 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 ⊆ 𝑈) → 𝑊 ∈ LMod) |
11 | 7, 2, 10, 6, 1 | lsatssn0 38606 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑇 ⊆ 𝑈) → 𝑈 ≠ { 0 }) |
12 | lsatcmp2.u | . . . . . . . . 9 ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∨ 𝑈 = { 0 })) | |
13 | 12 | ord 862 | . . . . . . . 8 ⊢ (𝜑 → (¬ 𝑈 ∈ 𝐴 → 𝑈 = { 0 })) |
14 | 13 | necon1ad 2946 | . . . . . . 7 ⊢ (𝜑 → (𝑈 ≠ { 0 } → 𝑈 ∈ 𝐴)) |
15 | 14 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑇 ⊆ 𝑈) → (𝑈 ≠ { 0 } → 𝑈 ∈ 𝐴)) |
16 | 11, 15 | mpd 15 | . . . . 5 ⊢ ((𝜑 ∧ 𝑇 ⊆ 𝑈) → 𝑈 ∈ 𝐴) |
17 | 2, 4, 6, 16 | lsatcmp 38607 | . . . 4 ⊢ ((𝜑 ∧ 𝑇 ⊆ 𝑈) → (𝑇 ⊆ 𝑈 ↔ 𝑇 = 𝑈)) |
18 | 1, 17 | mpbid 231 | . . 3 ⊢ ((𝜑 ∧ 𝑇 ⊆ 𝑈) → 𝑇 = 𝑈) |
19 | 18 | ex 411 | . 2 ⊢ (𝜑 → (𝑇 ⊆ 𝑈 → 𝑇 = 𝑈)) |
20 | eqimss 4035 | . 2 ⊢ (𝑇 = 𝑈 → 𝑇 ⊆ 𝑈) | |
21 | 19, 20 | impbid1 224 | 1 ⊢ (𝜑 → (𝑇 ⊆ 𝑈 ↔ 𝑇 = 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ⊆ wss 3944 {csn 4630 ‘cfv 6549 0gc0g 17429 LModclmod 20760 LVecclvec 21004 LSAtomsclsa 38578 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-nn 12251 df-2 12313 df-3 12314 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17189 df-ress 17218 df-plusg 17254 df-mulr 17255 df-0g 17431 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-grp 18906 df-minusg 18907 df-sbg 18908 df-cmn 19754 df-abl 19755 df-mgp 20092 df-rng 20110 df-ur 20139 df-ring 20192 df-oppr 20290 df-dvdsr 20313 df-unit 20314 df-invr 20344 df-drng 20643 df-lmod 20762 df-lss 20833 df-lsp 20873 df-lvec 21005 df-lsatoms 38580 |
This theorem is referenced by: mapdrvallem2 41250 |
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