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Mirrors > Home > MPE Home > Th. List > lspsnsubn0 | Structured version Visualization version GIF version |
Description: Unequal singleton spans imply nonzero vector subtraction. (Contributed by NM, 19-Mar-2015.) |
Ref | Expression |
---|---|
lspsnsubn0.v | ⊢ 𝑉 = (Base‘𝑊) |
lspsnsubn0.o | ⊢ 0 = (0g‘𝑊) |
lspsnsubn0.m | ⊢ − = (-g‘𝑊) |
lspsnsubn0.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lspsnsubn0.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lspsnsubn0.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
lspsnsubn0.e | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
Ref | Expression |
---|---|
lspsnsubn0 | ⊢ (𝜑 → (𝑋 − 𝑌) ≠ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspsnsubn0.e | . 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
2 | lspsnsubn0.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
3 | lspsnsubn0.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
4 | lspsnsubn0.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
5 | lspsnsubn0.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
6 | lspsnsubn0.o | . . . . . 6 ⊢ 0 = (0g‘𝑊) | |
7 | lspsnsubn0.m | . . . . . 6 ⊢ − = (-g‘𝑊) | |
8 | 5, 6, 7 | lmodsubeq0 20676 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝑋 − 𝑌) = 0 ↔ 𝑋 = 𝑌)) |
9 | 2, 3, 4, 8 | syl3anc 1370 | . . . 4 ⊢ (𝜑 → ((𝑋 − 𝑌) = 0 ↔ 𝑋 = 𝑌)) |
10 | sneq 4638 | . . . . 5 ⊢ (𝑋 = 𝑌 → {𝑋} = {𝑌}) | |
11 | 10 | fveq2d 6895 | . . . 4 ⊢ (𝑋 = 𝑌 → (𝑁‘{𝑋}) = (𝑁‘{𝑌})) |
12 | 9, 11 | syl6bi 253 | . . 3 ⊢ (𝜑 → ((𝑋 − 𝑌) = 0 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))) |
13 | 12 | necon3d 2960 | . 2 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) → (𝑋 − 𝑌) ≠ 0 )) |
14 | 1, 13 | mpd 15 | 1 ⊢ (𝜑 → (𝑋 − 𝑌) ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 {csn 4628 ‘cfv 6543 (class class class)co 7412 Basecbs 17149 0gc0g 17390 -gcsg 18858 LModclmod 20615 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7979 df-2nd 7980 df-0g 17392 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-grp 18859 df-minusg 18860 df-sbg 18861 df-lmod 20617 |
This theorem is referenced by: mapdpglem4N 40851 |
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