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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 20675 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝑋 − 𝑌) = 0 ↔ 𝑋 = 𝑌)) |
9 | 2, 3, 4, 8 | syl3anc 1369 | . . . 4 ⊢ (𝜑 → ((𝑋 − 𝑌) = 0 ↔ 𝑋 = 𝑌)) |
10 | sneq 4637 | . . . . 5 ⊢ (𝑋 = 𝑌 → {𝑋} = {𝑌}) | |
11 | 10 | fveq2d 6894 | . . . 4 ⊢ (𝑋 = 𝑌 → (𝑁‘{𝑋}) = (𝑁‘{𝑌})) |
12 | 9, 11 | syl6bi 252 | . . 3 ⊢ (𝜑 → ((𝑋 − 𝑌) = 0 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))) |
13 | 12 | necon3d 2959 | . 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 1539 ∈ wcel 2104 ≠ wne 2938 {csn 4627 ‘cfv 6542 (class class class)co 7411 Basecbs 17148 0gc0g 17389 -gcsg 18857 LModclmod 20614 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-0g 17391 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18858 df-minusg 18859 df-sbg 18860 df-lmod 20616 |
This theorem is referenced by: mapdpglem4N 40850 |
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