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Mirrors > Home > ILE Home > Th. List > grpsubval | GIF version |
Description: Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 13-Dec-2014.) |
Ref | Expression |
---|---|
grpsubval.b | ⊢ 𝐵 = (Base‘𝐺) |
grpsubval.p | ⊢ + = (+g‘𝐺) |
grpsubval.i | ⊢ 𝐼 = (invg‘𝐺) |
grpsubval.m | ⊢ − = (-g‘𝐺) |
Ref | Expression |
---|---|
grpsubval | ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋 + (𝐼‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpsubval.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
2 | 1 | a1i 9 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐵 = (Base‘𝐺)) |
3 | simpl 109 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
4 | 2, 3 | basmexd 12491 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐺 ∈ V) |
5 | grpsubval.p | . . . 4 ⊢ + = (+g‘𝐺) | |
6 | grpsubval.i | . . . 4 ⊢ 𝐼 = (invg‘𝐺) | |
7 | grpsubval.m | . . . 4 ⊢ − = (-g‘𝐺) | |
8 | 1, 5, 6, 7 | grpsubfvalg 12795 | . . 3 ⊢ (𝐺 ∈ V → − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦)))) |
9 | 4, 8 | syl 14 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦)))) |
10 | oveq1 5875 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 + (𝐼‘𝑦)) = (𝑋 + (𝐼‘𝑦))) | |
11 | fveq2 5510 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝐼‘𝑦) = (𝐼‘𝑌)) | |
12 | 11 | oveq2d 5884 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑋 + (𝐼‘𝑦)) = (𝑋 + (𝐼‘𝑌))) |
13 | 10, 12 | sylan9eq 2230 | . . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥 + (𝐼‘𝑦)) = (𝑋 + (𝐼‘𝑌))) |
14 | 13 | adantl 277 | . 2 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑥 + (𝐼‘𝑦)) = (𝑋 + (𝐼‘𝑌))) |
15 | simpr 110 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | |
16 | plusgslid 12538 | . . . . . 6 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ) | |
17 | 16 | slotex 12459 | . . . . 5 ⊢ (𝐺 ∈ V → (+g‘𝐺) ∈ V) |
18 | 4, 17 | syl 14 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (+g‘𝐺) ∈ V) |
19 | 5, 18 | eqeltrid 2264 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → + ∈ V) |
20 | eqid 2177 | . . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
21 | 1, 5, 20, 6 | grpinvfvalg 12792 | . . . . . 6 ⊢ (𝐺 ∈ V → 𝐼 = (𝑧 ∈ 𝐵 ↦ (℩𝑤 ∈ 𝐵 (𝑤 + 𝑧) = (0g‘𝐺)))) |
22 | 4, 21 | syl 14 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐼 = (𝑧 ∈ 𝐵 ↦ (℩𝑤 ∈ 𝐵 (𝑤 + 𝑧) = (0g‘𝐺)))) |
23 | basfn 12489 | . . . . . . . 8 ⊢ Base Fn V | |
24 | funfvex 5527 | . . . . . . . . 9 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V) | |
25 | 24 | funfni 5311 | . . . . . . . 8 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V) |
26 | 23, 4, 25 | sylancr 414 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (Base‘𝐺) ∈ V) |
27 | 1, 26 | eqeltrid 2264 | . . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐵 ∈ V) |
28 | 27 | mptexd 5738 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑧 ∈ 𝐵 ↦ (℩𝑤 ∈ 𝐵 (𝑤 + 𝑧) = (0g‘𝐺))) ∈ V) |
29 | 22, 28 | eqeltrd 2254 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐼 ∈ V) |
30 | fvexg 5529 | . . . 4 ⊢ ((𝐼 ∈ V ∧ 𝑌 ∈ 𝐵) → (𝐼‘𝑌) ∈ V) | |
31 | 29, 30 | sylancom 420 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐼‘𝑌) ∈ V) |
32 | ovexg 5902 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ + ∈ V ∧ (𝐼‘𝑌) ∈ V) → (𝑋 + (𝐼‘𝑌)) ∈ V) | |
33 | 3, 19, 31, 32 | syl3anc 1238 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + (𝐼‘𝑌)) ∈ V) |
34 | 9, 14, 3, 15, 33 | ovmpod 5995 | 1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋 + (𝐼‘𝑌))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 Vcvv 2737 ↦ cmpt 4061 Fn wfn 5206 ‘cfv 5211 ℩crio 5823 (class class class)co 5868 ∈ cmpo 5870 Basecbs 12432 +gcplusg 12505 0gc0g 12640 invgcminusg 12755 -gcsg 12756 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1re 7883 ax-addrcl 7886 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1st 6134 df-2nd 6135 df-inn 8896 df-2 8954 df-ndx 12435 df-slot 12436 df-base 12438 df-plusg 12518 df-minusg 12758 df-sbg 12759 |
This theorem is referenced by: grpsubinv 12819 grpsubrcan 12827 grpinvsub 12828 grpinvval2 12829 grpsubid 12830 grpsubid1 12831 grpsubeq0 12832 grpsubadd0sub 12833 grpsubadd 12834 grpsubsub 12835 grpaddsubass 12836 grpnpcan 12838 mulgsubdir 12898 ablsub2inv 12928 ablsub4 12930 ablsubsub4 12936 ringsubdi 13046 rngsubdir 13047 |
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