![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > gsum2dlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for gsum2d 19934. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 8-Jun-2019.) |
Ref | Expression |
---|---|
gsum2d.b | ⊢ 𝐵 = (Base‘𝐺) |
gsum2d.z | ⊢ 0 = (0g‘𝐺) |
gsum2d.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsum2d.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
gsum2d.r | ⊢ (𝜑 → Rel 𝐴) |
gsum2d.d | ⊢ (𝜑 → 𝐷 ∈ 𝑊) |
gsum2d.s | ⊢ (𝜑 → dom 𝐴 ⊆ 𝐷) |
gsum2d.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
gsum2d.w | ⊢ (𝜑 → 𝐹 finSupp 0 ) |
Ref | Expression |
---|---|
gsum2dlem1 | ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsum2d.b | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
2 | gsum2d.z | . 2 ⊢ 0 = (0g‘𝐺) | |
3 | gsum2d.g | . 2 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
4 | gsum2d.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
5 | imaexg 7927 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 “ {𝑗}) ∈ V) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → (𝐴 “ {𝑗}) ∈ V) |
7 | vex 3477 | . . . . 5 ⊢ 𝑗 ∈ V | |
8 | vex 3477 | . . . . 5 ⊢ 𝑘 ∈ V | |
9 | 7, 8 | elimasn 6098 | . . . 4 ⊢ (𝑘 ∈ (𝐴 “ {𝑗}) ↔ ⟨𝑗, 𝑘⟩ ∈ 𝐴) |
10 | df-ov 7429 | . . . . 5 ⊢ (𝑗𝐹𝑘) = (𝐹‘⟨𝑗, 𝑘⟩) | |
11 | gsum2d.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
12 | 11 | ffvelcdmda 7099 | . . . . 5 ⊢ ((𝜑 ∧ ⟨𝑗, 𝑘⟩ ∈ 𝐴) → (𝐹‘⟨𝑗, 𝑘⟩) ∈ 𝐵) |
13 | 10, 12 | eqeltrid 2833 | . . . 4 ⊢ ((𝜑 ∧ ⟨𝑗, 𝑘⟩ ∈ 𝐴) → (𝑗𝐹𝑘) ∈ 𝐵) |
14 | 9, 13 | sylan2b 592 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 “ {𝑗})) → (𝑗𝐹𝑘) ∈ 𝐵) |
15 | 14 | fmpttd 7130 | . 2 ⊢ (𝜑 → (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)):(𝐴 “ {𝑗})⟶𝐵) |
16 | gsum2d.w | . . . . 5 ⊢ (𝜑 → 𝐹 finSupp 0 ) | |
17 | 16 | fsuppimpd 9401 | . . . 4 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin) |
18 | rnfi 9367 | . . . 4 ⊢ ((𝐹 supp 0 ) ∈ Fin → ran (𝐹 supp 0 ) ∈ Fin) | |
19 | 17, 18 | syl 17 | . . 3 ⊢ (𝜑 → ran (𝐹 supp 0 ) ∈ Fin) |
20 | 9 | biimpi 215 | . . . . . . 7 ⊢ (𝑘 ∈ (𝐴 “ {𝑗}) → ⟨𝑗, 𝑘⟩ ∈ 𝐴) |
21 | 7, 8 | opelrn 5949 | . . . . . . . 8 ⊢ (⟨𝑗, 𝑘⟩ ∈ (𝐹 supp 0 ) → 𝑘 ∈ ran (𝐹 supp 0 )) |
22 | 21 | con3i 154 | . . . . . . 7 ⊢ (¬ 𝑘 ∈ ran (𝐹 supp 0 ) → ¬ ⟨𝑗, 𝑘⟩ ∈ (𝐹 supp 0 )) |
23 | 20, 22 | anim12i 611 | . . . . . 6 ⊢ ((𝑘 ∈ (𝐴 “ {𝑗}) ∧ ¬ 𝑘 ∈ ran (𝐹 supp 0 )) → (⟨𝑗, 𝑘⟩ ∈ 𝐴 ∧ ¬ ⟨𝑗, 𝑘⟩ ∈ (𝐹 supp 0 ))) |
24 | eldif 3959 | . . . . . 6 ⊢ (𝑘 ∈ ((𝐴 “ {𝑗}) ∖ ran (𝐹 supp 0 )) ↔ (𝑘 ∈ (𝐴 “ {𝑗}) ∧ ¬ 𝑘 ∈ ran (𝐹 supp 0 ))) | |
25 | eldif 3959 | . . . . . 6 ⊢ (⟨𝑗, 𝑘⟩ ∈ (𝐴 ∖ (𝐹 supp 0 )) ↔ (⟨𝑗, 𝑘⟩ ∈ 𝐴 ∧ ¬ ⟨𝑗, 𝑘⟩ ∈ (𝐹 supp 0 ))) | |
26 | 23, 24, 25 | 3imtr4i 291 | . . . . 5 ⊢ (𝑘 ∈ ((𝐴 “ {𝑗}) ∖ ran (𝐹 supp 0 )) → ⟨𝑗, 𝑘⟩ ∈ (𝐴 ∖ (𝐹 supp 0 ))) |
27 | ssidd 4005 | . . . . . . 7 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 )) | |
28 | 2 | fvexi 6916 | . . . . . . . 8 ⊢ 0 ∈ V |
29 | 28 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ V) |
30 | 11, 27, 4, 29 | suppssr 8207 | . . . . . 6 ⊢ ((𝜑 ∧ ⟨𝑗, 𝑘⟩ ∈ (𝐴 ∖ (𝐹 supp 0 ))) → (𝐹‘⟨𝑗, 𝑘⟩) = 0 ) |
31 | 10, 30 | eqtrid 2780 | . . . . 5 ⊢ ((𝜑 ∧ ⟨𝑗, 𝑘⟩ ∈ (𝐴 ∖ (𝐹 supp 0 ))) → (𝑗𝐹𝑘) = 0 ) |
32 | 26, 31 | sylan2 591 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐴 “ {𝑗}) ∖ ran (𝐹 supp 0 ))) → (𝑗𝐹𝑘) = 0 ) |
33 | 32, 6 | suppss2 8212 | . . 3 ⊢ (𝜑 → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) supp 0 ) ⊆ ran (𝐹 supp 0 )) |
34 | 19, 33 | ssfid 9298 | . 2 ⊢ (𝜑 → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) supp 0 ) ∈ Fin) |
35 | 1, 2, 3, 6, 15, 34 | gsumcl2 19876 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3473 ∖ cdif 3946 ⊆ wss 3949 {csn 4632 ⟨cop 4638 class class class wbr 5152 ↦ cmpt 5235 dom cdm 5682 ran crn 5683 “ cima 5685 Rel wrel 5687 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 supp csupp 8171 Fincfn 8970 finSupp cfsupp 9393 Basecbs 17187 0gc0g 17428 Σg cgsu 17429 CMndccmn 19742 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-fzo 13668 df-seq 14007 df-hash 14330 df-0g 17430 df-gsum 17431 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-cntz 19275 df-cmn 19744 |
This theorem is referenced by: gsum2dlem2 19933 gsum2d 19934 |
Copyright terms: Public domain | W3C validator |