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Mirrors > Home > MPE Home > Th. List > gsum2dlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for gsum2d 20004. (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 7935 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 “ {𝑗}) ∈ V) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → (𝐴 “ {𝑗}) ∈ V) |
7 | vex 3481 | . . . . 5 ⊢ 𝑗 ∈ V | |
8 | vex 3481 | . . . . 5 ⊢ 𝑘 ∈ V | |
9 | 7, 8 | elimasn 6109 | . . . 4 ⊢ (𝑘 ∈ (𝐴 “ {𝑗}) ↔ 〈𝑗, 𝑘〉 ∈ 𝐴) |
10 | df-ov 7433 | . . . . 5 ⊢ (𝑗𝐹𝑘) = (𝐹‘〈𝑗, 𝑘〉) | |
11 | gsum2d.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
12 | 11 | ffvelcdmda 7103 | . . . . 5 ⊢ ((𝜑 ∧ 〈𝑗, 𝑘〉 ∈ 𝐴) → (𝐹‘〈𝑗, 𝑘〉) ∈ 𝐵) |
13 | 10, 12 | eqeltrid 2842 | . . . 4 ⊢ ((𝜑 ∧ 〈𝑗, 𝑘〉 ∈ 𝐴) → (𝑗𝐹𝑘) ∈ 𝐵) |
14 | 9, 13 | sylan2b 594 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 “ {𝑗})) → (𝑗𝐹𝑘) ∈ 𝐵) |
15 | 14 | fmpttd 7134 | . 2 ⊢ (𝜑 → (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)):(𝐴 “ {𝑗})⟶𝐵) |
16 | gsum2d.w | . . . . 5 ⊢ (𝜑 → 𝐹 finSupp 0 ) | |
17 | 16 | fsuppimpd 9406 | . . . 4 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin) |
18 | rnfi 9377 | . . . 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 216 | . . . . . . 7 ⊢ (𝑘 ∈ (𝐴 “ {𝑗}) → 〈𝑗, 𝑘〉 ∈ 𝐴) |
21 | 7, 8 | opelrn 5956 | . . . . . . . 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 613 | . . . . . 6 ⊢ ((𝑘 ∈ (𝐴 “ {𝑗}) ∧ ¬ 𝑘 ∈ ran (𝐹 supp 0 )) → (〈𝑗, 𝑘〉 ∈ 𝐴 ∧ ¬ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 ))) |
24 | eldif 3972 | . . . . . 6 ⊢ (𝑘 ∈ ((𝐴 “ {𝑗}) ∖ ran (𝐹 supp 0 )) ↔ (𝑘 ∈ (𝐴 “ {𝑗}) ∧ ¬ 𝑘 ∈ ran (𝐹 supp 0 ))) | |
25 | eldif 3972 | . . . . . 6 ⊢ (〈𝑗, 𝑘〉 ∈ (𝐴 ∖ (𝐹 supp 0 )) ↔ (〈𝑗, 𝑘〉 ∈ 𝐴 ∧ ¬ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 ))) | |
26 | 23, 24, 25 | 3imtr4i 292 | . . . . 5 ⊢ (𝑘 ∈ ((𝐴 “ {𝑗}) ∖ ran (𝐹 supp 0 )) → 〈𝑗, 𝑘〉 ∈ (𝐴 ∖ (𝐹 supp 0 ))) |
27 | ssidd 4018 | . . . . . . 7 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 )) | |
28 | 2 | fvexi 6920 | . . . . . . . 8 ⊢ 0 ∈ V |
29 | 28 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ V) |
30 | 11, 27, 4, 29 | suppssr 8218 | . . . . . 6 ⊢ ((𝜑 ∧ 〈𝑗, 𝑘〉 ∈ (𝐴 ∖ (𝐹 supp 0 ))) → (𝐹‘〈𝑗, 𝑘〉) = 0 ) |
31 | 10, 30 | eqtrid 2786 | . . . . 5 ⊢ ((𝜑 ∧ 〈𝑗, 𝑘〉 ∈ (𝐴 ∖ (𝐹 supp 0 ))) → (𝑗𝐹𝑘) = 0 ) |
32 | 26, 31 | sylan2 593 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐴 “ {𝑗}) ∖ ran (𝐹 supp 0 ))) → (𝑗𝐹𝑘) = 0 ) |
33 | 32, 6 | suppss2 8223 | . . 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 19946 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ∖ cdif 3959 ⊆ wss 3962 {csn 4630 〈cop 4636 class class class wbr 5147 ↦ cmpt 5230 dom cdm 5688 ran crn 5689 “ cima 5691 Rel wrel 5693 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 supp csupp 8183 Fincfn 8983 finSupp cfsupp 9398 Basecbs 17244 0gc0g 17485 Σg cgsu 17486 CMndccmn 19812 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-fzo 13691 df-seq 14039 df-hash 14366 df-0g 17487 df-gsum 17488 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-cntz 19347 df-cmn 19814 |
This theorem is referenced by: gsum2dlem2 20003 gsum2d 20004 |
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