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Mirrors > Home > MPE Home > Th. List > gsum2dlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for gsum2d 19085. (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 7602 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 “ {𝑗}) ∈ V) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → (𝐴 “ {𝑗}) ∈ V) |
7 | vex 3444 | . . . . 5 ⊢ 𝑗 ∈ V | |
8 | vex 3444 | . . . . 5 ⊢ 𝑘 ∈ V | |
9 | 7, 8 | elimasn 5921 | . . . 4 ⊢ (𝑘 ∈ (𝐴 “ {𝑗}) ↔ 〈𝑗, 𝑘〉 ∈ 𝐴) |
10 | df-ov 7138 | . . . . 5 ⊢ (𝑗𝐹𝑘) = (𝐹‘〈𝑗, 𝑘〉) | |
11 | gsum2d.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
12 | 11 | ffvelrnda 6828 | . . . . 5 ⊢ ((𝜑 ∧ 〈𝑗, 𝑘〉 ∈ 𝐴) → (𝐹‘〈𝑗, 𝑘〉) ∈ 𝐵) |
13 | 10, 12 | eqeltrid 2894 | . . . 4 ⊢ ((𝜑 ∧ 〈𝑗, 𝑘〉 ∈ 𝐴) → (𝑗𝐹𝑘) ∈ 𝐵) |
14 | 9, 13 | sylan2b 596 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 “ {𝑗})) → (𝑗𝐹𝑘) ∈ 𝐵) |
15 | 14 | fmpttd 6856 | . 2 ⊢ (𝜑 → (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)):(𝐴 “ {𝑗})⟶𝐵) |
16 | gsum2d.w | . . . . 5 ⊢ (𝜑 → 𝐹 finSupp 0 ) | |
17 | 16 | fsuppimpd 8824 | . . . 4 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin) |
18 | rnfi 8791 | . . . 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 219 | . . . . . . 7 ⊢ (𝑘 ∈ (𝐴 “ {𝑗}) → 〈𝑗, 𝑘〉 ∈ 𝐴) |
21 | 7, 8 | opelrn 5777 | . . . . . . . 8 ⊢ (〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 ) → 𝑘 ∈ ran (𝐹 supp 0 )) |
22 | 21 | con3i 157 | . . . . . . 7 ⊢ (¬ 𝑘 ∈ ran (𝐹 supp 0 ) → ¬ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 )) |
23 | 20, 22 | anim12i 615 | . . . . . 6 ⊢ ((𝑘 ∈ (𝐴 “ {𝑗}) ∧ ¬ 𝑘 ∈ ran (𝐹 supp 0 )) → (〈𝑗, 𝑘〉 ∈ 𝐴 ∧ ¬ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 ))) |
24 | eldif 3891 | . . . . . 6 ⊢ (𝑘 ∈ ((𝐴 “ {𝑗}) ∖ ran (𝐹 supp 0 )) ↔ (𝑘 ∈ (𝐴 “ {𝑗}) ∧ ¬ 𝑘 ∈ ran (𝐹 supp 0 ))) | |
25 | eldif 3891 | . . . . . 6 ⊢ (〈𝑗, 𝑘〉 ∈ (𝐴 ∖ (𝐹 supp 0 )) ↔ (〈𝑗, 𝑘〉 ∈ 𝐴 ∧ ¬ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 ))) | |
26 | 23, 24, 25 | 3imtr4i 295 | . . . . 5 ⊢ (𝑘 ∈ ((𝐴 “ {𝑗}) ∖ ran (𝐹 supp 0 )) → 〈𝑗, 𝑘〉 ∈ (𝐴 ∖ (𝐹 supp 0 ))) |
27 | ssidd 3938 | . . . . . . 7 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 )) | |
28 | 2 | fvexi 6659 | . . . . . . . 8 ⊢ 0 ∈ V |
29 | 28 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ V) |
30 | 11, 27, 4, 29 | suppssr 7844 | . . . . . 6 ⊢ ((𝜑 ∧ 〈𝑗, 𝑘〉 ∈ (𝐴 ∖ (𝐹 supp 0 ))) → (𝐹‘〈𝑗, 𝑘〉) = 0 ) |
31 | 10, 30 | syl5eq 2845 | . . . . 5 ⊢ ((𝜑 ∧ 〈𝑗, 𝑘〉 ∈ (𝐴 ∖ (𝐹 supp 0 ))) → (𝑗𝐹𝑘) = 0 ) |
32 | 26, 31 | sylan2 595 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐴 “ {𝑗}) ∖ ran (𝐹 supp 0 ))) → (𝑗𝐹𝑘) = 0 ) |
33 | 32, 6 | suppss2 7847 | . . 3 ⊢ (𝜑 → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) supp 0 ) ⊆ ran (𝐹 supp 0 )) |
34 | 19, 33 | ssfid 8725 | . 2 ⊢ (𝜑 → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) supp 0 ) ∈ Fin) |
35 | 1, 2, 3, 6, 15, 34 | gsumcl2 19027 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∖ cdif 3878 ⊆ wss 3881 {csn 4525 〈cop 4531 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5519 ran crn 5520 “ cima 5522 Rel wrel 5524 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 supp csupp 7813 Fincfn 8492 finSupp cfsupp 8817 Basecbs 16475 0gc0g 16705 Σg cgsu 16706 CMndccmn 18898 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-0g 16707 df-gsum 16708 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-cntz 18439 df-cmn 18900 |
This theorem is referenced by: gsum2dlem2 19084 gsum2d 19085 |
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