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Mirrors > Home > MPE Home > Th. List > gsum2dlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for gsum2d 19092. (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 7620 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 “ {𝑗}) ∈ V) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → (𝐴 “ {𝑗}) ∈ V) |
7 | vex 3497 | . . . . 5 ⊢ 𝑗 ∈ V | |
8 | vex 3497 | . . . . 5 ⊢ 𝑘 ∈ V | |
9 | 7, 8 | elimasn 5954 | . . . 4 ⊢ (𝑘 ∈ (𝐴 “ {𝑗}) ↔ 〈𝑗, 𝑘〉 ∈ 𝐴) |
10 | df-ov 7159 | . . . . 5 ⊢ (𝑗𝐹𝑘) = (𝐹‘〈𝑗, 𝑘〉) | |
11 | gsum2d.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
12 | 11 | ffvelrnda 6851 | . . . . 5 ⊢ ((𝜑 ∧ 〈𝑗, 𝑘〉 ∈ 𝐴) → (𝐹‘〈𝑗, 𝑘〉) ∈ 𝐵) |
13 | 10, 12 | eqeltrid 2917 | . . . 4 ⊢ ((𝜑 ∧ 〈𝑗, 𝑘〉 ∈ 𝐴) → (𝑗𝐹𝑘) ∈ 𝐵) |
14 | 9, 13 | sylan2b 595 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 “ {𝑗})) → (𝑗𝐹𝑘) ∈ 𝐵) |
15 | 14 | fmpttd 6879 | . 2 ⊢ (𝜑 → (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)):(𝐴 “ {𝑗})⟶𝐵) |
16 | gsum2d.w | . . . . 5 ⊢ (𝜑 → 𝐹 finSupp 0 ) | |
17 | 16 | fsuppimpd 8840 | . . . 4 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin) |
18 | rnfi 8807 | . . . 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 218 | . . . . . . 7 ⊢ (𝑘 ∈ (𝐴 “ {𝑗}) → 〈𝑗, 𝑘〉 ∈ 𝐴) |
21 | 7, 8 | opelrn 5813 | . . . . . . . 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 614 | . . . . . 6 ⊢ ((𝑘 ∈ (𝐴 “ {𝑗}) ∧ ¬ 𝑘 ∈ ran (𝐹 supp 0 )) → (〈𝑗, 𝑘〉 ∈ 𝐴 ∧ ¬ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 ))) |
24 | eldif 3946 | . . . . . 6 ⊢ (𝑘 ∈ ((𝐴 “ {𝑗}) ∖ ran (𝐹 supp 0 )) ↔ (𝑘 ∈ (𝐴 “ {𝑗}) ∧ ¬ 𝑘 ∈ ran (𝐹 supp 0 ))) | |
25 | eldif 3946 | . . . . . 6 ⊢ (〈𝑗, 𝑘〉 ∈ (𝐴 ∖ (𝐹 supp 0 )) ↔ (〈𝑗, 𝑘〉 ∈ 𝐴 ∧ ¬ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 ))) | |
26 | 23, 24, 25 | 3imtr4i 294 | . . . . 5 ⊢ (𝑘 ∈ ((𝐴 “ {𝑗}) ∖ ran (𝐹 supp 0 )) → 〈𝑗, 𝑘〉 ∈ (𝐴 ∖ (𝐹 supp 0 ))) |
27 | ssidd 3990 | . . . . . . 7 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 )) | |
28 | 2 | fvexi 6684 | . . . . . . . 8 ⊢ 0 ∈ V |
29 | 28 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ V) |
30 | 11, 27, 4, 29 | suppssr 7861 | . . . . . 6 ⊢ ((𝜑 ∧ 〈𝑗, 𝑘〉 ∈ (𝐴 ∖ (𝐹 supp 0 ))) → (𝐹‘〈𝑗, 𝑘〉) = 0 ) |
31 | 10, 30 | syl5eq 2868 | . . . . 5 ⊢ ((𝜑 ∧ 〈𝑗, 𝑘〉 ∈ (𝐴 ∖ (𝐹 supp 0 ))) → (𝑗𝐹𝑘) = 0 ) |
32 | 26, 31 | sylan2 594 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐴 “ {𝑗}) ∖ ran (𝐹 supp 0 ))) → (𝑗𝐹𝑘) = 0 ) |
33 | 32, 6 | suppss2 7864 | . . 3 ⊢ (𝜑 → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) supp 0 ) ⊆ ran (𝐹 supp 0 )) |
34 | 19, 33 | ssfid 8741 | . 2 ⊢ (𝜑 → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) supp 0 ) ∈ Fin) |
35 | 1, 2, 3, 6, 15, 34 | gsumcl2 19034 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∖ cdif 3933 ⊆ wss 3936 {csn 4567 〈cop 4573 class class class wbr 5066 ↦ cmpt 5146 dom cdm 5555 ran crn 5556 “ cima 5558 Rel wrel 5560 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 supp csupp 7830 Fincfn 8509 finSupp cfsupp 8833 Basecbs 16483 0gc0g 16713 Σg cgsu 16714 CMndccmn 18906 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-seq 13371 df-hash 13692 df-0g 16715 df-gsum 16716 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-cntz 18447 df-cmn 18908 |
This theorem is referenced by: gsum2dlem2 19091 gsum2d 19092 |
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