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Mirrors > Home > MPE Home > Th. List > gsum2dlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for gsum2d 19832. (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 7901 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 “ {𝑗}) ∈ V) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → (𝐴 “ {𝑗}) ∈ V) |
7 | vex 3479 | . . . . 5 ⊢ 𝑗 ∈ V | |
8 | vex 3479 | . . . . 5 ⊢ 𝑘 ∈ V | |
9 | 7, 8 | elimasn 6085 | . . . 4 ⊢ (𝑘 ∈ (𝐴 “ {𝑗}) ↔ 〈𝑗, 𝑘〉 ∈ 𝐴) |
10 | df-ov 7407 | . . . . 5 ⊢ (𝑗𝐹𝑘) = (𝐹‘〈𝑗, 𝑘〉) | |
11 | gsum2d.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
12 | 11 | ffvelcdmda 7082 | . . . . 5 ⊢ ((𝜑 ∧ 〈𝑗, 𝑘〉 ∈ 𝐴) → (𝐹‘〈𝑗, 𝑘〉) ∈ 𝐵) |
13 | 10, 12 | eqeltrid 2838 | . . . 4 ⊢ ((𝜑 ∧ 〈𝑗, 𝑘〉 ∈ 𝐴) → (𝑗𝐹𝑘) ∈ 𝐵) |
14 | 9, 13 | sylan2b 595 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 “ {𝑗})) → (𝑗𝐹𝑘) ∈ 𝐵) |
15 | 14 | fmpttd 7110 | . 2 ⊢ (𝜑 → (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)):(𝐴 “ {𝑗})⟶𝐵) |
16 | gsum2d.w | . . . . 5 ⊢ (𝜑 → 𝐹 finSupp 0 ) | |
17 | 16 | fsuppimpd 9365 | . . . 4 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin) |
18 | rnfi 9331 | . . . 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 5940 | . . . . . . . 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 614 | . . . . . 6 ⊢ ((𝑘 ∈ (𝐴 “ {𝑗}) ∧ ¬ 𝑘 ∈ ran (𝐹 supp 0 )) → (〈𝑗, 𝑘〉 ∈ 𝐴 ∧ ¬ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 ))) |
24 | eldif 3957 | . . . . . 6 ⊢ (𝑘 ∈ ((𝐴 “ {𝑗}) ∖ ran (𝐹 supp 0 )) ↔ (𝑘 ∈ (𝐴 “ {𝑗}) ∧ ¬ 𝑘 ∈ ran (𝐹 supp 0 ))) | |
25 | eldif 3957 | . . . . . 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 4004 | . . . . . . 7 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 )) | |
28 | 2 | fvexi 6902 | . . . . . . . 8 ⊢ 0 ∈ V |
29 | 28 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ V) |
30 | 11, 27, 4, 29 | suppssr 8176 | . . . . . 6 ⊢ ((𝜑 ∧ 〈𝑗, 𝑘〉 ∈ (𝐴 ∖ (𝐹 supp 0 ))) → (𝐹‘〈𝑗, 𝑘〉) = 0 ) |
31 | 10, 30 | eqtrid 2785 | . . . . 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 8180 | . . 3 ⊢ (𝜑 → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) supp 0 ) ⊆ ran (𝐹 supp 0 )) |
34 | 19, 33 | ssfid 9263 | . 2 ⊢ (𝜑 → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) supp 0 ) ∈ Fin) |
35 | 1, 2, 3, 6, 15, 34 | gsumcl2 19774 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∖ cdif 3944 ⊆ wss 3947 {csn 4627 〈cop 4633 class class class wbr 5147 ↦ cmpt 5230 dom cdm 5675 ran crn 5676 “ cima 5678 Rel wrel 5680 ⟶wf 6536 ‘cfv 6540 (class class class)co 7404 supp csupp 8141 Fincfn 8935 finSupp cfsupp 9357 Basecbs 17140 0gc0g 17381 Σg cgsu 17382 CMndccmn 19641 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-supp 8142 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-fzo 13624 df-seq 13963 df-hash 14287 df-0g 17383 df-gsum 17384 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-cntz 19175 df-cmn 19643 |
This theorem is referenced by: gsum2dlem2 19831 gsum2d 19832 |
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