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Mirrors > Home > MPE Home > Th. List > psrbagev1OLD | Structured version Visualization version GIF version |
Description: Obsolete version of psrbagev1 21949 as of 7-Aug-2024. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 18-Jul-2019.) (Revised by AV, 11-Apr-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
psrbagev1.d | ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} |
psrbagev1.c | ⊢ 𝐶 = (Base‘𝑇) |
psrbagev1.x | ⊢ · = (.g‘𝑇) |
psrbagev1.z | ⊢ 0 = (0g‘𝑇) |
psrbagev1.t | ⊢ (𝜑 → 𝑇 ∈ CMnd) |
psrbagev1.b | ⊢ (𝜑 → 𝐵 ∈ 𝐷) |
psrbagev1.g | ⊢ (𝜑 → 𝐺:𝐼⟶𝐶) |
psrbagev1.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
Ref | Expression |
---|---|
psrbagev1OLD | ⊢ (𝜑 → ((𝐵 ∘f · 𝐺):𝐼⟶𝐶 ∧ (𝐵 ∘f · 𝐺) finSupp 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psrbagev1.t | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ CMnd) | |
2 | 1 | cmnmndd 19720 | . . . 4 ⊢ (𝜑 → 𝑇 ∈ Mnd) |
3 | psrbagev1.c | . . . . . 6 ⊢ 𝐶 = (Base‘𝑇) | |
4 | psrbagev1.x | . . . . . 6 ⊢ · = (.g‘𝑇) | |
5 | 3, 4 | mulgnn0cl 19013 | . . . . 5 ⊢ ((𝑇 ∈ Mnd ∧ 𝑦 ∈ ℕ0 ∧ 𝑧 ∈ 𝐶) → (𝑦 · 𝑧) ∈ 𝐶) |
6 | 5 | 3expb 1119 | . . . 4 ⊢ ((𝑇 ∈ Mnd ∧ (𝑦 ∈ ℕ0 ∧ 𝑧 ∈ 𝐶)) → (𝑦 · 𝑧) ∈ 𝐶) |
7 | 2, 6 | sylan 579 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ ℕ0 ∧ 𝑧 ∈ 𝐶)) → (𝑦 · 𝑧) ∈ 𝐶) |
8 | psrbagev1.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
9 | psrbagev1.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
10 | psrbagev1.d | . . . . 5 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
11 | 10 | psrbagfOLD 21782 | . . . 4 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝐵 ∈ 𝐷) → 𝐵:𝐼⟶ℕ0) |
12 | 8, 9, 11 | syl2anc 583 | . . 3 ⊢ (𝜑 → 𝐵:𝐼⟶ℕ0) |
13 | psrbagev1.g | . . 3 ⊢ (𝜑 → 𝐺:𝐼⟶𝐶) | |
14 | inidm 4218 | . . 3 ⊢ (𝐼 ∩ 𝐼) = 𝐼 | |
15 | 7, 12, 13, 8, 8, 14 | off 7692 | . 2 ⊢ (𝜑 → (𝐵 ∘f · 𝐺):𝐼⟶𝐶) |
16 | ovexd 7447 | . . 3 ⊢ (𝜑 → (𝐵 ∘f · 𝐺) ∈ V) | |
17 | 12 | ffnd 6718 | . . . 4 ⊢ (𝜑 → 𝐵 Fn 𝐼) |
18 | 13 | ffnd 6718 | . . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐼) |
19 | 17, 18, 8, 8 | offun 7688 | . . 3 ⊢ (𝜑 → Fun (𝐵 ∘f · 𝐺)) |
20 | psrbagev1.z | . . . . 5 ⊢ 0 = (0g‘𝑇) | |
21 | 20 | fvexi 6905 | . . . 4 ⊢ 0 ∈ V |
22 | 21 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ V) |
23 | 10 | psrbagfsuppOLD 21784 | . . . . 5 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝐼 ∈ 𝑊) → 𝐵 finSupp 0) |
24 | 9, 8, 23 | syl2anc 583 | . . . 4 ⊢ (𝜑 → 𝐵 finSupp 0) |
25 | 24 | fsuppimpd 9375 | . . 3 ⊢ (𝜑 → (𝐵 supp 0) ∈ Fin) |
26 | ssidd 4005 | . . . 4 ⊢ (𝜑 → (𝐵 supp 0) ⊆ (𝐵 supp 0)) | |
27 | 3, 20, 4 | mulg0 19000 | . . . . 5 ⊢ (𝑧 ∈ 𝐶 → (0 · 𝑧) = 0 ) |
28 | 27 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (0 · 𝑧) = 0 ) |
29 | c0ex 11215 | . . . . 5 ⊢ 0 ∈ V | |
30 | 29 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
31 | 26, 28, 12, 13, 8, 30 | suppssof1 8190 | . . 3 ⊢ (𝜑 → ((𝐵 ∘f · 𝐺) supp 0 ) ⊆ (𝐵 supp 0)) |
32 | suppssfifsupp 9384 | . . 3 ⊢ ((((𝐵 ∘f · 𝐺) ∈ V ∧ Fun (𝐵 ∘f · 𝐺) ∧ 0 ∈ V) ∧ ((𝐵 supp 0) ∈ Fin ∧ ((𝐵 ∘f · 𝐺) supp 0 ) ⊆ (𝐵 supp 0))) → (𝐵 ∘f · 𝐺) finSupp 0 ) | |
33 | 16, 19, 22, 25, 31, 32 | syl32anc 1377 | . 2 ⊢ (𝜑 → (𝐵 ∘f · 𝐺) finSupp 0 ) |
34 | 15, 33 | jca 511 | 1 ⊢ (𝜑 → ((𝐵 ∘f · 𝐺):𝐼⟶𝐶 ∧ (𝐵 ∘f · 𝐺) finSupp 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 {crab 3431 Vcvv 3473 ⊆ wss 3948 class class class wbr 5148 ◡ccnv 5675 “ cima 5679 Fun wfun 6537 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ∘f cof 7672 supp csupp 8151 ↑m cmap 8826 Fincfn 8945 finSupp cfsupp 9367 0cc0 11116 ℕcn 12219 ℕ0cn0 12479 Basecbs 17151 0gc0g 17392 Mndcmnd 18665 .gcmg 18993 CMndccmn 19696 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-n0 12480 df-z 12566 df-uz 12830 df-fz 13492 df-seq 13974 df-0g 17394 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-mulg 18994 df-cmn 19698 |
This theorem is referenced by: psrbagev2OLD 21952 |
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