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Mirrors > Home > MPE Home > Th. List > psraddclOLD | Structured version Visualization version GIF version |
Description: Obsolete version of psraddcl 21885 as of 12-Apr-2025. (Contributed by Mario Carneiro, 28-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
psraddclOLD.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
psraddclOLD.b | ⊢ 𝐵 = (Base‘𝑆) |
psraddclOLD.p | ⊢ + = (+g‘𝑆) |
psraddclOLD.r | ⊢ (𝜑 → 𝑅 ∈ Grp) |
psraddclOLD.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
psraddclOLD.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
psraddclOLD | ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psraddclOLD.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Grp) | |
2 | eqid 2725 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | eqid 2725 | . . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
4 | 2, 3 | grpcl 18900 | . . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅)) |
5 | 4 | 3expb 1117 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅)) |
6 | 1, 5 | sylan 578 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅)) |
7 | psraddclOLD.s | . . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
8 | eqid 2725 | . . . . 5 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
9 | psraddclOLD.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑆) | |
10 | psraddclOLD.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
11 | 7, 2, 8, 9, 10 | psrelbas 21881 | . . . 4 ⊢ (𝜑 → 𝑋:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅)) |
12 | psraddclOLD.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
13 | 7, 2, 8, 9, 12 | psrelbas 21881 | . . . 4 ⊢ (𝜑 → 𝑌:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅)) |
14 | ovex 7448 | . . . . . 6 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
15 | 14 | rabex 5329 | . . . . 5 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V |
16 | 15 | a1i 11 | . . . 4 ⊢ (𝜑 → {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V) |
17 | inidm 4213 | . . . 4 ⊢ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∩ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
18 | 6, 11, 13, 16, 16, 17 | off 7699 | . . 3 ⊢ (𝜑 → (𝑋 ∘f (+g‘𝑅)𝑌):{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅)) |
19 | fvex 6904 | . . . 4 ⊢ (Base‘𝑅) ∈ V | |
20 | 19, 15 | elmap 8886 | . . 3 ⊢ ((𝑋 ∘f (+g‘𝑅)𝑌) ∈ ((Base‘𝑅) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ↔ (𝑋 ∘f (+g‘𝑅)𝑌):{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅)) |
21 | 18, 20 | sylibr 233 | . 2 ⊢ (𝜑 → (𝑋 ∘f (+g‘𝑅)𝑌) ∈ ((Base‘𝑅) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) |
22 | psraddclOLD.p | . . 3 ⊢ + = (+g‘𝑆) | |
23 | 7, 9, 3, 22, 10, 12 | psradd 21884 | . 2 ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑋 ∘f (+g‘𝑅)𝑌)) |
24 | reldmpsr 21849 | . . . . . 6 ⊢ Rel dom mPwSer | |
25 | 24, 7, 9 | elbasov 17184 | . . . . 5 ⊢ (𝑋 ∈ 𝐵 → (𝐼 ∈ V ∧ 𝑅 ∈ V)) |
26 | 10, 25 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐼 ∈ V ∧ 𝑅 ∈ V)) |
27 | 26 | simpld 493 | . . 3 ⊢ (𝜑 → 𝐼 ∈ V) |
28 | 7, 2, 8, 9, 27 | psrbas 21880 | . 2 ⊢ (𝜑 → 𝐵 = ((Base‘𝑅) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) |
29 | 21, 23, 28 | 3eltr4d 2840 | 1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 {crab 3419 Vcvv 3463 ◡ccnv 5671 “ cima 5675 ⟶wf 6538 ‘cfv 6542 (class class class)co 7415 ∘f cof 7679 ↑m cmap 8841 Fincfn 8960 ℕcn 12240 ℕ0cn0 12500 Basecbs 17177 +gcplusg 17230 Grpcgrp 18892 mPwSer cmps 21839 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7681 df-om 7868 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-map 8843 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fsupp 9384 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-uz 12851 df-fz 13515 df-struct 17113 df-slot 17148 df-ndx 17160 df-base 17178 df-plusg 17243 df-mulr 17244 df-sca 17246 df-vsca 17247 df-tset 17249 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-grp 18895 df-psr 21844 |
This theorem is referenced by: (None) |
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