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Mirrors > Home > MPE Home > Th. List > psraddclOLD | Structured version Visualization version GIF version |
Description: Obsolete version of psraddcl 21976 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 2735 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | eqid 2735 | . . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
4 | 2, 3 | grpcl 18972 | . . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅)) |
5 | 4 | 3expb 1119 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅)) |
6 | 1, 5 | sylan 580 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅)) |
7 | psraddclOLD.s | . . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
8 | eqid 2735 | . . . . 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 21972 | . . . 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 21972 | . . . 4 ⊢ (𝜑 → 𝑌:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅)) |
14 | ovex 7464 | . . . . . 6 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
15 | 14 | rabex 5345 | . . . . 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 4235 | . . . 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 7715 | . . 3 ⊢ (𝜑 → (𝑋 ∘f (+g‘𝑅)𝑌):{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅)) |
19 | fvex 6920 | . . . 4 ⊢ (Base‘𝑅) ∈ V | |
20 | 19, 15 | elmap 8910 | . . 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 234 | . 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 21975 | . 2 ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑋 ∘f (+g‘𝑅)𝑌)) |
24 | reldmpsr 21952 | . . . . . 6 ⊢ Rel dom mPwSer | |
25 | 24, 7, 9 | elbasov 17252 | . . . . 5 ⊢ (𝑋 ∈ 𝐵 → (𝐼 ∈ V ∧ 𝑅 ∈ V)) |
26 | 10, 25 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐼 ∈ V ∧ 𝑅 ∈ V)) |
27 | 26 | simpld 494 | . . 3 ⊢ (𝜑 → 𝐼 ∈ V) |
28 | 7, 2, 8, 9, 27 | psrbas 21971 | . 2 ⊢ (𝜑 → 𝐵 = ((Base‘𝑅) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) |
29 | 21, 23, 28 | 3eltr4d 2854 | 1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {crab 3433 Vcvv 3478 ◡ccnv 5688 “ cima 5692 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ∘f cof 7695 ↑m cmap 8865 Fincfn 8984 ℕcn 12264 ℕ0cn0 12524 Basecbs 17245 +gcplusg 17298 Grpcgrp 18964 mPwSer cmps 21942 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-struct 17181 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-tset 17317 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-psr 21947 |
This theorem is referenced by: (None) |
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