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Mirrors > Home > MPE Home > Th. List > prdsbaslem | Structured version Visualization version GIF version |
Description: Lemma for prdsbas 17504 and similar theorems. (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 12-Jul-2024.) |
Ref | Expression |
---|---|
prdsbaslem.u | ⊢ (𝜑 → 𝑈 = (({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), , 〉}) ∪ ({〈(TopSet‘ndx), 𝑂〉, 〈(le‘ndx), 𝐿〉, 〈(dist‘ndx), 𝐷〉} ∪ {〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), ∙ 〉}))) |
prdsbaslem.1 | ⊢ 𝐴 = (𝐸‘𝑈) |
prdsbaslem.2 | ⊢ 𝐸 = Slot (𝐸‘ndx) |
prdsbaslem.3 | ⊢ (𝜑 → 𝑇 ∈ 𝑉) |
prdsbaslem.4 | ⊢ {〈(𝐸‘ndx), 𝑇〉} ⊆ (({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), , 〉}) ∪ ({〈(TopSet‘ndx), 𝑂〉, 〈(le‘ndx), 𝐿〉, 〈(dist‘ndx), 𝐷〉} ∪ {〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), ∙ 〉})) |
Ref | Expression |
---|---|
prdsbaslem | ⊢ (𝜑 → 𝐴 = 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prdsbaslem.u | . 2 ⊢ (𝜑 → 𝑈 = (({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), , 〉}) ∪ ({〈(TopSet‘ndx), 𝑂〉, 〈(le‘ndx), 𝐿〉, 〈(dist‘ndx), 𝐷〉} ∪ {〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), ∙ 〉}))) | |
2 | prdsvalstr 17499 | . 2 ⊢ (({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), , 〉}) ∪ ({〈(TopSet‘ndx), 𝑂〉, 〈(le‘ndx), 𝐿〉, 〈(dist‘ndx), 𝐷〉} ∪ {〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), ∙ 〉})) Struct 〈1, ;15〉 | |
3 | prdsbaslem.2 | . 2 ⊢ 𝐸 = Slot (𝐸‘ndx) | |
4 | prdsbaslem.4 | . 2 ⊢ {〈(𝐸‘ndx), 𝑇〉} ⊆ (({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), , 〉}) ∪ ({〈(TopSet‘ndx), 𝑂〉, 〈(le‘ndx), 𝐿〉, 〈(dist‘ndx), 𝐷〉} ∪ {〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), ∙ 〉})) | |
5 | prdsbaslem.3 | . 2 ⊢ (𝜑 → 𝑇 ∈ 𝑉) | |
6 | prdsbaslem.1 | . 2 ⊢ 𝐴 = (𝐸‘𝑈) | |
7 | 1, 2, 3, 4, 5, 6 | strfv3 17239 | 1 ⊢ (𝜑 → 𝐴 = 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∪ cun 3961 ⊆ wss 3963 {csn 4631 {cpr 4633 {ctp 4635 〈cop 4637 ‘cfv 6563 1c1 11154 5c5 12322 ;cdc 12731 Slot cslot 17215 ndxcnx 17227 Basecbs 17245 +gcplusg 17298 .rcmulr 17299 Scalarcsca 17301 ·𝑠 cvsca 17302 ·𝑖cip 17303 TopSetcts 17304 lecple 17305 distcds 17307 Hom chom 17309 compcco 17310 |
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-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-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 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-dec 12732 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-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 |
This theorem is referenced by: prdssca 17503 prdsbas 17504 prdsplusg 17505 prdsmulr 17506 prdsvsca 17507 prdsip 17508 prdsle 17509 prdsds 17511 prdstset 17513 prdshom 17514 prdsco 17515 |
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