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Mirrors > Home > MPE Home > Th. List > cnmpt1mulr | Structured version Visualization version GIF version |
Description: Continuity of ring multiplication; analogue of cnmpt12f 22535 which cannot be used directly because .r is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
mulrcn.j | ⊢ 𝐽 = (TopOpen‘𝑅) |
cnmpt1mulr.t | ⊢ · = (.r‘𝑅) |
cnmpt1mulr.r | ⊢ (𝜑 → 𝑅 ∈ TopRing) |
cnmpt1mulr.k | ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋)) |
cnmpt1mulr.a | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐾 Cn 𝐽)) |
cnmpt1mulr.b | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐾 Cn 𝐽)) |
Ref | Expression |
---|---|
cnmpt1mulr | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝐾 Cn 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2734 | . . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
2 | mulrcn.j | . . 3 ⊢ 𝐽 = (TopOpen‘𝑅) | |
3 | 1, 2 | mgptopn 19485 | . 2 ⊢ 𝐽 = (TopOpen‘(mulGrp‘𝑅)) |
4 | cnmpt1mulr.t | . . 3 ⊢ · = (.r‘𝑅) | |
5 | 1, 4 | mgpplusg 19480 | . 2 ⊢ · = (+g‘(mulGrp‘𝑅)) |
6 | cnmpt1mulr.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ TopRing) | |
7 | 1 | trgtmd 23034 | . . 3 ⊢ (𝑅 ∈ TopRing → (mulGrp‘𝑅) ∈ TopMnd) |
8 | 6, 7 | syl 17 | . 2 ⊢ (𝜑 → (mulGrp‘𝑅) ∈ TopMnd) |
9 | cnmpt1mulr.k | . 2 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋)) | |
10 | cnmpt1mulr.a | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐾 Cn 𝐽)) | |
11 | cnmpt1mulr.b | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐾 Cn 𝐽)) | |
12 | 3, 5, 8, 9, 10, 11 | cnmpt1plusg 22956 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝐾 Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ↦ cmpt 5124 ‘cfv 6369 (class class class)co 7202 .rcmulr 16768 TopOpenctopn 16898 mulGrpcmgp 19476 TopOnctopon 21779 Cn ccn 22093 TopMndctmd 22939 TopRingctrg 23025 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-er 8380 df-map 8499 df-en 8616 df-dom 8617 df-sdom 8618 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-plusg 16780 df-tset 16786 df-rest 16899 df-topn 16900 df-topgen 16920 df-plusf 18085 df-mgp 19477 df-top 21763 df-topon 21780 df-topsp 21802 df-bases 21815 df-cn 22096 df-tx 22431 df-tmd 22941 df-trg 23029 |
This theorem is referenced by: (None) |
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