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Mirrors > Home > MPE Home > Th. List > cnmptre | Structured version Visualization version GIF version |
Description: Lemma for iirevcn 22537 and related functions. (Contributed by Mario Carneiro, 6-Jun-2014.) |
Ref | Expression |
---|---|
cnmptre.1 | ⊢ 𝑅 = (TopOpen‘ℂfld) |
cnmptre.2 | ⊢ 𝐽 = ((topGen‘ran (,)) ↾t 𝐴) |
cnmptre.3 | ⊢ 𝐾 = ((topGen‘ran (,)) ↾t 𝐵) |
cnmptre.4 | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
cnmptre.5 | ⊢ (𝜑 → 𝐵 ⊆ ℝ) |
cnmptre.6 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ 𝐵) |
cnmptre.7 | ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ 𝐹) ∈ (𝑅 Cn 𝑅)) |
Ref | Expression |
---|---|
cnmptre | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐽 Cn 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . . . . 5 ⊢ (𝑅 ↾t 𝐴) = (𝑅 ↾t 𝐴) | |
2 | cnmptre.1 | . . . . . . 7 ⊢ 𝑅 = (TopOpen‘ℂfld) | |
3 | 2 | cnfldtopon 22396 | . . . . . 6 ⊢ 𝑅 ∈ (TopOn‘ℂ) |
4 | 3 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ (TopOn‘ℂ)) |
5 | cnmptre.4 | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
6 | ax-resscn 9872 | . . . . . 6 ⊢ ℝ ⊆ ℂ | |
7 | 5, 6 | syl6ss 3580 | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℂ) |
8 | cnmptre.7 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ 𝐹) ∈ (𝑅 Cn 𝑅)) | |
9 | 1, 4, 7, 8 | cnmpt1res 21289 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ ((𝑅 ↾t 𝐴) Cn 𝑅)) |
10 | eqid 2610 | . . . . . . . 8 ⊢ (topGen‘ran (,)) = (topGen‘ran (,)) | |
11 | 2, 10 | rerest 22415 | . . . . . . 7 ⊢ (𝐴 ⊆ ℝ → (𝑅 ↾t 𝐴) = ((topGen‘ran (,)) ↾t 𝐴)) |
12 | 5, 11 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝑅 ↾t 𝐴) = ((topGen‘ran (,)) ↾t 𝐴)) |
13 | cnmptre.2 | . . . . . 6 ⊢ 𝐽 = ((topGen‘ran (,)) ↾t 𝐴) | |
14 | 12, 13 | syl6eqr 2662 | . . . . 5 ⊢ (𝜑 → (𝑅 ↾t 𝐴) = 𝐽) |
15 | 14 | oveq1d 6564 | . . . 4 ⊢ (𝜑 → ((𝑅 ↾t 𝐴) Cn 𝑅) = (𝐽 Cn 𝑅)) |
16 | 9, 15 | eleqtrd 2690 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐽 Cn 𝑅)) |
17 | cnmptre.6 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ 𝐵) | |
18 | eqid 2610 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝐹) | |
19 | 17, 18 | fmptd 6292 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹):𝐴⟶𝐵) |
20 | frn 5966 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐹):𝐴⟶𝐵 → ran (𝑥 ∈ 𝐴 ↦ 𝐹) ⊆ 𝐵) | |
21 | 19, 20 | syl 17 | . . . 4 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐹) ⊆ 𝐵) |
22 | cnmptre.5 | . . . . 5 ⊢ (𝜑 → 𝐵 ⊆ ℝ) | |
23 | 22, 6 | syl6ss 3580 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℂ) |
24 | cnrest2 20900 | . . . 4 ⊢ ((𝑅 ∈ (TopOn‘ℂ) ∧ ran (𝑥 ∈ 𝐴 ↦ 𝐹) ⊆ 𝐵 ∧ 𝐵 ⊆ ℂ) → ((𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐽 Cn 𝑅) ↔ (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐽 Cn (𝑅 ↾t 𝐵)))) | |
25 | 4, 21, 23, 24 | syl3anc 1318 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐽 Cn 𝑅) ↔ (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐽 Cn (𝑅 ↾t 𝐵)))) |
26 | 16, 25 | mpbid 221 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐽 Cn (𝑅 ↾t 𝐵))) |
27 | 2, 10 | rerest 22415 | . . . . 5 ⊢ (𝐵 ⊆ ℝ → (𝑅 ↾t 𝐵) = ((topGen‘ran (,)) ↾t 𝐵)) |
28 | 22, 27 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑅 ↾t 𝐵) = ((topGen‘ran (,)) ↾t 𝐵)) |
29 | cnmptre.3 | . . . 4 ⊢ 𝐾 = ((topGen‘ran (,)) ↾t 𝐵) | |
30 | 28, 29 | syl6eqr 2662 | . . 3 ⊢ (𝜑 → (𝑅 ↾t 𝐵) = 𝐾) |
31 | 30 | oveq2d 6565 | . 2 ⊢ (𝜑 → (𝐽 Cn (𝑅 ↾t 𝐵)) = (𝐽 Cn 𝐾)) |
32 | 26, 31 | eleqtrd 2690 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐽 Cn 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ↦ cmpt 4643 ran crn 5039 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 (,)cioo 12046 ↾t crest 15904 TopOpenctopn 15905 topGenctg 15921 ℂfldccnfld 19567 TopOnctopon 20518 Cn ccn 20838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fi 8200 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-fz 12198 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-mulr 15782 df-starv 15783 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-rest 15906 df-topn 15907 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cn 20841 df-xms 21935 df-ms 21936 |
This theorem is referenced by: iirevcn 22537 iihalf1cn 22539 iihalf2cn 22541 pcoass 22632 |
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