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Mirrors > Home > MPE Home > Th. List > axcontlem11 | Structured version Visualization version GIF version |
Description: Lemma for axcont 25856. Eliminate the hypotheses from axcontlem10 25853. (Contributed by Scott Fenton, 20-Jun-2013.) |
Ref | Expression |
---|---|
axcontlem11 | ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn 〈𝑍, 𝑦〉)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑏 Btwn 〈𝑥, 𝑦〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq2 4403 | . . . . 5 ⊢ (𝑞 = 𝑝 → 〈𝑍, 𝑞〉 = 〈𝑍, 𝑝〉) | |
2 | 1 | breq2d 4665 | . . . 4 ⊢ (𝑞 = 𝑝 → (𝑈 Btwn 〈𝑍, 𝑞〉 ↔ 𝑈 Btwn 〈𝑍, 𝑝〉)) |
3 | breq1 4656 | . . . 4 ⊢ (𝑞 = 𝑝 → (𝑞 Btwn 〈𝑍, 𝑈〉 ↔ 𝑝 Btwn 〈𝑍, 𝑈〉)) | |
4 | 2, 3 | orbi12d 746 | . . 3 ⊢ (𝑞 = 𝑝 → ((𝑈 Btwn 〈𝑍, 𝑞〉 ∨ 𝑞 Btwn 〈𝑍, 𝑈〉) ↔ (𝑈 Btwn 〈𝑍, 𝑝〉 ∨ 𝑝 Btwn 〈𝑍, 𝑈〉))) |
5 | 4 | cbvrabv 3199 | . 2 ⊢ {𝑞 ∈ (𝔼‘𝑁) ∣ (𝑈 Btwn 〈𝑍, 𝑞〉 ∨ 𝑞 Btwn 〈𝑍, 𝑈〉)} = {𝑝 ∈ (𝔼‘𝑁) ∣ (𝑈 Btwn 〈𝑍, 𝑝〉 ∨ 𝑝 Btwn 〈𝑍, 𝑈〉)} |
6 | eqid 2622 | . . 3 ⊢ {〈𝑧, 𝑟〉 ∣ (𝑧 ∈ {𝑞 ∈ (𝔼‘𝑁) ∣ (𝑈 Btwn 〈𝑍, 𝑞〉 ∨ 𝑞 Btwn 〈𝑍, 𝑈〉)} ∧ (𝑟 ∈ (0[,)+∞) ∧ ∀𝑗 ∈ (1...𝑁)(𝑧‘𝑗) = (((1 − 𝑟) · (𝑍‘𝑗)) + (𝑟 · (𝑈‘𝑗)))))} = {〈𝑧, 𝑟〉 ∣ (𝑧 ∈ {𝑞 ∈ (𝔼‘𝑁) ∣ (𝑈 Btwn 〈𝑍, 𝑞〉 ∨ 𝑞 Btwn 〈𝑍, 𝑈〉)} ∧ (𝑟 ∈ (0[,)+∞) ∧ ∀𝑗 ∈ (1...𝑁)(𝑧‘𝑗) = (((1 − 𝑟) · (𝑍‘𝑗)) + (𝑟 · (𝑈‘𝑗)))))} | |
7 | 6 | axcontlem1 25844 | . 2 ⊢ {〈𝑧, 𝑟〉 ∣ (𝑧 ∈ {𝑞 ∈ (𝔼‘𝑁) ∣ (𝑈 Btwn 〈𝑍, 𝑞〉 ∨ 𝑞 Btwn 〈𝑍, 𝑈〉)} ∧ (𝑟 ∈ (0[,)+∞) ∧ ∀𝑗 ∈ (1...𝑁)(𝑧‘𝑗) = (((1 − 𝑟) · (𝑍‘𝑗)) + (𝑟 · (𝑈‘𝑗)))))} = {〈𝑥, 𝑡〉 ∣ (𝑥 ∈ {𝑞 ∈ (𝔼‘𝑁) ∣ (𝑈 Btwn 〈𝑍, 𝑞〉 ∨ 𝑞 Btwn 〈𝑍, 𝑈〉)} ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑍‘𝑖)) + (𝑡 · (𝑈‘𝑖)))))} |
8 | 5, 7 | axcontlem10 25853 | 1 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn 〈𝑍, 𝑦〉)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑏 Btwn 〈𝑥, 𝑦〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 383 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 {crab 2916 ⊆ wss 3574 ∅c0 3915 〈cop 4183 class class class wbr 4653 {copab 4712 ‘cfv 5888 (class class class)co 6650 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 +∞cpnf 10071 − cmin 10266 ℕcn 11020 [,)cico 12177 ...cfz 12326 𝔼cee 25768 Btwn cbtwn 25769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-z 11378 df-uz 11688 df-ico 12181 df-icc 12182 df-fz 12327 df-ee 25771 df-btwn 25772 |
This theorem is referenced by: axcontlem12 25855 |
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