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Mirrors > Home > MPE Home > Th. List > axcontlem11 | Structured version Visualization version GIF version |
Description: Lemma for axcont 26764. Eliminate the hypotheses from axcontlem10 26761. (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 4806 | . . . . 5 ⊢ (𝑞 = 𝑝 → 〈𝑍, 𝑞〉 = 〈𝑍, 𝑝〉) | |
2 | 1 | breq2d 5080 | . . . 4 ⊢ (𝑞 = 𝑝 → (𝑈 Btwn 〈𝑍, 𝑞〉 ↔ 𝑈 Btwn 〈𝑍, 𝑝〉)) |
3 | breq1 5071 | . . . 4 ⊢ (𝑞 = 𝑝 → (𝑞 Btwn 〈𝑍, 𝑈〉 ↔ 𝑝 Btwn 〈𝑍, 𝑈〉)) | |
4 | 2, 3 | orbi12d 915 | . . 3 ⊢ (𝑞 = 𝑝 → ((𝑈 Btwn 〈𝑍, 𝑞〉 ∨ 𝑞 Btwn 〈𝑍, 𝑈〉) ↔ (𝑈 Btwn 〈𝑍, 𝑝〉 ∨ 𝑝 Btwn 〈𝑍, 𝑈〉))) |
5 | 4 | cbvrabv 3493 | . 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 2823 | . . 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 26752 | . 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 26761 | 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 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 {crab 3144 ⊆ wss 3938 ∅c0 4293 〈cop 4575 class class class wbr 5068 {copab 5130 ‘cfv 6357 (class class class)co 7158 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 +∞cpnf 10674 − cmin 10872 ℕcn 11640 [,)cico 12743 ...cfz 12895 𝔼cee 26676 Btwn cbtwn 26677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-z 11985 df-uz 12247 df-ico 12747 df-icc 12748 df-fz 12896 df-ee 26679 df-btwn 26680 |
This theorem is referenced by: axcontlem12 26763 |
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