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Mirrors > Home > MPE Home > Th. List > axcontlem11 | Structured version Visualization version GIF version |
Description: Lemma for axcont 27925. Eliminate the hypotheses from axcontlem10 27922. (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 4831 | . . . . 5 ⊢ (𝑞 = 𝑝 → 〈𝑍, 𝑞〉 = 〈𝑍, 𝑝〉) | |
2 | 1 | breq2d 5117 | . . . 4 ⊢ (𝑞 = 𝑝 → (𝑈 Btwn 〈𝑍, 𝑞〉 ↔ 𝑈 Btwn 〈𝑍, 𝑝〉)) |
3 | breq1 5108 | . . . 4 ⊢ (𝑞 = 𝑝 → (𝑞 Btwn 〈𝑍, 𝑈〉 ↔ 𝑝 Btwn 〈𝑍, 𝑈〉)) | |
4 | 2, 3 | orbi12d 917 | . . 3 ⊢ (𝑞 = 𝑝 → ((𝑈 Btwn 〈𝑍, 𝑞〉 ∨ 𝑞 Btwn 〈𝑍, 𝑈〉) ↔ (𝑈 Btwn 〈𝑍, 𝑝〉 ∨ 𝑝 Btwn 〈𝑍, 𝑈〉))) |
5 | 4 | cbvrabv 3417 | . 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 2736 | . . 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 27913 | . 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 27922 | 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 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 {crab 3407 ⊆ wss 3910 ∅c0 4282 〈cop 4592 class class class wbr 5105 {copab 5167 ‘cfv 6496 (class class class)co 7357 0cc0 11051 1c1 11052 + caddc 11054 · cmul 11056 +∞cpnf 11186 − cmin 11385 ℕcn 12153 [,)cico 13266 ...cfz 13424 𝔼cee 27837 Btwn cbtwn 27838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-z 12500 df-uz 12764 df-ico 13270 df-icc 13271 df-fz 13425 df-ee 27840 df-btwn 27841 |
This theorem is referenced by: axcontlem12 27924 |
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