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Mirrors > Home > MPE Home > Th. List > Mathboxes > cncfioobdlem | Structured version Visualization version GIF version |
Description: 𝐺 actually extends 𝐹. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
cncfioobdlem.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
cncfioobdlem.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
cncfioobdlem.f | ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶𝑉) |
cncfioobdlem.g | ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))) |
cncfioobdlem.c | ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) |
Ref | Expression |
---|---|
cncfioobdlem | ⊢ (𝜑 → (𝐺‘𝐶) = (𝐹‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cncfioobdlem.g | . . 3 ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))))) |
3 | cncfioobdlem.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | 3 | adantr 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐴 ∈ ℝ) |
5 | cncfioobdlem.c | . . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) | |
6 | 3 | rexrd 11213 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
7 | cncfioobdlem.b | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
8 | 7 | rexrd 11213 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
9 | elioo2 13314 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) | |
10 | 6, 8, 9 | syl2anc 585 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) |
11 | 5, 10 | mpbid 231 | . . . . . . . . 9 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵)) |
12 | 11 | simp2d 1144 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 < 𝐶) |
13 | 12 | adantr 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐴 < 𝐶) |
14 | eqcom 2740 | . . . . . . . . 9 ⊢ (𝑥 = 𝐶 ↔ 𝐶 = 𝑥) | |
15 | 14 | biimpi 215 | . . . . . . . 8 ⊢ (𝑥 = 𝐶 → 𝐶 = 𝑥) |
16 | 15 | adantl 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐶 = 𝑥) |
17 | 13, 16 | breqtrd 5135 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐴 < 𝑥) |
18 | 4, 17 | gtned 11298 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝑥 ≠ 𝐴) |
19 | 18 | neneqd 2945 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → ¬ 𝑥 = 𝐴) |
20 | 19 | iffalsed 4501 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) |
21 | simpr 486 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝑥 = 𝐶) | |
22 | 5 | elioored 43877 | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
23 | 22 | adantr 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐶 ∈ ℝ) |
24 | 21, 23 | eqeltrd 2834 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝑥 ∈ ℝ) |
25 | 11 | simp3d 1145 | . . . . . . . 8 ⊢ (𝜑 → 𝐶 < 𝐵) |
26 | 25 | adantr 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐶 < 𝐵) |
27 | 21, 26 | eqbrtrd 5131 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝑥 < 𝐵) |
28 | 24, 27 | ltned 11299 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝑥 ≠ 𝐵) |
29 | 28 | neneqd 2945 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → ¬ 𝑥 = 𝐵) |
30 | 29 | iffalsed 4501 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)) = (𝐹‘𝑥)) |
31 | 21 | fveq2d 6850 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → (𝐹‘𝑥) = (𝐹‘𝐶)) |
32 | 20, 30, 31 | 3eqtrd 2777 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = (𝐹‘𝐶)) |
33 | ioossicc 13359 | . . 3 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) | |
34 | 33, 5 | sselid 3946 | . 2 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) |
35 | cncfioobdlem.f | . . 3 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶𝑉) | |
36 | 35, 5 | ffvelcdmd 7040 | . 2 ⊢ (𝜑 → (𝐹‘𝐶) ∈ 𝑉) |
37 | 2, 32, 34, 36 | fvmptd 6959 | 1 ⊢ (𝜑 → (𝐺‘𝐶) = (𝐹‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ifcif 4490 class class class wbr 5109 ↦ cmpt 5192 ⟶wf 6496 ‘cfv 6500 (class class class)co 7361 ℝcr 11058 ℝ*cxr 11196 < clt 11197 (,)cioo 13273 [,]cicc 13276 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-pre-lttri 11133 ax-pre-lttrn 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-po 5549 df-so 5550 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-1st 7925 df-2nd 7926 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-ioo 13277 df-icc 13280 |
This theorem is referenced by: cncfioobd 44228 |
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