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Mirrors > Home > MPE Home > Th. List > Mathboxes > metakunt23 | Structured version Visualization version GIF version |
Description: B coincides on the union of bijections of functions. (Contributed by metakunt, 28-May-2024.) |
Ref | Expression |
---|---|
metakunt23.1 | ⊢ (𝜑 → 𝑀 ∈ ℕ) |
metakunt23.2 | ⊢ (𝜑 → 𝐼 ∈ ℕ) |
metakunt23.3 | ⊢ (𝜑 → 𝐼 ≤ 𝑀) |
metakunt23.4 | ⊢ 𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀 − 𝐼)), (𝑥 + (1 − 𝐼))))) |
metakunt23.5 | ⊢ 𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀 − 𝐼))) |
metakunt23.6 | ⊢ 𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼))) |
metakunt23.7 | ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) |
Ref | Expression |
---|---|
metakunt23 | ⊢ (𝜑 → (𝐵‘𝑋) = (((𝐶 ∪ 𝐷) ∪ {〈𝑀, 𝑀〉})‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metakunt23.1 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
2 | 1 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑀) → 𝑀 ∈ ℕ) |
3 | metakunt23.2 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ ℕ) | |
4 | 3 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑀) → 𝐼 ∈ ℕ) |
5 | metakunt23.3 | . . . 4 ⊢ (𝜑 → 𝐼 ≤ 𝑀) | |
6 | 5 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑀) → 𝐼 ≤ 𝑀) |
7 | metakunt23.4 | . . 3 ⊢ 𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀 − 𝐼)), (𝑥 + (1 − 𝐼))))) | |
8 | metakunt23.5 | . . 3 ⊢ 𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀 − 𝐼))) | |
9 | metakunt23.6 | . . 3 ⊢ 𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼))) | |
10 | metakunt23.7 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) | |
11 | 10 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑀) → 𝑋 ∈ (1...𝑀)) |
12 | simpr 486 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑀) → 𝑋 = 𝑀) | |
13 | 2, 4, 6, 7, 8, 9, 11, 12 | metakunt20 40942 | . 2 ⊢ ((𝜑 ∧ 𝑋 = 𝑀) → (𝐵‘𝑋) = (((𝐶 ∪ 𝐷) ∪ {〈𝑀, 𝑀〉})‘𝑋)) |
14 | 1 | ad2antrr 725 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀) ∧ 𝑋 < 𝐼) → 𝑀 ∈ ℕ) |
15 | 3 | ad2antrr 725 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀) ∧ 𝑋 < 𝐼) → 𝐼 ∈ ℕ) |
16 | 5 | ad2antrr 725 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀) ∧ 𝑋 < 𝐼) → 𝐼 ≤ 𝑀) |
17 | 10 | ad2antrr 725 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀) ∧ 𝑋 < 𝐼) → 𝑋 ∈ (1...𝑀)) |
18 | simplr 768 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀) ∧ 𝑋 < 𝐼) → ¬ 𝑋 = 𝑀) | |
19 | simpr 486 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀) ∧ 𝑋 < 𝐼) → 𝑋 < 𝐼) | |
20 | 14, 15, 16, 7, 8, 9, 17, 18, 19 | metakunt21 40943 | . . 3 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀) ∧ 𝑋 < 𝐼) → (𝐵‘𝑋) = (((𝐶 ∪ 𝐷) ∪ {〈𝑀, 𝑀〉})‘𝑋)) |
21 | 1 | ad2antrr 725 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀) ∧ ¬ 𝑋 < 𝐼) → 𝑀 ∈ ℕ) |
22 | 3 | ad2antrr 725 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀) ∧ ¬ 𝑋 < 𝐼) → 𝐼 ∈ ℕ) |
23 | 5 | ad2antrr 725 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀) ∧ ¬ 𝑋 < 𝐼) → 𝐼 ≤ 𝑀) |
24 | 10 | ad2antrr 725 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀) ∧ ¬ 𝑋 < 𝐼) → 𝑋 ∈ (1...𝑀)) |
25 | simplr 768 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀) ∧ ¬ 𝑋 < 𝐼) → ¬ 𝑋 = 𝑀) | |
26 | simpr 486 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀) ∧ ¬ 𝑋 < 𝐼) → ¬ 𝑋 < 𝐼) | |
27 | 21, 22, 23, 7, 8, 9, 24, 25, 26 | metakunt22 40944 | . . 3 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀) ∧ ¬ 𝑋 < 𝐼) → (𝐵‘𝑋) = (((𝐶 ∪ 𝐷) ∪ {〈𝑀, 𝑀〉})‘𝑋)) |
28 | 20, 27 | pm2.61dan 812 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀) → (𝐵‘𝑋) = (((𝐶 ∪ 𝐷) ∪ {〈𝑀, 𝑀〉})‘𝑋)) |
29 | 13, 28 | pm2.61dan 812 | 1 ⊢ (𝜑 → (𝐵‘𝑋) = (((𝐶 ∪ 𝐷) ∪ {〈𝑀, 𝑀〉})‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∪ cun 3945 ifcif 4527 {csn 4627 〈cop 4633 class class class wbr 5147 ↦ cmpt 5230 ‘cfv 6540 (class class class)co 7404 1c1 11107 + caddc 11109 < clt 11244 ≤ cle 11245 − cmin 11440 ℕcn 12208 ...cfz 13480 |
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 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-fz 13481 df-fzo 13624 |
This theorem is referenced by: metakunt25 40947 |
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