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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 40642 | . 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 40643 | . . 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 40644 | . . 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 3909 ifcif 4487 {csn 4587 ⟨cop 4593 class class class wbr 5106 ↦ cmpt 5189 ‘cfv 6497 (class class class)co 7358 1c1 11057 + caddc 11059 < clt 11194 ≤ cle 11195 − cmin 11390 ℕcn 12158 ...cfz 13430 |
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 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
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-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-n0 12419 df-z 12505 df-uz 12769 df-rp 12921 df-fz 13431 df-fzo 13574 |
This theorem is referenced by: metakunt25 40647 |
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