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Mirrors > Home > HSE Home > Th. List > cdj3lem3 | Structured version Visualization version GIF version |
Description: Lemma for cdj3i 28684. Value of the second-component function 𝑇. (Contributed by NM, 23-May-2005.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cdj3lem2.1 | ⊢ 𝐴 ∈ Sℋ |
cdj3lem2.2 | ⊢ 𝐵 ∈ Sℋ |
cdj3lem3.3 | ⊢ 𝑇 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤))) |
Ref | Expression |
---|---|
cdj3lem3 | ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑇‘(𝐶 +ℎ 𝐷)) = 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 3767 | . . . 4 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | |
2 | 1 | eqeq1i 2615 | . . 3 ⊢ ((𝐴 ∩ 𝐵) = 0ℋ ↔ (𝐵 ∩ 𝐴) = 0ℋ) |
3 | cdj3lem2.2 | . . . . . . . 8 ⊢ 𝐵 ∈ Sℋ | |
4 | 3 | sheli 27455 | . . . . . . 7 ⊢ (𝐷 ∈ 𝐵 → 𝐷 ∈ ℋ) |
5 | cdj3lem2.1 | . . . . . . . 8 ⊢ 𝐴 ∈ Sℋ | |
6 | 5 | sheli 27455 | . . . . . . 7 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ ℋ) |
7 | ax-hvcom 27242 | . . . . . . 7 ⊢ ((𝐷 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐷 +ℎ 𝐶) = (𝐶 +ℎ 𝐷)) | |
8 | 4, 6, 7 | syl2an 493 | . . . . . 6 ⊢ ((𝐷 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐷 +ℎ 𝐶) = (𝐶 +ℎ 𝐷)) |
9 | 8 | fveq2d 6107 | . . . . 5 ⊢ ((𝐷 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴) → (𝑇‘(𝐷 +ℎ 𝐶)) = (𝑇‘(𝐶 +ℎ 𝐷))) |
10 | 9 | 3adant3 1074 | . . . 4 ⊢ ((𝐷 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ (𝐵 ∩ 𝐴) = 0ℋ) → (𝑇‘(𝐷 +ℎ 𝐶)) = (𝑇‘(𝐶 +ℎ 𝐷))) |
11 | cdj3lem3.3 | . . . . . 6 ⊢ 𝑇 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤))) | |
12 | 3, 5 | shscomi 27606 | . . . . . . 7 ⊢ (𝐵 +ℋ 𝐴) = (𝐴 +ℋ 𝐵) |
13 | 3 | sheli 27455 | . . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝐵 → 𝑤 ∈ ℋ) |
14 | 5 | sheli 27455 | . . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ ℋ) |
15 | ax-hvcom 27242 | . . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑤 +ℎ 𝑧) = (𝑧 +ℎ 𝑤)) | |
16 | 13, 14, 15 | syl2an 493 | . . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴) → (𝑤 +ℎ 𝑧) = (𝑧 +ℎ 𝑤)) |
17 | 16 | eqeq2d 2620 | . . . . . . . . 9 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴) → (𝑥 = (𝑤 +ℎ 𝑧) ↔ 𝑥 = (𝑧 +ℎ 𝑤))) |
18 | 17 | rexbidva 3031 | . . . . . . . 8 ⊢ (𝑤 ∈ 𝐵 → (∃𝑧 ∈ 𝐴 𝑥 = (𝑤 +ℎ 𝑧) ↔ ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤))) |
19 | 18 | riotabiia 6528 | . . . . . . 7 ⊢ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑤 +ℎ 𝑧)) = (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤)) |
20 | 12, 19 | mpteq12i 4670 | . . . . . 6 ⊢ (𝑥 ∈ (𝐵 +ℋ 𝐴) ↦ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑤 +ℎ 𝑧))) = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤))) |
21 | 11, 20 | eqtr4i 2635 | . . . . 5 ⊢ 𝑇 = (𝑥 ∈ (𝐵 +ℋ 𝐴) ↦ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑤 +ℎ 𝑧))) |
22 | 3, 5, 21 | cdj3lem2 28678 | . . . 4 ⊢ ((𝐷 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ (𝐵 ∩ 𝐴) = 0ℋ) → (𝑇‘(𝐷 +ℎ 𝐶)) = 𝐷) |
23 | 10, 22 | eqtr3d 2646 | . . 3 ⊢ ((𝐷 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ (𝐵 ∩ 𝐴) = 0ℋ) → (𝑇‘(𝐶 +ℎ 𝐷)) = 𝐷) |
24 | 2, 23 | syl3an3b 1356 | . 2 ⊢ ((𝐷 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑇‘(𝐶 +ℎ 𝐷)) = 𝐷) |
25 | 24 | 3com12 1261 | 1 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑇‘(𝐶 +ℎ 𝐷)) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ∩ cin 3539 ↦ cmpt 4643 ‘cfv 5804 ℩crio 6510 (class class class)co 6549 ℋchil 27160 +ℎ cva 27161 Sℋ csh 27169 +ℋ cph 27172 0ℋc0h 27176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-hilex 27240 ax-hfvadd 27241 ax-hvcom 27242 ax-hvass 27243 ax-hv0cl 27244 ax-hvaddid 27245 ax-hfvmul 27246 ax-hvmulid 27247 ax-hvmulass 27248 ax-hvdistr1 27249 ax-hvdistr2 27250 ax-hvmul0 27251 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-grpo 26731 df-ablo 26783 df-hvsub 27212 df-sh 27448 df-ch0 27494 df-shs 27551 |
This theorem is referenced by: cdj3lem3a 28682 cdj3i 28684 |
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