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Mirrors > Home > MPE Home > Th. List > itgvallem | Structured version Visualization version GIF version |
Description: Substitution lemma. (Contributed by Mario Carneiro, 7-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
Ref | Expression |
---|---|
itgvallem.1 | ā¢ (iāš¾) = š |
Ref | Expression |
---|---|
itgvallem | ā¢ (š = š¾ ā (ā«2ā(š„ ā ā ā¦ if((š„ ā š“ ā§ 0 ā¤ (āā(šµ / (iāš)))), (āā(šµ / (iāš))), 0))) = (ā«2ā(š„ ā ā ā¦ if((š„ ā š“ ā§ 0 ā¤ (āā(šµ / š))), (āā(šµ / š)), 0)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7369 | . . . . . . . . 9 ā¢ (š = š¾ ā (iāš) = (iāš¾)) | |
2 | itgvallem.1 | . . . . . . . . 9 ā¢ (iāš¾) = š | |
3 | 1, 2 | eqtrdi 2789 | . . . . . . . 8 ā¢ (š = š¾ ā (iāš) = š) |
4 | 3 | oveq2d 7377 | . . . . . . 7 ā¢ (š = š¾ ā (šµ / (iāš)) = (šµ / š)) |
5 | 4 | fveq2d 6850 | . . . . . 6 ā¢ (š = š¾ ā (āā(šµ / (iāš))) = (āā(šµ / š))) |
6 | 5 | breq2d 5121 | . . . . 5 ā¢ (š = š¾ ā (0 ā¤ (āā(šµ / (iāš))) ā 0 ā¤ (āā(šµ / š)))) |
7 | 6 | anbi2d 630 | . . . 4 ā¢ (š = š¾ ā ((š„ ā š“ ā§ 0 ā¤ (āā(šµ / (iāš)))) ā (š„ ā š“ ā§ 0 ā¤ (āā(šµ / š))))) |
8 | 7, 5 | ifbieq1d 4514 | . . 3 ā¢ (š = š¾ ā if((š„ ā š“ ā§ 0 ā¤ (āā(šµ / (iāš)))), (āā(šµ / (iāš))), 0) = if((š„ ā š“ ā§ 0 ā¤ (āā(šµ / š))), (āā(šµ / š)), 0)) |
9 | 8 | mpteq2dv 5211 | . 2 ā¢ (š = š¾ ā (š„ ā ā ā¦ if((š„ ā š“ ā§ 0 ā¤ (āā(šµ / (iāš)))), (āā(šµ / (iāš))), 0)) = (š„ ā ā ā¦ if((š„ ā š“ ā§ 0 ā¤ (āā(šµ / š))), (āā(šµ / š)), 0))) |
10 | 9 | fveq2d 6850 | 1 ā¢ (š = š¾ ā (ā«2ā(š„ ā ā ā¦ if((š„ ā š“ ā§ 0 ā¤ (āā(šµ / (iāš)))), (āā(šµ / (iāš))), 0))) = (ā«2ā(š„ ā ā ā¦ if((š„ ā š“ ā§ 0 ā¤ (āā(šµ / š))), (āā(šµ / š)), 0)))) |
Colors of variables: wff setvar class |
Syntax hints: ā wi 4 ā§ wa 397 = wceq 1542 ā wcel 2107 ifcif 4490 class class class wbr 5109 ā¦ cmpt 5192 ācfv 6500 (class class class)co 7361 ācr 11058 0cc0 11059 ici 11061 ā¤ cle 11198 / cdiv 11820 ācexp 13976 ācre 14991 ā«2citg2 25003 |
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-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-iota 6452 df-fv 6508 df-ov 7364 |
This theorem is referenced by: iblcnlem1 25175 itgcnlem 25177 |
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